diff --git a/.github/workflows/np-prolate-1-arb.yml b/.github/workflows/np-prolate-1-arb.yml
new file mode 100644
index 00000000..e86ad5ac
--- /dev/null
+++ b/.github/workflows/np-prolate-1-arb.yml
@@ -0,0 +1,37 @@
+name: NP-PROLATE-1 Arb certificate
+on:
+  pull_request:
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  np-prolate-1:
+    strategy:
+      fail-fast: false
+      matrix:
+        sector: [even, odd]
+    runs-on: ubuntu-latest
+    timeout-minutes: 60
+    steps:
+      - uses: actions/checkout@11bd71901bbe5b1630ceea73d27597364c9af683 # v4.2.2
+        with:
+          ref: ${{ github.event.pull_request.head.sha || github.sha }}
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.13'
+      - name: Install exact numerical dependency
+        run: python -m pip install python-flint==0.9.0
+      - name: Run NP-PROLATE-1 exact-head certificate
+        run: |
+          set -euo pipefail
+          git rev-parse HEAD
+          git hash-object scripts/check_np_prolate_1_arb.py
+          python --version
+          python scripts/check_np_prolate_1_arb.py "${{ matrix.sector }}" | tee "np_prolate_1_${{ matrix.sector }}.log"
+          grep -q "NP-PROLATE-1 ${{ matrix.sector == 'even' && 'EVEN' || 'ODD' }} PASS ZERTIFIZIERT" "np_prolate_1_${{ matrix.sector }}.log"
+          grep -q "ALL REQUESTED NP-PROLATE-1 SECTORS PASS ZERTIFIZIERT" "np_prolate_1_${{ matrix.sector }}.log"
+      - uses: actions/upload-artifact@ea165f8d65b6e75b540449e92b4886f43607fa02 # v4
+        with:
+          name: np-prolate-1-${{ matrix.sector }}
+          path: np_prolate_1_${{ matrix.sector }}.log
diff --git a/00-uebersicht/HARD_AUDIT_SURVIVOR_STATE.md b/00-uebersicht/HARD_AUDIT_SURVIVOR_STATE.md
new file mode 100644
index 00000000..8bb05083
--- /dev/null
+++ b/00-uebersicht/HARD_AUDIT_SURVIVOR_STATE.md
@@ -0,0 +1,370 @@
+# Objekt X — Hard-Audit Survivor State
+
+**Stand:** 16. September 2026  
+**Rolle:** kanonische Einstiegsebene fuer den Forschungsstand dieses Astes.  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**PR:** #116  
+**Registry:** unveraendert.  
+**Nonclaim:** kein RH-Beweis, kein Object-X-Abschluss, kein Publikationsneuheitsclaim.
+
+> Diese Datei ist Navigation, kein neuer Beweis. Bei Konflikten gilt der verlinkte Primaeraudit im dort ausgewiesenen Scope.
+
+---
+
+## 0. Ein-Satz-Stand
+
+Nach destruktivem Gegencheck bleibt als einzige ernsthafte Object-X-Front:
+
+```text
+relative cancellation first, Critical-half output second
+```
+
+Prime- und Continuum-Zustaende muessen zuerst auf der sicheren Root-Skala `Lambda(n)/n` relativ gekoppelt werden; eine getrennte diagonale Anhebung auf die Weil-Skala `Lambda(n)/sqrt(n)` ist unbeschraenkt und ausgeschlossen. Ein erfolgreicher Kandidat muss die relative Ausloeschung intern vollziehen und erst den endlichen relativen Boundary-Output durch den Critical-half-Port fuehren.
+
+---
+
+# 1. Was nach dem Hard Audit steht
+
+## S1 — NULLPOL = Critical-half Green-Range `✓[M]`
+
+```math
+C_c^\infty(-a,a)\cap\ker M(0)\cap\ker M(1)
+=
+(-\partial_x^2+1/4)C_c^\infty(-a,a).
+```
+
+Green-Kern:
+
+```math
+G_{1/2}(x,y)=e^{-|x-y|/2}.
+```
+
+Die zwei Mellinbedingungen loeschen exakt die beiden aeusseren Green-Schwanzladungen.
+
+Primaerquelle: `audits/P11_CRITICAL_HALF_GREEN_TREE_BRIDGE_2026-09-13.md`.
+
+## S2 — P11 Prime-Power-Ledger = sampled Critical-half OU/AR(1) `✓[M]`
+
+Nach Weil-Diagonalnormalisierung:
+
+```math
+R_q(j,k)=q^{|j-k|},\qquad q=p^{-1/2}.
+```
+
+Hub + Innovationen ergeben den stationaeren AR(1)-Kern. Dies ist eine Aussage ueber den Kanalindex-Ledger, nicht automatisch ueber den vollen raeumlichen P11-Operator.
+
+Primaerquelle: `audits/P11_CRITICAL_HALF_GREEN_TREE_BRIDGE_2026-09-13.md`.
+
+## S3 — P11-Masken = stopped OU filtration `✓[M]`
+
+Mit
+
+```math
+L_R(x)=2(R-|x|),\qquad
+m_p(x)=\lfloor L_R(x)/\log p\rfloor
+```
+
+ist die lokale normalisierte Kovarianz
+
+```math
+R_q^{(m)}(j,k)=q^{j+k-2\min(j,k,m)}.
+```
+
+Jedes Tiefeninkrement ist positiv Rang eins.
+
+Universeller Kernel:
+
+```math
+G_L(s,t)=\exp(-(s+t)/2+\min(s,t,L)).
+```
+
+P11 ist dessen logarithmische Gitterabtastung.
+
+Primaerquellen: `audits/P11_SPATIAL_STOPPED_OU_FILTRATION_2026-09-15.md`, `audits/P11_UNIVERSAL_STOPPED_OU_PRECISION_2026-09-15.md`.
+
+## S4 — Boundary-tail shorting `✓[M]`
+
+Diskret:
+
+```math
+Y_m=\sqrt{1-q^2}\sum_{k\ge m}q^{k-m}X_k
+```
+
+und
+
+```math
+\sum_{r=1}^{m-1}\frac{\|Y_r-qY_{r+1}\|^2}{1-q^2}+\|Y_m\|^2
+=
+\sum_{r<m}\|X_r\|^2+\|Y_m\|^2.
+```
+
+Kontinuierlich gilt dieselbe Shorting-Identitaet fuer
+
+```math
+Y(u)=\int_u^\infty e^{-(t-u)/2}X(t)dt.
+```
+
+Innen wird die nackte Energie exakt rekonstruiert; der aeussere Tail wird auf einen normierten exponentiellen Boundary-Modus komprimiert, das orthogonale Restdefizit bleibt positiv.
+
+Primaerquelle: `audits/P11_BOUNDARY_TAIL_SHORTING_AND_UNIVERSAL_SAMPLING_2026-09-15.md`.
+
+## S5 — Universelles Jump-Feld `✓[M]`
+
+```math
+X_v(t)=e^{-t/4}K_tv,
+```
+
+und fuer `h=log p`:
+
+```math
+\sqrt{w_{p,k}}K_{kh}v
+=\sqrt h\,X_v(kh).
+```
+
+Alle Primzahlen tasten dasselbe kontinuierliche Feld auf unterschiedlichen logarithmischen Gittern ab.
+
+Primaerquelle: `audits/P11_BOUNDARY_TAIL_SHORTING_AND_UNIVERSAL_SAMPLING_2026-09-15.md`.
+
+## S6 — Prime-Pole-Diskrepanz und Gamma-Grund-Cancellation `✓[M]`
+
+```math
+\Delta
+=\sum_{n=p^k}\frac{\Lambda(n)}{\sqrt n}\delta_{\log n}-e^{t/2}dt.
+```
+
+Auf NULLPOL hebt sich der glatte Prime-Hauptterm gegen den Gamma-Grundmodus auf. Die grosse fensterabhaengige Diagonalmasse verschwindet aus der fensterfreien Darstellung.
+
+Primaerquelle: `audits/P11_NULLPOL_PRIME_DISCREPANCY_CANCELLATION_2026-09-15.md`.
+
+## S7 — Critical-half finite-part transfer `✓[M]`
+
+```math
+J_\Delta(L)
+=e^{L/2}\left(L-\gamma-\sum_{\log n<L}\frac{\Lambda(n)}n\right),
+```
+
+```math
+J_\Delta(0)=-\gamma,
+\qquad
+d\Delta=\frac12J_\Delta dL-dJ_\Delta.
+```
+
+Dadurch koppelt die Diskrepanz exakt an
+
+```math
+D_+=\partial_x+1/2.
+```
+
+Primaerquelle: `audits/P11_CRITICAL_HALF_FINITE_PART_TRANSFER_2026-09-15.md`.
+
+## S8 — Kausale Volterra-Normalform `✓[M]` als Identitaet
+
+Mit
+
+```math
+(V_Jv)(x)=\int_{-\infty}^xJ_\Delta(x-y)v(y)dy
+```
+
+und `D_-=D_+^*`:
+
+```math
+D_-V_J+V_J^*D_+=2\gamma I+\mathcal C_\Delta.
+```
+
+Die hoehere Gamma-Schicht ist eine positive stabile Filterbank. Auf NULLPOL:
+
+```math
+Q_W(v)
+=
+\sum_{m\ge1}\frac2{\mu_m}\|y_m'\|^2
+-c_*\|v\|^2
+-2\operatorname{Re}\langle D_+v,V_Jv\rangle.
+```
+
+Positivitaet bleibt offen.
+
+Primaerquelle: `audits/P11_CAUSAL_VOLterra_PASSIVITY_NORMAL_FORM_2026-09-15.md`.
+
+## S9 — Eulerfaktor = P11 AR(1)-Resolvent `✓[M]`
+
+```math
+T_q^*=\sqrt{1-q^2}(I-qS)^{-1}.
+```
+
+Am Root-Impuls ist die Transferfunktion `1/(1-qz)`. Mit
+
+```math
+q=p^{-1/2},\qquad z=e^{-s\log p}
+```
+
+wird dies exakt
+
+```math
+(1-p^{-(s+1/2)})^{-1}.
+```
+
+Der Eulerfaktor ist klassisch; projektintern ist relevant, dass exakt derselbe Resolvent im vorhandenen P11-Choleskyfaktor steckt.
+
+Primaerquelle: `audits/P11_EULER_AR1_RESOLVENT_BRIDGE_2026-09-15.md`.
+
+## S10 — Jede Euler-Zelle wird von Gamma-Grund dominiert `✓[M]`
+
+Fuer
+
+```math
+A_h=\sum_{k\ge1}h e^{-kh/2}K_{kh}^*K_{kh}
+```
+
+und
+
+```math
+A_0=\int_0^\infty e^{-t/2}K_t^*K_tdt
+=4D^2(D^2+1/4)^{-1}
+```
+
+gilt fuer jedes `h>0`:
+
+```math
+0\preceq A_h\preceq A_0.
+```
+
+Auf Transferniveau existiert daher ein kanonischer lokaler Schur-Faktor
+
+```math
+b_h=\Phi_h b_0,
+\qquad \|\Phi_h\|_{H^\infty(\Re s>0)}\le1.
+```
+
+Lokale Prime-/Euler-Zellen sind somit passive Kontraktionen der Gamma-Grundzelle. Der globale Engpass liegt nicht mehr lokal pro Primzahl, sondern in der Skalenkopplung.
+
+Primaerquelle: `audits/P11_EULER_CELL_DOMINATED_BY_GAMMA_GROUND_2026-09-16.md`.
+
+## S11 — Root-Trace-Bilanz `✓[M]`
+
+P11-Rootmasse bis Tiefe `L`:
+
+```math
+M_P(L)=\sum_{\log n<L}\frac{\Lambda(n)}n.
+```
+
+Riemann-R- und Gamma-Grund-Referenz:
+
+```math
+M_R(L)=L-1+e^{-L},
+\qquad
+M_{\Gamma,0}(L)=1-e^{-L},
+```
+
+also
+
+```math
+M_R(L)+M_{\Gamma,0}(L)=L.
+```
+
+Damit
+
+```math
+e^{-L/2}J_\Delta(L)
+=M_R(L)+M_{\Gamma,0}(L)-M_P(L)-\gamma.
+```
+
+Der kritische Volterra-Kern ist damit direkt die relative gestoppte Root-Storage-Bilanz.
+
+Primaerquelle: `audits/P11_CRITICAL_TRANSFER_AS_STOPPED_ROOT_TRACE_DEFECT_2026-09-16.md`.
+
+## S12 — Monotone Root-Mass-Transportkopplung `✓[K/M]`
+
+Ordne die Prime-Power-Tiefen `h_j=log n_j` und setze
+
+```math
+a_j=\Lambda(n_j)/n_j,
+\quad
+s_0=\gamma,
+\quad
+s_j=\gamma+\sum_{i\le j}a_i,
+\quad
+I_0=[0,\gamma),\ I_j=[s_{j-1},s_j).
+```
+
+Dann `|I_j|=a_j` und aus dem klassischen PNT-Fehler folgt
+
+```math
+\sup_{L\in I_j}|L-h_j|\to0,
+\qquad
+\sum_j a_j\sup_{L\in I_j}|L-h_j|<\infty.
+```
+
+Damit entsteht ein kanonischer positiver Massentransport vom diskreten Root-Raum in `L^2(dL)`, und fuer jedes feste `s` mit `Re s>=0` konvergiert der relative Zelltransfer absolut:
+
+```math
+\gamma-\int_0^\gamma e^{-sL}dL
++
+\sum_j\left[a_je^{-sh_j}-\int_{I_j}e^{-sL}dL\right].
+```
+
+Primaerquellen: `audits/P11_ROOT_MASS_MONOTONE_TRANSPORT_2026-09-16.md`, `audits/P11_ROOT_TRANSPORT_SAFE_TRANSFER_AND_CRITICAL_LIFT_FIREWALL_2026-09-16.md`.
+
+## S13 — Plain Critical-half lift of root storage is impossible `×[M]`
+
+Der Schritt
+
+```math
+a_j=\Lambda(n_j)/n_j
+\mapsto
+e^{h_j/2}a_j=\Lambda(n_j)/\sqrt{n_j}
+```
+
+ist auf dem Root-Storage eine unbeschraenkte exponentielle Multiplikation. Fuer jedes nichtverschwindende kompakt getragene `v` gilt fuer grosse `t`
+
+```math
+\|K_tv\|^2=2\|v\|^2,
+```
+
+sodass die getrennte Critical-half-Verstaerkung des Prime- bzw. Continuum-Tails divergiert. Auch endlich viele Source-Ableitungen beheben dies nicht.
+
+Folge:
+
+```text
+prime/continuum cancellation MUST occur before critical amplification.
+```
+
+Primaerquelle: `audits/P11_ROOT_TRANSPORT_SAFE_TRANSFER_AND_CRITICAL_LIFT_FIREWALL_2026-09-16.md`.
+
+---
+
+# 2. Was NICHT als Neuheit behauptet wird
+
+Nicht neu beanspruchen:
+
+- RH als Positive-Real/Herglotz-Kriterium fuer `xi'/xi`;
+- Lagarias-Positivitaet;
+- Zeta-bezogene Kontroll-/Transfer-Systeme allgemein;
+- Eulerprodukt, Poisson-, OU-/AR(1)-, Green-, de-Branges-, Paley-Wiener-, Toeplitz-, Prolate-Theorie an sich;
+- passive/KYP-Realisierungen an sich;
+- klassische PNT-Fehlerabschaetzungen.
+
+Der moegliche projektspezifische Wert kann nur in der konkreten konstruktiven Glueung der bereits vorhandenen P11-/stopped-OU-/Root-Transport-/Gamma-Komponenten liegen.
+
+---
+
+# 3. Aktueller harter Gate
+
+```text
+RELATIVE-COLLIGATION / CANCEL-FIRST
+```
+
+Gesucht ist eine coefficient-free positive bzw. konservative `2x2`-relative Kolligation auf den Transportzellen `I_j`, die
+
+1. diskrete Prime-Power-Rootzustande und positive Continuum-Zellen koppelt;
+2. deren divergent kritische Anteile intern ausloescht, solange sie noch auf der sicheren `Lambda(n)/n`-Skala liegen;
+3. nur den endlichen relativen Boundary-State nach aussen gibt;
+4. erst diesen Output durch den Critical-half-Port hebt;
+5. die bereits fixierten lokalen Schur-Zellen `Phi_h` und Gamma-Filter benutzt;
+6. keine Koeffizienten nach Sicht auf die Weilform oder Zeta-Nullen fitten darf;
+7. zuerst den exakten `R=1` Prime-2 mixed witness reproduzieren muss.
+
+PASS: erstmals ein substantieller vorwaerts konstruierter Object-X-Mechanismus.
+
+FAIL: die gesamte verbliebene stopped-OU / relative-storage Architekturklasse wird geschlossen; die Survivor-Identitaeten bleiben als eigenstaendige Mathematik erhalten.
+
+Naechster Einstieg: `00-uebersicht/NEXT_GATE.md`.
diff --git a/00-uebersicht/NEXT_GATE.md b/00-uebersicht/NEXT_GATE.md
new file mode 100644
index 00000000..41eee201
--- /dev/null
+++ b/00-uebersicht/NEXT_GATE.md
@@ -0,0 +1,288 @@
+# Objekt X — NEXT GATE
+
+**Stand:** 16. September 2026  
+**Gate-ID:** `RELATIVE-COLLIGATION / CANCEL-FIRST`  
+**Status:** `?[O]`  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**PR:** #116  
+**Memory-layer parent:** `b3a74a7497d9c92c4dcb289d72a395b341deb727`  
+**Canonical memory-layer commit:** see the Git commit that contains this file; do not replace by a floating branch assumption.
+
+> Dies ist die einzige aktuelle Hauptfront dieses Forschungsastes. Keine neue Proxy-Suche, kein weiterer Methodenwechsel und keine Wiedereroeffnung gesperrter Klassen, solange dieser Gate nicht PASS oder FAIL erhalten hat.
+
+---
+
+## 0. Ziel in einem Satz
+
+Konstruiere **vorwaerts**, coefficient-free und ohne Weil-Positivitaet/RH als Input eine positive bzw. konservative relative Kolligation, die Prime- und Continuum-Rootzustaende zuerst auf der sicheren `Lambda(n)/n`-Skala intern ausloescht und erst den endlichen relativen Boundary-Output durch den Critical-half-Port auf die Weil-Skala hebt.
+
+Leitregel:
+
+```text
+RELATIVE CANCELLATION FIRST.
+CRITICAL-HALF OUTPUT SECOND.
+```
+
+---
+
+# 1. Fest vorgegebene Daten — nicht fitten
+
+## 1.1 Lokale Euler-/Gamma-Zellen
+
+Fuer `h>0`, `q=e^{-h/2}`:
+
+```math
+A_h
+=\sum_{k\ge1}h e^{-kh/2}K_{kh}^*K_{kh},
+```
+
+```math
+A_0
+=\int_0^\infty e^{-t/2}K_t^*K_tdt
+=4D^2(D^2+1/4)^{-1}.
+```
+
+Bereits bewiesen:
+
+```math
+0\preceq A_h\preceq A_0.
+```
+
+Daher existiert ein kanonischer lokaler Schur-Faktor
+
+```math
+b_h=\Phi_h b_0,
+\qquad \|\Phi_h\|_{H^\infty(\Re s>0)}\le1.
+```
+
+Diese lokale Metrik ist fix.
+
+## 1.2 Prime-Root-Massen
+
+Prime-Power-Tiefen und sichere Root-Masse:
+
+```math
+h_j=\log n_j,
+\qquad
+a_j=\frac{\Lambda(n_j)}{n_j}.
+```
+
+## 1.3 Monotone Transportzellen
+
+```math
+s_0=\gamma,
+\qquad
+s_j=\gamma+\sum_{i\le j}a_i,
+```
+
+```math
+I_0=[0,\gamma),
+\qquad
+I_j=[s_{j-1},s_j),
+\qquad |I_j|=a_j.
+```
+
+Der Transport `a_j delta_{h_j} -> 1_{I_j}(L)dL` ist fix und positiv.
+
+Bereits bewiesen:
+
+```math
+\sum_j a_j\sup_{L\in I_j}|L-h_j|<\infty.
+```
+
+## 1.4 Relative Root-Transferfunktion
+
+Fuer `Re s>=0` konvergiert direkt und absolut:
+
+```math
+\gamma-\int_0^\gamma e^{-sL}dL
++
+\sum_j\left[
+ a_je^{-sh_j}-\int_{I_j}e^{-sL}dL
+\right].
+```
+
+Dies ist die sichere, bereits konstruierte relative Stufe.
+
+## 1.5 Critical-half Port
+
+Der endliche relative Output ist mit
+
+```math
+J_\Delta(L)
+=e^{L/2}\left(L-\gamma-\sum_{\log n<L}\frac{\Lambda(n)}n\right)
+```
+
+und
+
+```math
+D_+=\partial_x+1/2
+```
+
+gekoppelt.
+
+Die beiden Seiten Prime/Continuum duerfen **nicht getrennt** durch `e^{L/2}` verstaerkt werden.
+
+## 1.6 Hoehere Gamma-Reservoirs
+
+```math
+\mu_m=2m+1/2,\qquad m\ge1,
+```
+
+mit den bereits fixierten stabilen first-order Filterzustaenden. Diese sind positive Reservoirs, keine freien Fit-Parameter.
+
+---
+
+# 2. Minimale Kandidatenklasse
+
+Der erste Kandidat soll bewusst eng sein.
+
+Pro Transportzelle `I_j` verwende einen `2x2`-relativen Zustand
+
+```text
+[ discrete prime/root state ]
+[ matched continuum cell   ]
+```
+
+mit folgenden Regeln:
+
+1. Diagonalmetriken stammen ausschliesslich aus den bereits bewiesenen lokalen OU/Gamma-Zellen.
+2. Die Prime-/Continuum-Massen sind exakt `a_j`; keine Gewichtsveraenderung.
+3. Die Lagekopplung ist ausschliesslich durch `h_j` und `I_j` bestimmt.
+4. Cross-Terme muessen aus der kanonischen overlap-cone / stopped-OU incidence folgen.
+5. Kein Koeffizient darf anhand des Weil-Zielwertes, des Prime-2-Zeugen oder von Zeta-Nullstellen angepasst werden.
+6. Vor Critical-half-Ausgabe muss der gemeinsame divergente Modus intern ausgeloescht sein.
+7. Der ausgegebene relative Boundary-State muss in einen echten Hilbertraum fallen.
+
+Die Kandidatenklasse darf spaeter vergroessert werden, aber nur nach einem expliziten FAIL dieser minimalen Klasse.
+
+---
+
+# 3. Was als PASS zaehlt
+
+Ein PASS benoetigt **alle** folgenden Punkte:
+
+### P1 — Positiver Parent
+
+Expliziter Hilbert-/Storage-Raum `H_rel` und positive Energie
+
+```math
+E(z)=\langle Pz,z\rangle,\qquad P\succeq0,
+```
+
+vor jeder Verwendung von RH/Weil-Positivitaet.
+
+### P2 — Coefficient-free coupling
+
+Alle Kopplungen folgen aus P11-/OU-/Transport-/Gamma-Daten; keine nachtraegliche Kalibration.
+
+### P3 — Relative cancellation is internal
+
+Der unbeschraenkte gemeinsame Critical-half-Hauptteil darf nicht als Differenz zweier separat divergenter Outputnormen definiert werden. Die Ausloeschung muss als interner State-/Boundary-Mechanismus vor der Outputverstarkung stattfinden.
+
+### P4 — Exact Prime-2 mixed calibration
+
+Auf dem bereits fixierten `R=1` PR-#91-Zeugen muss exakt die bekannte gemischte Weil-Korrektur reproduziert werden:
+
+```math
+B_W(a,b)-c_\Gamma(a,b)=-\frac{\log2}{\sqrt2}.
+```
+
+Kein numerischer Naeherungstreffer genuegt.
+
+### P5 — Full local identities preserved
+
+Der Kandidat darf S1–S13 aus `HARD_AUDIT_SURVIVOR_STATE.md` nicht verletzen.
+
+### P6 — Noncircular right-half-plane storage
+
+Erst wenn P1–P5 vorwaerts stehen, darf untersucht werden, ob die resultierende Transferfunktion die bekannte Lagarias-/`xi'/xi`-Positive-Real-Struktur liefert.
+
+Ein PASS waere der erste substanzielle Object-X-Mechanismus dieses Astes.
+
+---
+
+# 4. Was als FAIL zaehlt
+
+Die minimale Klasse ist FAIL, sobald einer dieser Punkte rigoros gezeigt wird:
+
+1. Positivitaet des Parent-Blocks erzwingt eine Prime-2-Kalibration, die vom exakten Ziel getrennt ist.
+2. Die interne relative Ausloeschung laesst notwendigerweise einen unbeschraenkten Critical-half-Output zurueck.
+3. Der kanonische Cross-Block ist auf dem Root-Transportraum nicht closable/beschraenkt in der benoetigten Energy-Topologie.
+4. Ein Rang-/Totalitaetsargument reduziert den relativen Output auf null oder auf eine bereits gesperrte Klasse.
+5. Die Konstruktion ist algebraisch nur eine Rueckdefinition aus der Weilform/`xi'/xi` und daher zirkulaer.
+
+Bei FAIL:
+
+```text
+close RELATIVE-COLLIGATION minimal class;
+keep all survivor identities;
+do not immediately invent fitted coefficients.
+```
+
+---
+
+# 5. Erster konkreter Arbeitszug
+
+Nicht global starten. Zuerst `R=1`, Prime-2-Witness.
+
+### Schritt A — Transportzellen im endlichen Witness-Scope
+
+Bestimme exakt, welche `I_j` und Prime-Power-Tiefen durch den Witness ueberhaupt aktiv werden.
+
+### Schritt B — Kanonischen 2x2 Cross-Block ableiten
+
+Nur aus:
+
+```text
+- equal-mass transport I_j,
+- local Schur factor Phi_h,
+- stopped OU boundary mode,
+- overlap-cone incidence,
+- fixed D_+=partial+1/2 port.
+```
+
+Keine freien Skalare.
+
+### Schritt C — Mixed pairing symbolisch ausrechnen
+
+Berechne die induced pairing `Q_rel(a,b)` exakt.
+
+### Schritt D — Destruktiver Vergleich
+
+Vergleiche mit
+
+```math
+-\log2/\sqrt2.
+```
+
+- exakt gleich -> Kandidatenklasse ueberlebt und wird auf weitere Witnesses/radii erweitert;
+- exakt getrennt -> Class No-Go dokumentieren;
+- Ausdruck nicht wohldefiniert/unbeschraenkt -> Functional-analytic No-Go dokumentieren.
+
+---
+
+# 6. Governance
+
+- PR #116 bleibt Draft.
+- Keine Registry-Promotion durch einen einzelnen lokalen PASS.
+- Jede neue Identitaet bekommt Primaeraudit + Scope + Nonclaim.
+- Jeder No-Go wird in `WHAT_NOT_TO_REOPEN.md` aufgenommen.
+- `RESEARCH_STATE.yaml` muss bei Frontverschiebung aktualisiert werden.
+- `HARD_AUDIT_SURVIVOR_STATE.md` bleibt die menschliche kanonische Einstiegsebene.
+
+---
+
+# 7. Erfolgskriterium fuer den naechsten Agenten
+
+Ein neuer Agent soll nach Lesen dieser Datei **nicht** fragen: „Welche Methode probieren wir als Naechstes?“
+
+Er soll genau eine Frage bearbeiten:
+
+```text
+Can the canonical equal-mass Prime/continuum transport cells support a
+positive 2x2 relative colligation which cancels before the Critical-half lift
+and reproduces the exact R=1 Prime-2 mixed Weil pairing without fitted data?
+```
+
+Alles andere ist vorerst Nebenfront.
diff --git a/00-uebersicht/RESEARCH_STATE.yaml b/00-uebersicht/RESEARCH_STATE.yaml
new file mode 100644
index 00000000..55c5c31a
--- /dev/null
+++ b/00-uebersicht/RESEARCH_STATE.yaml
@@ -0,0 +1,167 @@
+version: 1
+updated: 2026-09-16
+branch: research/critical-half-green-tree-bridge-2026-09-13
+pr: 116
+registry_unchanged: true
+nonclaims:
+  - no_rh_proof
+  - no_object_x_completion
+  - no_registry_promotion
+  - no_publication_novelty_claim
+
+status_codes:
+  proven: "theorem-level exact identity/result in stated scope"
+  partial: "substantive exact reduction/bridge but not closure of the target gate"
+  nogo: "architecture/class excluded in stated scope"
+  open: "unproved current research gate"
+  diagnostic: "numerical or heuristic only"
+
+canonical_entrypoints:
+  human: 00-uebersicht/HARD_AUDIT_SURVIVOR_STATE.md
+  machine: 00-uebersicht/RESEARCH_STATE.yaml
+  closed_routes: 00-uebersicht/WHAT_NOT_TO_REOPEN.md
+  next_gate: 00-uebersicht/NEXT_GATE.md
+  detailed_survivor_ledger: audits/P11_HARD_AUDIT_SURVIVOR_LEDGER_2026-09-15.md
+
+claims:
+  - id: S1_NULLPOL_GREEN_RANGE
+    status: proven
+    statement: "NULLPOL on compact smooth windows equals Range(-d^2+1/4); Mellin constraints are zero exterior Green charge."
+    proof: audits/P11_CRITICAL_HALF_GREEN_TREE_BRIDGE_2026-09-13.md
+    dependencies: []
+
+  - id: S2_P11_AR1_LEDGER
+    status: proven
+    statement: "The windowless P11 prime-power channel ledger is the sampled Critical-half OU/AR(1) Gram with q=p^-1/2."
+    proof: audits/P11_CRITICAL_HALF_GREEN_TREE_BRIDGE_2026-09-13.md
+    firewall: "channel-index ledger only; not automatically the full spatial P11 operator"
+    dependencies: []
+
+  - id: S3_STOPPED_OU_FILTRATION
+    status: proven
+    statement: "P11 spatial masks implement a stopped OU filtration with L_R(x)=2(R-|x|) and positive rank-one depth increments."
+    proof: audits/P11_SPATIAL_STOPPED_OU_FILTRATION_2026-09-15.md
+    dependencies: [S2_P11_AR1_LEDGER]
+
+  - id: S4_UNIVERSAL_STOPPED_KERNEL
+    status: proven
+    statement: "G_L(s,t)=exp(-(s+t)/2+min(s,t,L)) is the universal stopped Critical-half OU kernel sampled by P11."
+    proof: audits/P11_UNIVERSAL_STOPPED_OU_PRECISION_2026-09-15.md
+    dependencies: [S3_STOPPED_OU_FILTRATION]
+
+  - id: S5_BOUNDARY_TAIL_SHORTING
+    status: proven
+    statement: "Stopped OU precision reconstructs all visible bare channels exactly and compresses the exterior tail to one normalized exponential boundary mode with positive discarded complement."
+    proof: audits/P11_BOUNDARY_TAIL_SHORTING_AND_UNIVERSAL_SAMPLING_2026-09-15.md
+    dependencies: [S4_UNIVERSAL_STOPPED_KERNEL]
+
+  - id: S6_UNIVERSAL_JUMP_FIELD
+    status: proven
+    statement: "X_v(t)=e^-t/4 K_t v is sampled by every weighted prime channel: sqrt(w_p,k)K_klogp v=sqrt(log p) X_v(k log p)."
+    proof: audits/P11_BOUNDARY_TAIL_SHORTING_AND_UNIVERSAL_SAMPLING_2026-09-15.md
+    dependencies: []
+
+  - id: S7_PRIME_DISCREPANCY_CANCELLATION
+    status: proven
+    statement: "On NULLPOL the smooth prime main density cancels the Gamma ground mode; arithmetic remainder is Delta=sum Lambda(n)/sqrt(n) delta_log n - e^(t/2)dt."
+    proof: audits/P11_NULLPOL_PRIME_DISCREPANCY_CANCELLATION_2026-09-15.md
+    dependencies: [S1_NULLPOL_GREEN_RANGE]
+
+  - id: S8_CRITICAL_FINITE_PART_TRANSFER
+    status: proven
+    statement: "J_Delta(L)=e^(L/2)(L-gamma-sum_{log n<L}Lambda(n)/n), J_Delta(0)=-gamma, and dDelta=(1/2)J_Delta dL-dJ_Delta."
+    proof: audits/P11_CRITICAL_HALF_FINITE_PART_TRANSFER_2026-09-15.md
+    dependencies: [S7_PRIME_DISCREPANCY_CANCELLATION]
+
+  - id: S9_CAUSAL_VOLTERRA_NORMAL_FORM
+    status: proven
+    statement: "The discrepancy operator admits a causal Volterra factorization with D_+=partial+1/2; together with higher Gamma filters this yields the exact causal NULLPOL Weil normal form."
+    proof: audits/P11_CAUSAL_VOLterra_PASSIVITY_NORMAL_FORM_2026-09-15.md
+    dependencies: [S8_CRITICAL_FINITE_PART_TRANSFER]
+    firewall: "identity only; positivity remains open"
+
+  - id: S10_EULER_AR1_RESOLVENT
+    status: proven
+    statement: "The adjoint P11 AR(1) Cholesky factor has transfer 1/(1-qz), exactly the local Euler factor for q=p^-1/2 and z=e^-slogp."
+    proof: audits/P11_EULER_AR1_RESOLVENT_BRIDGE_2026-09-15.md
+    dependencies: [S2_P11_AR1_LEDGER]
+
+  - id: S11_EULER_CELL_DOMINATION
+    status: proven
+    statement: "For every h>0, 0 <= A_h <= A_0 where A_h is the complete Euler/AR(1) cell and A_0 the Gamma ground operator; equivalently b_h=Phi_h b_0 with Phi_h Schur."
+    proof: audits/P11_EULER_CELL_DOMINATED_BY_GAMMA_GROUND_2026-09-16.md
+    dependencies: [S10_EULER_AR1_RESOLVENT]
+    consequence: "local prime-cell storage solved; global scale coupling remains"
+
+  - id: S12_ROOT_TRACE_BALANCE
+    status: proven
+    statement: "e^-L/2 J_Delta(L)=M_R(L)+M_Gamma0(L)-M_P(L)-gamma, with M_P(L)=sum_{log n<L}Lambda(n)/n."
+    proof: audits/P11_CRITICAL_TRANSFER_AS_STOPPED_ROOT_TRACE_DEFECT_2026-09-16.md
+    dependencies: [S8_CRITICAL_FINITE_PART_TRANSFER, S3_STOPPED_OU_FILTRATION]
+
+  - id: S13_ROOT_MASS_TRANSPORT
+    status: partial
+    statement: "Prime-power root masses a_j=Lambda(n_j)/n_j admit a monotone positive transport into L2(dL) via equal-mass cells I_j, with summable displacement and absolutely convergent relative boundary transfer."
+    proof:
+      - audits/P11_ROOT_MASS_MONOTONE_TRANSPORT_2026-09-16.md
+      - audits/P11_ROOT_TRANSPORT_SAFE_TRANSFER_AND_CRITICAL_LIFT_FIREWALL_2026-09-16.md
+    dependencies: [S12_ROOT_TRACE_BALANCE]
+    external_input: "classical unconditional PNT error"
+
+  - id: N1_PLAIN_CRITICAL_LIFT
+    status: nogo
+    statement: "Plain root storage cannot be lifted to Weil weights by bounded diagonal critical-half amplification; cancellation must occur before amplification."
+    proof: audits/P11_ROOT_TRANSPORT_SAFE_TRANSFER_AND_CRITICAL_LIFT_FIREWALL_2026-09-16.md
+    dependencies: [S13_ROOT_MASS_TRANSPORT]
+
+  - id: N2_POINTWISE_SCATTERING_POSITIVITY
+    status: nogo
+    statement: "Pointwise nonnegative scattering/Wigner-Smith multiplier factorization is impossible because tau_a changes sign."
+    proof: audits/P11_SCATTERING_POINTWISE_POSITIVITY_NOGO_2026-09-13.md
+    dependencies: []
+
+  - id: N3_PRIMEWISE_INNOVATION_SHORTING
+    status: nogo
+    statement: "Direct primewise contractive or reverse-contractive shorting between P11 innovations and bare COMMON-JUMP channels fails on the actual translation-generated range."
+    proof: audits/P11_PRIMEWISE_INNOVATION_SHORTING_NOGO_2026-09-15.md
+    dependencies: [S2_P11_AR1_LEDGER]
+
+  - id: N4_COMMON_DEPTH_SHORTING
+    status: nogo
+    statement: "Ordinary Gram embedding of prime and continuum OU cells into a common depth L2 followed by orthogonal shorting kills the arithmetic side because the continuum cells are total."
+    proof: audits/P11_COMMON_DEPTH_SHORTING_TOTALITY_NOGO_2026-09-15.md
+    dependencies: [S4_UNIVERSAL_STOPPED_KERNEL]
+
+  - id: N5_PASSIVE_EULER_CASCADE
+    status: nogo
+    statement: "Ordinary infinite passive cascade of normalized local Euler Schur cells collapses to zero and loses arithmetic information."
+    proof: audits/P11_PASSIVE_EULER_CASCADE_COLLAPSE_NOGO_2026-09-15.md
+    dependencies: [S10_EULER_AR1_RESOLVENT]
+
+  - id: N6_STATIC_GAMMA_FRAME_REPAIR
+    status: nogo
+    statement: "Higher Gamma exponentials are dense in the lost OU tail complement but do not provide a radius-uniform static lower frame."
+    proof: audits/P11_GAMMA_TAIL_FRAME_NOGO_2026-09-15.md
+    dependencies: []
+
+  - id: N7_SCALAR_HUB_REPAIR
+    status: nogo
+    statement: "A scalar/root rank-one correction cannot repair the higher-rank stopped-tail defect."
+    proof: audits/P11_PRIMEWISE_INNOVATION_SHORTING_NOGO_2026-09-15.md
+    dependencies: [S5_BOUNDARY_TAIL_SHORTING]
+
+  - id: O1_RELATIVE_COLLIGATION_CANCEL_FIRST
+    status: open
+    statement: "Construct a coefficient-free positive/conservative relative colligation that cancels discrete prime and continuum root states before critical-half amplification, then exposes only the finite relative boundary output."
+    spec: 00-uebersicht/NEXT_GATE.md
+    dependencies:
+      - S11_EULER_CELL_DOMINATION
+      - S12_ROOT_TRACE_BALANCE
+      - S13_ROOT_MASS_TRANSPORT
+      - N1_PLAIN_CRITICAL_LIFT
+    first_destructive_test: "exact R=1 Prime-2 mixed witness"
+
+frozen_interpretation:
+  main_front: O1_RELATIVE_COLLIGATION_CANCEL_FIRST
+  guiding_rule: "relative cancellation first, Critical-half output second"
+  do_not_claim: "that the surviving identities already constitute Object X"
diff --git a/00-uebersicht/WHAT_NOT_TO_REOPEN.md b/00-uebersicht/WHAT_NOT_TO_REOPEN.md
new file mode 100644
index 00000000..a36c6e6b
--- /dev/null
+++ b/00-uebersicht/WHAT_NOT_TO_REOPEN.md
@@ -0,0 +1,215 @@
+# Objekt X — WHAT NOT TO REOPEN
+
+**Stand:** 16. September 2026  
+**Zweck:** Sackgassen-Firewall fuer neue Agenten und spaetere Forschungsrunden.  
+**Regel:** Diese Klassen nur erneut oeffnen, wenn eine neue Hypothese den dokumentierten No-Go explizit umgeht. Nicht durch Umbenennung derselben Konstruktion reaktivieren.
+
+> Diese Datei schliesst nur die angegebenen Klassen im angegebenen Scope. Sie ist kein universelles Object-X-No-Go.
+
+---
+
+## 1. Pointwise scattering positivity — GESPERRT `×[M]`
+
+Nicht erneut versuchen:
+
+```math
+\tau_a(z)=|B_a(z)|^2
+```
+
+oder irgendeine Architektur, deren Positivitaet punktweise `tau_a(z)>=0` auf der ganzen reellen Achse verlangt.
+
+Grund:
+
+```math
+\tau_a(0)<0,
+\qquad
+\tau_a(z)\to+\infty.
+```
+
+Primaerquelle: `audits/P11_SCATTERING_POINTWISE_POSITIVITY_NOGO_2026-09-13.md`.
+
+Erlaubt bleibt: nichtlokale Paley-Wiener-/Boundary-/Storage-Kompression.
+
+---
+
+## 2. Direkte primeweise P11 -> COMMON-JUMP Kontraktion — GESPERRT `×[M]`
+
+Nicht erneut versuchen, den P11-Innovationstail pro Primzahl direkt kontraktiv oder reverse-kontraktiv mit den nackten Jump-Kanaelen zu identifizieren.
+
+Auf dem tatsaechlichen translation-generated Kanalbereich expandiert der P11-Tail niedrige Frequenzen und kontrahiert andere Richtungen. Es existiert keine einheitliche Ordnung in einer der beiden Richtungen.
+
+Primaerquelle: `audits/P11_PRIMEWISE_INNOVATION_SHORTING_NOGO_2026-09-15.md`.
+
+Erlaubt bleibt: covariance/precision duality und relative global coupling.
+
+---
+
+## 3. Scalar Hub repair — GESPERRT `×[M]`
+
+Nicht erneut versuchen, den gestoppten Taildefekt nur durch einen skalar gewichteten Root-/Hub-Term zu reparieren.
+
+Nach Stopp bleibt ein hoeher-/unendlich-rangiger Exterior-Defekt; ein Rang-1-Rootterm kann diesen universell nicht korrigieren.
+
+Primaerquelle: `audits/P11_PRIMEWISE_INNOVATION_SHORTING_NOGO_2026-09-15.md`.
+
+---
+
+## 4. Common-depth OU embedding + ordinary orthogonal shorting — GESPERRT `×[M]`
+
+Nicht Prime- und Riemann-R-Continuum-Zellen nur als Vektoren im selben `L^2(dt)` einbetten und anschliessend den Continuum-Block orthogonal herausschurten.
+
+Die Continuum-Zellen sind bereits total im gemeinsamen Tiefenraum; das Shorting entfernt die diskrete arithmetische Seite vollstaendig.
+
+Primaerquelle: `audits/P11_COMMON_DEPTH_SHORTING_TOTALITY_NOGO_2026-09-15.md`.
+
+Erlaubt bleibt: Kopplungen, die die Skalenvariable `h`, Overlap-Cone-Inzidenz oder relative Boundary-Daten erhalten.
+
+---
+
+## 5. Ordinary passive cascade of normalized Euler cells — GESPERRT `×[M]`
+
+Nicht die lokalen Schur-Zellen einfach ueber alle Primzahlen als gewoehnliche passive Kaskade multiplizieren.
+
+Fuer festes reelles `s>0` kollabiert das Produkt auf null; der Cayley-Transfer wird trivial und verliert die Arithmetik.
+
+Primaerquelle: `audits/P11_PASSIVE_EULER_CASCADE_COLLAPSE_NOGO_2026-09-15.md`.
+
+Erlaubt bleibt: relative diskret/continuum Renormierung innerhalb eines groesseren konservativen Systems.
+
+---
+
+## 6. Local positive finite-part Hamiltonian by common scalar subtraction — GESPERRT `×[M]`
+
+Nicht Prime- und Continuum-Hamiltonians lokal positiv addieren und danach nur einen gemeinsamen divergenten Skalar abziehen, um einen endlichen positiven kritischen Hamiltonian zu erhalten.
+
+Der renormierte Eigenwert enthaelt `J_Delta(L)` und ist bereits bei `L=0` negativ:
+
+```math
+J_\Delta(0)=-\gamma<0.
+```
+
+Primaerquelle: `audits/P11_CRITICAL_RATE_LOCAL_RENORMALIZATION_NOGO_2026-09-15.md`.
+
+Erlaubt bleibt: relative Rand-/Transfer-/Spektralverschiebung in einem unrenormiert positiven Elternsystem.
+
+---
+
+## 7. Static higher-Gamma frame repair — GESPERRT `×[M]`
+
+Nicht argumentieren:
+
+```text
+P11 behaelt den OU-Grundmodus;
+hoehere Gamma-Moden sind dicht;
+also kontrollieren sie den verlorenen Tail uniform.
+```
+
+Die hoeheren Gamma-Exponentialmoden sind zwar total/dicht im passenden Tailkomplement, besitzen aber keine radiusuniforme untere Frame-Schranke. Nach aussen verschobene Einheitsvektoren haben verschwindende Gamma-Koeffizienten.
+
+Primaerquelle: `audits/P11_GAMMA_TAIL_FRAME_NOGO_2026-09-15.md`.
+
+---
+
+## 8. Separate Critical-half amplification of Prime and Continuum — GESPERRT `×[M]`
+
+Dies ist die wichtigste aktuelle Firewall.
+
+Nicht zuerst den diskreten Root-Storage und den Continuum-Root-Storage getrennt von
+
+```math
+\Lambda(n)/n
+```
+
+auf
+
+```math
+\Lambda(n)/\sqrt n
+```
+
+anheben und erst danach subtrahieren.
+
+Die notwendige Energieverstarkung ist `e^{L/2}` und auf dem plain Root-Storage unbeschraenkt. Fuer nichtverschwindende kompakte Sources wird der grosse-Tail nach der kritischen Verstarkung konstant statt integrierbar.
+
+Primaerquelle: `audits/P11_ROOT_TRANSPORT_SAFE_TRANSFER_AND_CRITICAL_LIFT_FIREWALL_2026-09-16.md`.
+
+Verbindliche Folgerung:
+
+```text
+CANCEL FIRST, CRITICAL-LIFT SECOND.
+```
+
+---
+
+## 9. Fixed-window prolate as all-window strategy — DEMOTED
+
+Die shorted Paley-Wiener-/Prolate-Reduktion und das Trace-minus-Ritz-Lemma bleiben korrekt und nuetzlich als fixed-window Zertifikationswerkzeug.
+
+Nicht mehr als skalierbare globale Hauptstrategie behandeln: die relevante time-band-limiting Region waechst extrem stark mit dem Fenster und reproduziert eine bekannte lokale-Zertifikationsbarriere.
+
+Primaerquellen:
+
+- `audits/P11_NP_GAP_PROLATE_COMPRESSION_2026-09-13.md`
+- `audits/P11_NP_PROLATE_TRACE_RITZ_CERTIFICATE_2026-09-13.md`
+- `audits/P11_HARD_AUDIT_ERRATA_AND_PROLATE_BARRIER_2026-09-15.md`
+
+---
+
+## 10. Old orbit-chain exact formula — KORRIGIERT
+
+Nicht mehr schreiben
+
+```math
+N(t)=\lfloor L/t\rfloor+1
+```
+
+als exakte wesentliche maximale Kettenlaenge an ganzzahligen Verhaeltnissen `L/t`.
+
+Korrekt ist die wesentliche maximale Kettenlaenge
+
+```math
+N(t)=\lceil L/t\rceil
+```
+
+mit der ueblichen Randmengen-Firewall. Die alte daraus abgeleitete Lower-Bound war konservativ, aber die Exaktheitsbehauptung war falsch.
+
+Primaerquelle: `audits/P11_HARD_AUDIT_ERRATA_AND_PROLATE_BARRIER_2026-09-15.md`.
+
+---
+
+## 11. Zeugen-Koinzidenzen nicht verallgemeinern
+
+Nicht erneut aus einer einzigen Prime-2-Kalibration freie universelle Koeffizienten ableiten. Insbesondere die gefundene `1/4`-Hub-Korrektur am einzelnen Zeugen war eine Koinzidenz und scheitert als Operatorgesetz.
+
+Regel:
+
+```text
+kein freier Koeffizient darf nach Sicht auf den Prime-2 witness angepasst werden.
+```
+
+---
+
+## 12. Bekannte Theorie nicht als Projektneuheit verkaufen
+
+Nicht als Neuheit beanspruchen:
+
+- `xi'/xi` als Positive-Real/Herglotz RH-Kriterium;
+- Lagarias-Positivitaet;
+- Zeta-Transfer-/Kontrolltheorie allgemein;
+- Eulerprodukt als Determinanten-/Resolventensprache;
+- OU/AR(1), Green-, Poisson-, de-Branges-, Paley-Wiener-, Toeplitz-, Prolate- oder KYP-Theorie an sich;
+- klassische PNT-Fehlerabschaetzungen.
+
+Der moegliche Projektbeitrag kann nur in einer neuen, korrekt auditierten Glueung der konkreten P11-/Overlap-Cone-/Root-Transport-/Gamma-Komponenten liegen.
+
+---
+
+# Reopen-Protokoll
+
+Eine gesperrte Klasse darf nur reaktiviert werden, wenn eine neue Konstruktion schriftlich beantwortet:
+
+1. Welcher konkrete No-Go oben wird umgangen?
+2. Welche neue Hypothese/Struktur war dort nicht enthalten?
+3. Warum ist es nicht nur eine Umparametrisierung derselben gescheiterten Klasse?
+4. Welcher vorab festgelegte destruktive Test kann die neue Variante falsifizieren?
+
+Ohne diese vier Punkte: **nicht reopen**.
diff --git a/CURRENT-FRONT.md b/CURRENT-FRONT.md
index 4647dadf..f356cbee 100644
--- a/CURRENT-FRONT.md
+++ b/CURRENT-FRONT.md
@@ -1,218 +1,442 @@
-# CURRENT FRONT — Objekt X / NULLPOL-COMMON → NP-GAP
+# CURRENT FRONT — Objekt X / CRITICAL-HALF SCATTERING → NP-GAP
 
 > **Operative Kopfschicht — zuerst lesen.**  
 > **Stand:** 13. September 2026; keine Registry-Promotion.  
-> **Kanonischer neuer Audit:** [NULLPOL common jump-Gram](audits/P11_NP_COMMON_JUMP_GRAM_2026-09-13.md).  
-> Vorheriger strategischer Filter: [Nullpol-Reklassifikation](audits/P11_NULLPOLE_STRATEGIC_RECLASSIFICATION_2026-09-13.md).
+> **Basis:** NULLPOL-COMMON ist gemergt; dieser Branch konsolidiert die neue Critical-half-Green-/Poisson-/Scattering-Route.  
+> **Hauptengpass bleibt:** NP-GAP fuer alle Fenster.
 
-Diese Datei ordnet die Arbeit; sie beweist nichts. Registry und Arbeitsdefinition bleiben unverändert.
+Diese Datei ordnet die Arbeit; sie beweist nichts. Registry und Object-X-Arbeitsdefinition bleiben unverändert.
 
-## 1. Nullpol-Hauptklasse
+## 1. COMMON-JUMP bleibt die exakte Ausgangsnormalform `✓[M]`
 
 Mit
 
 ```math
-M(v)(s)=\int_{\mathbb R}v(x)e^{(s-1/2)x}\,dx
+K_t=T_{t/2}-T_{-t/2},
+\qquad
+h(t)=\frac{e^{-t/2}}{1-e^{-2t}},
+\qquad
+w_n=\frac{\Lambda(n)}{\sqrt n}
 ```
 
-gilt exakt
+und
+
+```math
+\mathcal X_av=
+\left([t\mapsto K_tv],
+[\sqrt{w_n}K_{\log n}v]_{\log n\le2a}\right)
+```
+
+gilt fuer jedes `a>0`
 
 ```math
-E_-(v)=M(v)(0),\qquad E_+(v)=M(v)(1).
+Q_W(v,w)
+=\langle\mathcal Ev,P\mathcal Ew\rangle
++\langle\mathcal X_av,\mathcal X_aw\rangle
+-\Gamma_a\langle v,w\rangle,
+```
+
+wobei
+
+```math
+\Gamma_a
+=2\sum_{\log n\le2a}\frac{\Lambda(n)}{\sqrt n}
++\kappa_*,
+\qquad
+\kappa_*=\log\pi-\psi(1/4).
 ```
 
 Auf
 
 ```math
-\mathscr D_{NP}=\ker M(0)\cap\ker M(1)
+\mathscr D_{NP}
+=\ker M(0)\cap\ker M(1)
+```
+
+verschwindet der Polblock und
+
+```math
+\boxed{Q_W|_{NP}=\mathcal X_a^*\mathcal X_a-\Gamma_aI.}
 ```
 
-verschwinden daher `R_0` und `\mathcal E`. Connes–Consani Proposition C.1 liefert global einen RH-äquivalenten Weil-Scope mit diesen Nullbedingungen. Keine fixed-`a`-Äquivalenz wird behauptet.
+Die Familie aller Fenster ist global RH-aequivalent; fixed-`a` wird nicht als RH-aequivalent behauptet.
 
-## 2. Der gemeinsame Prime-/archimedische Kanal `✓[M]`
+---
 
-Setze für `t>0`
+## 2. Ein einziger Critical-half Quellenoperator `✓[M]`
+
+Setze
 
 ```math
-\boxed{K_t:=T_{t/2}-T_{-t/2}.}
+\boxed{
+L_{1/2}:=-\partial_x^2+\frac14.
+}
 ```
 
-Dann
+Sein Green-Kern auf `R` ist
 
 ```math
-K_t^*K_t=2I-T_t-T_{-t}.
+\boxed{G_{1/2}(x,y)=e^{-|x-y|/2}.}
 ```
 
-Die archimedische positive Dichte ist
+Fuer jedes `a>0` gilt exakt
 
 ```math
 \boxed{
-h(t)=\frac{e^{-t/2}}{1-e^{-2t}}
-=\frac{e^{t/2}}{e^t-e^{-t}}>0.
+C_c^\infty(-a,a)\cap\ker M(0)\cap\ker M(1)
+=L_{1/2}C_c^\infty(-a,a).
 }
 ```
 
-Die Prime-Power-Gewichte sind
+Die Nullpolbedingungen sind genau die Bedingungen, welche die beiden exponentiellen Aussenschwaenze des Green-Inversen vernichten. Damit wird NP-GAP durch
 
 ```math
-w_n=\frac{\Lambda(n)}{\sqrt n}.
+v=L_{1/2}u,
+\qquad u\in C_c^\infty(-a,a)
 ```
 
-Für `supp(v),supp(w) subset [-a,a]` sei
+zu einer momentfreien Coercivity-Frage.
+
+---
+
+## 3. Pole = orientierter Außen-Transfer desselben Green-Kerns `✓[M]`
+
+Definiere
 
 ```math
-\mathcal P_a=\{n=p^k:\log n\le2a\}.
+E_+(f)=\int e^{x/2}f(x)dx,
+\qquad
+E_-(f)=\int e^{-x/2}f(x)dx.
 ```
 
-Definiere die positive gemischte Maßstruktur
+Sind die Traeger getrennt und liegt `f` links von `g`, dann
 
 ```math
 \boxed{
-\mu_a=h(t)dt+\sum_{n\in\mathcal P_a}w_n\delta_{\log n}.
+\langle f,G_{1/2}g\rangle
+=E_+(f)\overline{E_-(g)}.
 }
 ```
 
-**Archimedes = kontinuierlicher Teil, Primzahlpotenzen = atomarer Teil desselben `K_t`-Features.**
+In umgekehrter Orientierung:
 
-## 3. Positive Featureabbildung `✓[M]`
+```math
+\boxed{
+\langle f,G_{1/2}g\rangle
+=E_-(f)\overline{E_+(g)}.
+}
+```
 
-Ein natürlicher Hilbert-Zielraum ist
+Dies sind exakt die beiden Off-Diagonal-Amplituden des bisherigen Pole-pairings
 
 ```math
-\mathscr H_a=
-L^2((0,\infty),h(t)dt;L^2(\mathbb R))
-\oplus
-\bigoplus_{n\in\mathcal P_a}L^2(\mathbb R).
+\langle\mathcal Ef,P\mathcal Eg\rangle.
 ```
 
-Setze
+Daher bedeutet NULLPOL zugleich: **keine Critical-half Green-Außenladung**. Getrennte Translationen einer Nullpol-Funktion sind in der Green-Metrik exakt orthogonal.
+
+Firewall: Der `P`-Block bleibt indefinit; diese Identitaet macht ihn nicht selbst positiv.
+
+---
+
+## 4. P11-Prime-Ledger = logarithmischer OU-/Green-Baum `✓[M]`
+
+Auf einem verwurzelten Sternbaum mit einem Halbast pro Primzahl und Knoten
+
+```math
+x_{p,k}=k\log p
+```
+
+gibt es explizite Einheitsfeatures `Phi` mit
 
 ```math
 \boxed{
-\mathcal X_av=
-\left(
-[t\mapsto K_tv],
-[\sqrt{w_n}K_{\log n}v]_{n\in\mathcal P_a}
-\right).
+\langle\Phi_\alpha,\Phi_\beta\rangle
+=e^{-d(\alpha,\beta)/2}.
 }
 ```
 
-Dann
+Auf demselben Primast:
+
+```math
+p^{-|j-k|/2}
+=e^{-\frac12|j-k|\log p},
+```
+
+also exakt der aus P11 bekannte AR(1)-Kern. Nach Weil-Gewichtung
+
+```math
+\Psi_{p,k}=\sqrt{(\log p)p^{-k/2}}\,\Phi_{p,k\log p}
+```
+
+reproduziert der Gram den gesamten fensterlosen P11-Hub+Rest-Kanalindex-Ledger, inklusive cross-prime Root-Korrelation.
+
+Die bereits bekannte Faktorisierung
+
+```math
+R_q=T_q^*T_q+uu^*
+```
+
+ist exakt die Zerlegung in gemeinsame Root-Koordinate plus orthogonale Ast-Innovationen.
+
+Firewall: Fensterprojektionen und die raeumliche P11-Operatorgeometrie bleiben notwendig.
+
+---
+
+## 5. Critical-half rigidity `✓[M]`
+
+In der normierten OU-Familie
+
+```math
+\langle\Phi^{(\sigma)}_\alpha,
+\Phi^{(\sigma)}_\beta\rangle=e^{-\sigma d(\alpha,\beta)}
+```
+
+hat die gewichtete Prime-Rootamplitude den Exponenten
+
+```math
+\beta(\sigma)=\frac14+\sigma.
+```
+
+P11 erzwingt unabhaengig `beta=3/4`; daher
+
+```math
+\boxed{\sigma=1/2.}
+```
+
+Dieselbe Skala wird separat durch die Nullpol-Moden `e^{±x/2}` und durch den Grundmodus der Gamma-Schicht erzwungen. Die `1/2` ist in dieser Familie kein Fitparameter.
+
+---
+
+## 6. Gamma-Schicht = positive Resolventenleiter desselben Grundoperators `✓[M]`
+
+Aus
+
+```math
+h(t)=\sum_{m\ge0}e^{-\mu_mt},
+\qquad
+\mu_m=2m+\frac12
+```
+
+folgt
 
 ```math
 \boxed{
-\langle\mathcal X_av,\mathcal X_aw\rangle
-=
-\int_0^\infty h(t)\langle K_tv,K_tw\rangle dt
-+
-\sum_{n\in\mathcal P_a}w_n
-\langle K_{\log n}v,K_{\log n}w\rangle.
+\Phi_\infty(D)
+=\sum_{m\ge0}
+\frac{2}{\mu_m}D^2(D^2+\mu_m^2)^{-1}.
+}
+```
+
+Der Grundmodus ist
+
+```math
+\mu_0=1/2,
+\qquad
+D^2+\mu_0^2=L_{1/2}.
+```
+
+Nach dem Source-Lift `v=L_{1/2}u` wird dieser Grundmodus rein lokal:
+
+```math
+\boxed{
+4\|u''\|_2^2+\|u'\|_2^2.
 }
 ```
 
-Diese Form ist vorwärts konstruiert und positiv; keine Weil-Positivität und keine RH-Annahme gehen ein.
+Der einfache Schur-Test der Resolventenleiter liefert ausserdem einen vorwaerts bewiesenen kleinen NP-GAP-Bereich `0<a<=1/16`; dies ist kein Literaturrekord, sondern ein Mechanismus-Zertifikat.
+
+---
 
-## 4. Exakte archimedische Schwelle `✓[M]`
+## 7. Die gesamte Schwelle ist Diagonal-/finite-part-Masse derselben positiven Modi `✓[M]`
 
-Direkt aus dem Integralterm der expliziten Formel:
+Prime-Atom:
 
 ```math
-2\int_0^\infty h(t)(1-e^{-t/2})dt
-=\log2+\frac\pi2.
+w_nK_t^*K_t
+=2w_nI-w_n(T_t+T_{-t}).
 ```
 
-Daher
+Also ist `2w_n` exakt seine Diagonalmasse.
+
+Gamma-Modus:
+
+```math
+A_m(z)
+=\frac2{\mu_m}\frac{z^2}{z^2+\mu_m^2}
+=\frac2{\mu_m}-\frac{2\mu_m}{z^2+\mu_m^2}.
+```
+
+Seine Diagonal-/UV-Masse ist `2/mu_m`; die logarithmisch divergente Summe besitzt exakt die finite part
 
 ```math
 \boxed{
 \kappa_*
-=\log(8\pi)+\gamma+\frac\pi2
-=\log\pi-\psi(1/4).
+=
+\lim_{N\to\infty}
+\left[
+\sum_{m=0}^N\frac2{\mu_m}
+-
+\log\frac{N+1}{\pi}
+\right].
 }
 ```
 
-und
+Damit ist
 
 ```math
 \boxed{
-\Gamma_a=
-2\sum_{n\in\mathcal P_a}\frac{\Lambda(n)}{\sqrt n}
-+\log\pi-\psi(1/4).
+\Gamma_a
+=\text{endliche Prime-Diagonalmasse}
++\text{renormalisierte Gamma-Diagonalmasse}.
 }
 ```
 
-## 5. Exakte COMMON-JUMP-GRAM-Normalform `✓[M]`
+Der Skalar ist kein separater Object-X-Generator mehr. Offen ist nur die Positivitaet **nach dieser kanonischen Zentrierung**.
+
+---
+
+## 8. Prime towers = `flat - P11 AR(1) Poisson` `✓[M]`
+
+Fuer `q_p=p^{-1/2}` ist der Disk-Poisson-Kern
 
-Für glatte kompakt getragene `v,w` in `[-a,a]`, **für jedes `a>0`**, gilt exakt
+```math
+P_q^{\mathbb D}(\theta)
+=\frac{1-q^2}{1-2q\cos\theta+q^2}
+=1+2\sum_{k\ge1}q^k\cos(k\theta).
+```
+
+Der komplette zentrierte Weil-Beitrag des `p`-Power-Turms ist daher
 
 ```math
 \boxed{
-Q_W(v,w)
-=
-\langle\mathcal Ev,P\mathcal Ew\rangle
-+
-\langle\mathcal X_av,\mathcal X_aw\rangle
--
-\Gamma_a\langle v,w\rangle,
+-2\log p\sum_{k\ge1}p^{-k/2}\cos(kz\log p)
+=(\log p)
+\left[1-P_{p^{-1/2}}^{\mathbb D}(z\log p)\right].
 }
 ```
 
-mit
+`P_q^D` ist exakt das Toeplitz-Symbol des P11-AR(1)-Kerns `q^{|j-k|}`.
+
+Die unendliche Potenzergänzung ist auf einem kompakten Fenster formexakt: fuer `k log p>2a` verschwindet der zentrierte Jump-Summand wegen Trägertrennung identisch.
+
+---
+
+## 9. Gamma-Moden = Half-plane-Poisson; Prime und Archimedes sind dieselbe harmonische Kernelklasse `✓[M]`
+
+Mit
 
 ```math
-P=\begin{pmatrix}0&1\\1&0\end{pmatrix}.
+P_y^{\mathbb H}(x)=\frac1\pi\frac{y}{x^2+y^2}
 ```
 
-Auf Nullpol:
+gilt
 
 ```math
 \boxed{
-Q_W(v,w)
-=
-\langle\mathcal X_av,\mathcal X_aw\rangle
--
-\Gamma_a\langle v,w\rangle.
+A_m(z)-\frac2{\mu_m}
+=-2\pi P_{\mu_m}^{\mathbb H}(z).
 }
 ```
 
-Damit sind `log|D|`, `R_1`, Prime shifts und Exterior-shell-Masse nicht mehr getrennte Geometriefragen: sie liegen in derselben positiven `K_t`-Jump-Gram-Struktur.
+Somit ist die archimedische zentrierte Schicht ein finite-part-renormalisierter Half-plane-Poisson-Kamm, waehrend jeder Prime-Turm ein Disk-Poisson-Defekt ist. Disk und oberer Halbraum sind die beiden Standardmodelle derselben harmonischen Maßgeometrie (Cayley-Transformation).
+
+Es wird keine kuenstliche direkte Identifikation `q_p <-> mu_m` behauptet.
+
+---
+
+## 10. Exakte Euler/Gamma-Streuphase `✓[M]`
+
+Fuer jede Primzahl setze
+
+```math
+S_p(z)
+=
+\frac{1-p^{-1/2}e^{iz\log p}}
+     {1-p^{-1/2}e^{-iz\log p}},
+```
+
+und archimedisch
 
-## 6. Fourier-Gegencheck
+```math
+S_\infty(z)
+=
+\pi^{-iz}
+\frac{\Gamma(1/4+iz/2)}
+     {\Gamma(1/4-iz/2)}.
+```
 
-Die positive Featureform hat Symbol
+Fuer reelles `z` sind alle diese Faktoren unitär. Es gilt exakt
 
 ```math
-\Phi_a(z)=
-\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\psi(1/4)
-+2\sum_{n\in\mathcal P_a}w_n(1-\cos(z\log n))\ge0
+-i\overline{S_p}S_p'
+=(\log p)
+\left[1-P_{p^{-1/2}}^{\mathbb D}(z\log p)\right],
 ```
 
-für reelles `z`. Nach Abzug von `Gamma_a` erhält man exakt den nichtpolaren Suzuki-/Weil-Multiplikator.
+```math
+-i\overline{S_\infty}S_\infty'
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi.
+```
 
-## 7. Cutoff-Gauge jetzt geometrisiert `✓[M]`
+Fuer
 
-Für `0<a<b` und bereits in `[-a,a]` getragenes `v,w` liefern neue Prime-Atome mit `2a<log n<=2b`
+```math
+\boxed{
+S_a(z)=S_\infty(z)\prod_{p\le e^{2a}}S_p(z)
+}
+```
+
+ist daher der zentrierte Fensteroperator in quadratischer Form genau der skalare Wigner--Smith-/Phasenoperator
 
 ```math
-\langle K_{\log n}v,K_{\log n}w\rangle
-=2\langle v,w\rangle.
+\boxed{
+\tau_a(z)=-i\overline{S_a(z)}S_a'(z).
+}
 ```
 
-Daher wachsen positive Gramform und Schwelle um exakt denselben Skalar:
+Auf NULLPOL:
 
 ```math
 \boxed{
-\langle X_bv,X_bw\rangle-\Gamma_b\langle v,w\rangle
-=
-\langle X_av,X_aw\rangle-\Gamma_a\langle v,w\rangle.
+Q_W(v)
+=\int|\widehat v(z)|^2\tau_a(z)\,dz.
 }
 ```
 
-Der invariant sinnvolle Gegenstand ist also die **zentrierte Gram-Paarung**, nicht ein isolierter `c_a`-Wert.
+Die Gleichheit ist im Fenster-Testfunktionsraum gemeint; die in `S_p` enthaltenen hohen Prime powers jenseits des geometrischen Cutoffs integrieren dort exakt zu null.
+
+Allgemeine Zeta-/Scattering-Interpretationen sind aus der Literatur bekannt. Publikationsneuheit dieser spezifischen P11→AR(1)→Poisson→Fenster-Streuphase-Verklebung bleibt offen.
+
+---
+
+## 11. Kanonische Object-X-Architektur nach diesem Durchlauf `✓[M]_part`
+
+Die vorher getrennten Schichten liegen jetzt unter einem einzigen vorwaerts konstruierten Mechanismus:
+
+```text
+                    L_{1/2} = -d^2 + 1/4
+                            |
+        -------------------------------------------------
+        |                  |                |           |
+ pole / NULLPOL       Prime OU tree      Gamma ladder   threshold
+ Green transfer       AR(1) / Poisson    Poisson        finite part
+        |                  |                |           |
+        ---------------- Euler/Gamma scattering ---------
+                            |
+                       -i S_a^* S_a'
+                            |
+                         NP-GAP
+```
+
+Das ist ein echter struktureller Object-X-Fortschritt. Es ist **noch keine vollständige positive Object-X-Realisierung**, weil ein unitärer Streufaktor keine positive Wigner--Smith-Verzoegerung garantiert.
+
+---
 
-## 8. Neuer einziger Hauptengpass — NP-GAP `?[O]`
+## 12. Einziger globale Hauptengpass — NP-GAP `?[O]`
 
-Auf der global RH-äquivalenten Nullpolklasse wird RH zu der scharfen Lower-Frame-Bedingung
+Zu beweisen bleibt
 
 ```math
 \boxed{
@@ -220,44 +444,57 @@ Auf der global RH-äquivalenten Nullpolklasse wird RH zu der scharfen Lower-Fram
 }
 ```
 
-für jedes `a>0` und jede glatte Nullpol-Testfunktion mit Träger in `[-a,a]`.
+fuer jedes `a>0` und jedes Nullpol-`v` in `[-a,a]`.
 
-Äquivalent:
+Aequivalent nach dem neuen Source-Lift:
 
-```math
-\lambda_{NP}(a)
-:=
-\inf_{\substack{0\ne v\in C_c^\infty(-a,a)\\M(v)(0)=M(v)(1)=0}}
-\frac{\|\mathcal X_av\|^2}{\|v\|_2^2}
-\stackrel{?}{\ge}\Gamma_a.
+```text
+positivity of the canonically centered
+Euler/Gamma harmonic/scattering form
+on Ran(L_{1/2}) for every compact window.
 ```
 
-**Das ist jetzt der Default-Auftrag.** Ein Beweis für alle `a` wäre bereits RH; deshalb darf Weil-Positivität nicht rückwärts als Input verwendet werden.
+### Bevorzugter naechster Gate
 
-## 9. Strategische Statusänderung
+Suche **vorwaerts** eine positive conservative/harmonic-extension dilation, deren Compression/Schur-Komplement den Wigner--Smith-Operator `-iS_a^*S_a'` auf `Ran L_{1/2}` liefert. Erlaubte Daten sind ausschliesslich:
 
 ```text
-NP-R1 separate geometry question                       closed/subsumed ✓[M]
-NP-COMMON common Prime/archimedean feature family      ✓[M]
-NP-SCALAR cutoff-gauge covariance                      ✓[M]
-COMMON-JUMP exact normal form, all a>0                 ✓[M]
+L_{1/2} / Green transfer,
+OU/tree edges and innovations,
+Disk-/Half-plane Poisson extension,
+COMMON-JUMP incidence operators K_t,
+finite-part diagonal renormalization.
+```
+
+Keine rueckwaerts aus fertiger Weil-Positivitaet definierte Dilation ist zulaessig. Ein enger Klassen-No-Go zaehlt ebenfalls als Fortschritt.
+
+---
+
+## 13. Status / Firewalls
+
+```text
+COMMON-JUMP exact all-a normal form                    ✓[M]
+NULLPOL = Range(L_{1/2})                               ✓[M]
+pole amplitudes = critical-half Green transfer         ✓[M]
+P11 hub+rest = logarithmic OU/tree Gram                ✓[M]
+critical-half rigidity sigma=1/2                       ✓[M]
+Gamma positive resolvent ladder                        ✓[M]
+threshold = common diagonal / finite-part mass         ✓[M]
+Prime tower = flat - P11 AR(1) Poisson                 ✓[M]
+Gamma centered modes = half-plane Poisson              ✓[M]
+finite-window Euler/Gamma scattering phase             ✓[M]
 forward Object-X candidate architecture                ✓[M]_part
-sharp NULLPOL frame/spectral gap                       ?[O]
+sharp NP-GAP for all a                                 ?[O]
 full positive Object-X realization / RH                ?[O]
 publication novelty                                    ?[O]
 ```
 
-OX-GEN-A bleibt die exakte Polschicht. POS-DIL #101--#105 bleibt eine mathematisch gültige auxiliary full-class route.
-
-## 10. Firewalls
-
 Nicht behaupten:
 
-- `Q_W>=0` sei bewiesen;
-- der scharfe Lower-Frame-Bound sei bewiesen;
-- ein einzelnes fixes `a` sei bereits RH-äquivalent;
-- Publikationsneuheit sei geklärt;
-- die Theorem-Registry werde durch diese Navigation promoviert;
-- Object X oder RH seien gelöst.
-
-PR #91, PR #49 und R37/G4c bleiben separate Nebenfronten.
\ No newline at end of file
+- `tau_a(z)>=0` punktweise;
+- Unitaritaet impliziere Wigner--Smith-Positivitaet;
+- der Sternbaum allein sei der volle raeumliche P11-Operator;
+- ein globales Eulerprodukt werde auf `Re(s)=1/2` benutzt;
+- fixed-`a`-Positivitaet allein sei RH-aequivalent;
+- NP-GAP, Object X oder RH seien geloest;
+- Publikationsneuheit sei geklaert.
diff --git a/OFFENE_PROBLEME.md b/OFFENE_PROBLEME.md
index b7793a23..b2541779 100644
--- a/OFFENE_PROBLEME.md
+++ b/OFFENE_PROBLEME.md
@@ -1,69 +1,223 @@
-# Offene Probleme — COMMON-JUMP / NP-GAP
+# Offene Probleme — CRITICAL-HALF SCATTERING / NP-GAP
 
 > **Stand:** 13. September 2026.  
-> Operative Quellen: [CURRENT-FRONT](CURRENT-FRONT.md), [Common-Jump-Audit](audits/P11_NP_COMMON_JUMP_GRAM_2026-09-13.md), [Roadmap](00-uebersicht/FORSCHUNGS_ROADMAP_AKTUELL.md), [DAG](00-uebersicht/DAG.md).
+> Operative Quelle: [CURRENT-FRONT](CURRENT-FRONT.md).  
+> Registry und Arbeitsdefinition bleiben unveraendert.
 
 ## Neu geschlossen
 
-### `[NP-R1]` separate Geometriefrage — `✓[M]`
+### `[NP-R1]`, `[NP-COMMON]`, `[NP-SCALAR-GAUGE]` — `✓[M]`
 
-`R_1` ist nicht mehr als separater Generatorblock offen. Zusammen mit dem logarithmischen archimedischen Anteil wird er vom kontinuierlichen positiven Kanal
+Die exakte COMMON-JUMP-Normalform ist
 
 ```math
-K_t=T_{t/2}-T_{-t/2},
+Q_W(v,w)
+=\langle Ev,PEw\rangle
++\langle X_av,X_aw\rangle
+-\Gamma_a\langle v,w\rangle,
+```
+
+mit
+
+```math
+\Gamma_a
+=2\sum_{\log n\le2a}\frac{\Lambda(n)}{\sqrt n}
++\log\pi-\psi(1/4).
+```
+
+Auf NULLPOL:
+
+```math
+\boxed{Q_W|_{NP}=X_a^*X_a-\Gamma_aI.}
+```
+
+### `[CRIT-HALF-RANGE]` — `✓[M]`
+
+```math
+L_{1/2}=-\partial_x^2+1/4,
 \qquad
-h(t)dt=\frac{e^{-t/2}}{1-e^{-2t}}dt
+L_{1/2}^{-1}(x,y)=e^{-|x-y|/2},
+```
+
+und fuer jedes Fenster
+
+```math
+\boxed{
+C_c^\infty(-a,a)\cap\ker M(0)\cap\ker M(1)
+=L_{1/2}C_c^\infty(-a,a).
+}
 ```
 
-erzeugt.
+NULLPOL ist exakt die support-erhaltende Range dieses Operators.
+
+### `[CRIT-HALF-POLE]` — `✓[M]`
 
-### `[NP-COMMON]` — `✓[M]`
+Auf getrennten Traegern faktorisiert derselbe Green-Kern in die beiden Pole-Randladungen:
+
+```math
+\langle f,G_{1/2}g\rangle
+=E_+(f)\overline{E_-(g)}
+```
 
-Archimedischer Ort und Primzahlpotenzen liegen in derselben positiven Hilbert-Geometrie:
+oder in umgekehrter Orientierung
 
 ```math
-\mu_a=h(t)dt+
-\sum_{\log n\le2a}\frac{\Lambda(n)}{\sqrt n}\delta_{\log n}.
+\langle f,G_{1/2}g\rangle
+=E_-(f)\overline{E_+(g)}.
 ```
 
-Die Featureform ist `X_a^*X_a>=0`.
+Die OX-GEN-A-Pole-Amplituden sind somit Außen-Transferdaten desselben Critical-half-Propagators.
 
-### `[NP-SCALAR-GAUGE]` — `✓[M]`
+### `[CRIT-HALF-TREE]` und `[CRIT-HALF-RIGIDITY]` — `✓[M]`
+
+Der P11-AR(1)-Kern
 
 ```math
-\Gamma_a
-=2\sum_{\log n\le2a}\frac{\Lambda(n)}{\sqrt n}
-+\log\pi-\psi(1/4).
+p^{-|j-k|/2}
 ```
 
-Neue Außen-Prime-Atome erhöhen `X_a^*X_a` und `Gamma_a I` exakt gleich. Gaugeinvariant ist die zentrierte Paarung
+ist die logarithmische Abtastung von `e^{-|x-y|/2}` auf einem Prim-Sternbaum. Der gesamte fensterlose Hub+Rest-Kanalindex-Ledger besitzt eine explizite positive OU-/Tree-Gram-Realisierung.
+
+In der Familie `e^{-sigma d}` erzwingt die P11-Amplitude `beta=3/4` exakt
 
 ```math
-X_a^*X_a-\Gamma_aI.
+sigma=1/2,
 ```
 
-### `[COMMON-JUMP-NORMAL-FORM]` — `✓[M]`
+dieselbe Skala wie NULLPOL und der Gamma-Grundmodus.
+
+### `[NP-GAP-LIFT]` — `✓[M]`
 
-Für alle `a>0`:
+Jedes Nullpol-`v` ist eindeutig
 
 ```math
-Q_W(v,w)
-=\langle Ev,PEw\rangle
-+\langle X_av,X_aw\rangle
--\Gamma_a\langle v,w\rangle.
+v=L_{1/2}u,
+\qquad
+u\in C_c^\infty(-a,a).
 ```
 
-Auf Nullpol:
+Damit ist NP-GAP eine momentfreie Coercivity-Frage.
+
+### `[GAMMA-RESOLVENT]` — `✓[M]`
 
 ```math
-\boxed{Q_W|_{NP}=X_a^*X_a-\Gamma_aI.}
+\Phi_\infty(D)
+=\sum_{m\ge0}\frac2{\mu_m}D^2(D^2+\mu_m^2)^{-1},
+\qquad
+\mu_m=2m+1/2.
+```
+
+Der Grundmodus ist exakt `L_{1/2}`. Ein direkter Schur-Test liefert als Mechanismus-Zertifikat NP-GAP fuer `0<a<=1/16`; kein Neuheits-/Radiusrekord wird beansprucht.
+
+### `[NP-SCALAR-ORIGIN]` — `✓[M]`
+
+Die fruehere Frage nach dem „Restskalar“ ist geschlossen. Prime-Atome tragen exakt ihre endliche Diagonalmasse `2w_n`. Archimedisch gilt
+
+```math
+\boxed{
+\kappa_*
+=
+\lim_{N\to\infty}
+\left[
+\sum_{m=0}^N\frac2{\mu_m}
+-
+\log\frac{N+1}{\pi}
+\right].
+}
+```
+
+Damit ist `Gamma_a` genau
+
+```text
+finite Prime diagonal mass
++
+renormalized Gamma diagonal mass.
+```
+
+Es wird kein weiterer separater Skalar-Generator gesucht.
+
+### `[PRIME-AR1-POISSON]` — `✓[M]`
+
+Fuer `q_p=p^{-1/2}`:
+
+```math
+P_q^{\mathbb D}(\theta)
+=1+2\sum_{k\ge1}q^k\cos(k\theta).
+```
+
+Der komplette zentrierte `p`-Power-Weil-Turm ist exakt
+
+```math
+\boxed{
+(\log p)
+\left[1-P_{p^{-1/2}}^{\mathbb D}(z\log p)\right].
+}
+```
+
+`P_q^D` ist exakt das Toeplitz-Symbol der P11-AR(1)-Kovarianz `q^{|j-k|}`.
+
+### `[GAMMA-POISSON]` — `✓[M]`
+
+Mit
+
+```math
+P_y^{\mathbb H}(x)=\frac1\pi\frac{y}{x^2+y^2}
 ```
 
+gilt modeweise
+
+```math
+\boxed{
+A_m(z)-2/\mu_m=-2\pi P_{\mu_m}^{\mathbb H}(z).
+}
+```
+
+Prime und Archimedes liegen somit auch in derselben harmonischen Kernelklasse: Disk-/Half-plane-Poisson.
+
+### `[EULER-GAMMA-SCATTERING]` — `✓[M]`
+
+Die expliziten unitären Faktoren
+
+```math
+S_p(z)=
+\frac{1-p^{-1/2}e^{iz\log p}}
+     {1-p^{-1/2}e^{-iz\log p}},
+```
+
+```math
+S_\infty(z)=
+\pi^{-iz}
+\frac{\Gamma(1/4+iz/2)}
+     {\Gamma(1/4-iz/2)}
+```
+
+haben genau die zentrierten Prime-/Archimedes-Multiplikatoren als Phasenableitungen.
+
+Fuer
+
+```math
+S_a(z)=S_\infty(z)\prod_{p\le e^{2a}}S_p(z)
+```
+
+ist in der Fenster-Quadratikform
+
+```math
+\boxed{
+Q_W(v)
+=\int|\widehat v(z)|^2
+\left[-i\overline{S_a(z)}S_a'(z)\right]dz
+}
+```
+
+auf NULLPOL.
+
+Die in `S_p` enthaltenen hohen Prime powers jenseits des geometrischen Cutoffs sind auf dem Fenster-Testfunktionsraum exakt null in der zentrierten Form.
+
 ---
 
-## Priorität 0 — `[NP-GAP]` `?[O]`
+# Prioritaet 0 — `[NP-GAP]` `?[O]`
 
-Der neue einzige harte Hauptengpass ist
+Global bleibt
 
 ```math
 \boxed{
@@ -71,82 +225,100 @@ Der neue einzige harte Hauptengpass ist
 :=
 \inf_{\substack{0\ne v\in C_c^\infty(-a,a)\\M(v)(0)=M(v)(1)=0}}
 \frac{\|X_av\|^2}{\|v\|_2^2}
-\stackrel{?}{\ge}\Gamma_a.
+\stackrel?\ge\Gamma_a
 }
 ```
 
-Für die Familie aller Fenster ist dies die verbleibende RH-äquivalente Lower-Frame-/Spektralgap-Frage.
+fuer alle `a>0`.
 
-### `[NP-GAP-A]` support-erhaltende Momentfaktorisierung
+Fuer die Familie aller Fenster ist dies die verbleibende RH-aequivalente Lower-Frame-/Spektralgap-Frage.
 
-Nutze die Nullpolbedingungen als
+## `[NP-GAP-A]` Source-lift coercivity — Default analytischer Angriff
+
+Nutze
 
 ```math
-\widehat v(i/2)=\widehat v(-i/2)=0
+v=L_{1/2}u
 ```
 
-und untersuche die Faktorisierung
+vollstaendig aus:
+
+1. erste Gamma-Moden lokal/coerciv in der `L_{1/2}`-Metrik behandeln;
+2. Gamma-Rest per Resolventen-/Schurtechnik kontrollieren;
+3. Prime-Jumps `L_{1/2}K_{log n}u` nicht getrennt wegwerfen, sondern mit Tree-/OU-Innovationen koppeln;
+4. nur radiusstabile Mechanismen verfolgen, nicht Radiuskosmetik.
+
+## `[NP-SCATTER]` positive conservative scattering dilation — neuer bevorzugter Struktur-Gate
+
+Gesucht ist eine **vorwaerts definierte** positive conservative/harmonic-extension dilation, deren Compression oder Schur-Komplement auf `Ran L_{1/2}` den Fenster-Wigner--Smith-Operator
 
 ```math
-v=Q_0u,
-\qquad
-Q_0=-d^2/dx^2+1/4,
+-iS_a^*S_a'
 ```
 
-mit gleichem kompaktem Trägerscope.
+liefert.
+
+Zulaessige Ausgangsdaten:
+
+```text
+critical-half Green propagation L_{1/2}^{-1},
+Tree-root / OU innovations,
+Disk- und Half-plane-Poisson extension,
+COMMON-JUMP incidence operators K_t,
+canonical finite-part diagonal renormalization.
+```
 
-Ziel: echte coercivity von `X_a^*X_a` auf `Ran Q_0`, oder ein enger No-Go gegen diese Methode.
+Nicht zulaessig ist eine Dilation, die aus fertiger Weil-Positivitaet oder aus dem gewuenschten Bound rueckwaerts definiert wird.
 
-### `[NP-GAP-B]` Fourier-/Paley-Wiener
+## `[NP-SCATTER-NOGO]` naive positivity classes eliminieren
 
-Arbeite mit
+Vor einem globalen Konstruktionsversuch sollen enge No-Go-Klassen geprueft werden:
 
-```math
-\Phi_a(z)=
-\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\psi(1/4)
-+2\sum_{\log n\le2a}\frac{\Lambda(n)}{\sqrt n}(1-\cos(z\log n)).
-```
+- punktweise Positivitaet der Phasenableitung;
+- Unitaritaet `=>` positive time delay;
+- endliche Gamma-Moden ohne Prime-/Tree-Kopplung;
+- rein kanalweise Poincare-Schranken;
+- Dilationen, die die Paley-Wiener-/Fensterstruktur ignorieren.
 
-Die Schwierigkeit ist nicht `Phi_a>=0`, sondern die scharfe Untergrenze auf Funktionen mit den komplexen Nullbedingungen `z=±i/2`.
+Ein theorem-level No-Go gegen eine solche Klasse ist echter Fortschritt.
 
-### `[NP-GAP-C]` nonlocal Poincare / frame mechanism
+## `[NP-HARDY/DB]` Hardy-/Blaschke-/de-Branges-Route nur vorwaerts
 
-Teste nur vorab definierte, nichtzirkuläre Mechanismen:
+Die neuen Disk-/Half-plane-Poisson- und Blaschke-Faktoren machen Hardy-/de-Branges-Methoden natuerlich. Offen ist, ob eine kanonische positive Raumstruktur **aus den vorhandenen Faktoren** konstruiert werden kann.
 
-- nonlocal Poincare-/Dirichlet-form-Ungleichungen;
-- Paley-Wiener-/sampling-Struktur;
-- de-Branges-/reproduzierende-Kern-Methoden;
-- Schur-/Shorting-Mechanismen, sofern sie den Bound nicht aus fertiger Weil-Positivität zurückdefinieren.
+Firewall: keine bekannte RH-aequivalente de-Branges-Bedingung einfach umbenennen und als Fortschritt ausgeben.
 
 ---
 
-## Danach
+## Kanonische Architektur nach diesem Durchlauf
 
 ```text
-NP-GAP
-  |
-  v
-positive Weil form on global null-pole class
-  |
-  v
-RH
+critical-half L_{1/2}
+   |
+   +-- Pole/NULLPOL = Green boundary charges / zero exterior charge
+   +-- P11 = logarithmic OU tree / AR(1)
+   +-- Prime towers = flat - disk Poisson
+   +-- Gamma ladder = resolvents / half-plane Poisson
+   +-- Gamma_a = finite + finite-part diagonal mass
+   +-- finite-window Euler/Gamma unitary scattering S_a
+                         |
+                         v
+                     -i S_a^* S_a'
+                         |
+                         v
+                       NP-GAP
 ```
 
-Ein Beweis von NP-GAP für alle `a` wäre daher bereits RH. Hier ist besondere Zirkularitätskontrolle Pflicht.
-
-## Auxiliary / separate
-
-- OX-GEN-A: exakte Pole-layer geometry;
-- POS-DIL #101--#105: auxiliary full-class route;
-- Prime-Power AR(1): eigenständige positive Struktur;
-- R37/G4c, PR #91, PR #49 separat.
+---
 
 ## Firewalls
 
 Nicht behaupten:
 
-- COMMON-JUMP beweise den Lower-Frame-Bound;
-- die exakte Differenz `X_a^*X_a-Gamma_aI` sei bereits eine positive Gramdarstellung der Weilform;
-- fixed-`a`-Positivität allein sei RH-äquivalent;
-- Publikationsneuheit sei geklärt;
-- Object X oder RH seien bewiesen.
\ No newline at end of file
+- `-iS_a^*S_a'` sei punktweise positiv;
+- Unitaritaet impliziere Wigner--Smith-Positivitaet;
+- der abstrakte Prime-Sternbaum ersetze die raeumliche P11-Fenstergeometrie;
+- ein globales klassisches Eulerprodukt werde auf `Re(s)=1/2` gebildet;
+- fixed-`a`-Positivitaet allein sei RH-aequivalent;
+- Publikationsneuheit sei geklaert;
+- NP-GAP, full Object X oder RH seien bewiesen.
diff --git a/audits/P11_AR1_TO_GAMMA_GRAPH_NORM_RATE_2026-09-15.md b/audits/P11_AR1_TO_GAMMA_GRAPH_NORM_RATE_2026-09-15.md
new file mode 100644
index 00000000..24fd2e6a
--- /dev/null
+++ b/audits/P11_AR1_TO_GAMMA_GRAPH_NORM_RATE_2026-09-15.md
@@ -0,0 +1,119 @@
+# P11 Audit — Graph-norm rate for the AR(1)-to-Gamma cell limit
+
+**Datum:** 15. September 2026
+
+## Result
+
+For `mu>0`,
+
+```math
+A_{h,\mu}
+=h\sum_{k\ge1}e^{-\mu kh}K_{kh}^*K_{kh},
+```
+
+and
+
+```math
+A_{0,\mu}
+=\int_0^\infty e^{-\mu t}K_t^*K_tdt
+=\frac2\mu D^2(D^2+\mu^2)^{-1}.
+```
+
+Although
+
+```math
+\|A_{h,\mu}-A_{0,\mu}\|=2/\mu
+```
+
+for every `h>0`, one has the quantitative graph-norm estimate
+
+```math
+\boxed{
+\|(A_{h,\mu}-A_{0,\mu})(1+D^2)^{-1/2}\|
+\le\left(4+\frac2\mu\right)h.
+}
+```
+
+In particular, for the Critical-half ground mode `mu=1/2`,
+
+```math
+\boxed{
+\|(A_{h,1/2}-A_{0,1/2})(1+D^2)^{-1/2}\|
+\le8h.
+}
+```
+
+Thus the failure of bare `L^2` norm convergence is purely a high-frequency aliasing obstruction; after paying one derivative, the discrete-to-archimedean cell transition is operator-norm continuous with linear rate.
+
+**Status:** `✓[M]`.
+
+---
+
+## Proof
+
+At Fourier frequency `z`, define
+
+```math
+f_z(t)=2e^{-\mu t}(1-\cos zt).
+```
+
+Then the multipliers are
+
+```math
+a_{h,\mu}(z)=h\sum_{k\ge1}f_z(kh),
+\qquad
+a_{0,\mu}(z)=\int_0^\infty f_z(t)dt.
+```
+
+For an absolutely continuous function with `f_z(0)=0`, the right-Riemann-sum error satisfies
+
+```math
+\left|h\sum_{k\ge1}f_z(kh)-\int_0^\infty f_z(t)dt\right|
+\le h\int_0^\infty|f_z'(t)|dt.
+```
+
+Indeed, on each interval `((k-1)h,kh]`, integrate
+
+```math
+|f_z(kh)-f_z(t)|
+\le\int_t^{kh}|f_z'(u)|du
+```
+
+and sum.
+
+Now
+
+```math
+f_z'(t)
+=-2\mu e^{-\mu t}(1-\cos zt)
++2z e^{-\mu t}\sin zt,
+```
+
+so
+
+```math
+\int_0^\infty|f_z'(t)|dt
+\le4+\frac{2|z|}{\mu}.
+```
+
+Hence
+
+```math
+\frac{|a_{h,\mu}(z)-a_{0,\mu}(z)|}
+{\sqrt{1+z^2}}
+\le h\left(
+\frac4{\sqrt{1+z^2}}
++\frac{2|z|}{\mu\sqrt{1+z^2}}
+\right)
+\le h\left(4+\frac2\mu\right).
+```
+
+Taking the essential supremum proves the claim.
+
+---
+
+## Interpretation
+
+The strong-but-not-norm cell limit and the earlier Strong-Terminal behavior have the same geometry: raw `L^2` permits arbitrarily high-frequency vectors tuned to the lattice aliasing, whereas every fixed smooth source pays a derivative and sees a stable limit.
+
+This estimate does not prove any Weil positivity. It only hardens the finite-place/archimedean gluing topology and shows that the archimedean boundary is quantitatively accessible in natural graph norms.
diff --git a/audits/P11_ARCHIMEDEAN_AS_STRONG_H0_CELL_LIMIT_2026-09-15.md b/audits/P11_ARCHIMEDEAN_AS_STRONG_H0_CELL_LIMIT_2026-09-15.md
new file mode 100644
index 00000000..580d20e5
--- /dev/null
+++ b/audits/P11_ARCHIMEDEAN_AS_STRONG_H0_CELL_LIMIT_2026-09-15.md
@@ -0,0 +1,198 @@
+# P11 Audit — Archimedean ground as the strong h=0 cell limit
+
+**Datum:** 15. September 2026
+
+## 0. Result
+
+For `h>0`, put `q=e^{-h/2}` and
+
+```math
+A_h:=B_h^*B_h
+=\sum_{k\ge1}h e^{-kh/2}K_{kh}^*K_{kh}.
+```
+
+Then
+
+```math
+\boxed{
+A_h\xrightarrow[h\downarrow0]{\rm strong}
+A_0:=\int_0^\infty e^{-t/2}K_t^*K_tdt
+=4D^2(D^2+1/4)^{-1}.
+}
+```
+
+Thus `A_0` is exactly the Gamma ground mode.
+
+However the convergence is maximally non-uniform:
+
+```math
+\boxed{\|A_h-A_0\|=4\quad\text{for every }h>0.}
+```
+
+So the archimedean ground is a genuine strong boundary point of the same Euler/AR(1) cell family, but never an operator-norm limit.
+
+**Status:** `✓[M]`.
+
+---
+
+## 1. Multipliers
+
+Let `z` denote Fourier frequency. Since
+
+```math
+A_h=\sum_{k\ge1}h q^kK_{kh}^*K_{kh},
+```
+
+its multiplier is
+
+```math
+\boxed{
+a_h(z)
+=h\left[
+\frac{1+q}{1-q}-P_q(zh)
+\right],
+}
+```
+
+where
+
+```math
+P_q(\theta)=\frac{1-q^2}{1-2q\cos\theta+q^2}
+```
+
+is the disk Poisson kernel.
+
+For fixed `z`, Taylor expansion at `h=0` gives
+
+```math
+P_{e^{-h/2}}(zh)
+=\frac1{h(z^2+1/4)}+O(1),
+```
+
+and
+
+```math
+h\frac{1+q}{1-q}=4+O(h^2).
+```
+
+Hence
+
+```math
+\boxed{
+a_h(z)\to
+4-\frac1{z^2+1/4}
+=\frac{4z^2}{z^2+1/4}
+=:a_0(z).}
+```
+
+This is exactly the multiplier of the `mu_0=1/2` Gamma resolvent mode.
+
+---
+
+## 2. Uniform domination and strong convergence
+
+The Poisson kernel satisfies
+
+```math
+\frac{1-q}{1+q}\le P_q(\theta)\le\frac{1+q}{1-q}.
+```
+
+Therefore
+
+```math
+0\le a_h(z)
+\le h\left[
+\frac{1+q}{1-q}-\frac{1-q}{1+q}
+\right]
+=\frac{4hq}{1-q^2}
+=\frac{2h}{\sinh(h/2)}<4.
+```
+
+Also `0<=a_0(z)<4`. Pointwise convergence plus bounded dominated convergence yields
+
+```math
+\|(A_h-A_0)v\|_2\to0
+```
+
+for every `v in L^2(R)`.
+
+---
+
+## 3. Exact norm failure by aliasing
+
+At every alias frequency
+
+```math
+z_n=\frac{2\pi n}{h},
+```
+
+one has `P_q(z_nh)=(1+q)/(1-q)`, hence
+
+```math
+a_h(z_n)=0.
+```
+
+But
+
+```math
+a_0(z_n)\to4
+\qquad(n\to\infty).
+```
+
+Thus
+
+```math
+\|A_h-A_0\|\ge4.
+```
+
+The uniform bounds `0<=a_h,a_0<=4` give the reverse inequality, so
+
+```math
+\boxed{\|A_h-A_0\|=4.}
+```
+
+This is an exact sampling-aliasing obstruction, not a numerical effect.
+
+---
+
+## 4. Centered cells and the archimedean place
+
+The centered cell uses
+
+```math
+c(h)=\frac{h e^{-h/2}}{1-e^{-h/2}}\to2,
+```
+
+so for every fixed source
+
+```math
+C_v(h)=\langle v,A_hv\rangle-2c(h)\|v\|^2
+```
+
+has the strong boundary value
+
+```math
+\boxed{
+C_v(0)
+=\langle v,A_0v\rangle-4\|v\|^2.
+}
+```
+
+On NULLPOL the complete Weil form can therefore be written
+
+```math
+\boxed{
+Q_W(v)
+=C_v(0)+\sum_pC_v(\log p)
++E_{Gamma,>=1}(v)
+-(\kappa_*-4)\|v\|^2.
+}
+```
+
+The scale set `h=0` together with `h=log p` thus treats the Gamma ground and finite prime places as one strong cell family.
+
+---
+
+## 5. Interpretation firewall
+
+This is not an adelic proof and not an Object-X completion. The lack of norm continuity is essential: high-frequency aliasing prevents any uniform cell-limit argument. Any global construction must therefore use a strong/source-local topology or an additional smoothing/storage mechanism.
diff --git a/audits/P11_BOUNDARY_TAIL_SHORTING_AND_UNIVERSAL_SAMPLING_2026-09-15.md b/audits/P11_BOUNDARY_TAIL_SHORTING_AND_UNIVERSAL_SAMPLING_2026-09-15.md
new file mode 100644
index 00000000..b4a33181
--- /dev/null
+++ b/audits/P11_BOUNDARY_TAIL_SHORTING_AND_UNIVERSAL_SAMPLING_2026-09-15.md
@@ -0,0 +1,447 @@
+# P11 Audit — Boundary-tail shorting and universal prime-lattice sampling
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Randzustandsidentitaet fuer die gestoppte OU/P11-Praezision.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein Object-X-/Weil-/RH-Abschluss; keine Publikationsneuheit.
+
+---
+
+## 0. Kurzurteil
+
+Die endliche Praezisionsform eines gestoppten P11-AR(1)-Astes rekonstruiert **alle inneren nackten Weil-Jump-Kanaele exakt** und komprimiert den gesamten unsichtbaren Aussentail in einen einzigen kontraktiven exponentiellen Randzustand.
+
+Diskret, fuer `q in (0,1)` und
+
+```math
+Y_m=\sqrt{1-q^2}\sum_{k\ge m}q^{k-m}X_k,
+```
+
+gilt fuer die sichtbaren Zustaende `Y_1,...,Y_m`
+
+```math
+\boxed{
+\mathfrak p_m(Y)
+=
+\sum_{r=1}^{m-1}
+\frac{\|Y_r-qY_{r+1}\|^2}{1-q^2}
++\|Y_m\|^2
+=
+\sum_{r<m}\|X_r\|^2+\|Y_m\|^2.
+}
+```
+
+Dabei
+
+```math
+\boxed{
+0\le
+\sum_{k\ge m}\|X_k\|^2-\|Y_m\|^2.
+}
+```
+
+Kontinuierlich, fuer
+
+```math
+Y(u)=\int_u^\infty e^{-(t-u)/2}X(t)dt,
+```
+
+gilt auf `[0,L]`
+
+```math
+\boxed{
+\mathfrak p_L(Y)
+=
+\int_0^L\left\|Y'(u)-\frac12Y(u)\right\|^2du
++\|Y(L)\|^2
+=
+\int_0^L\|X(u)\|^2du+\|Y(L)\|^2,
+}
+```
+
+und
+
+```math
+\boxed{
+0\le
+\int_L^\infty\|X(t)\|^2dt-\|Y(L)\|^2.
+}
+```
+
+Der diskrete Randvektor und der kontinuierliche Randvektor sind beide exakt normiert.
+
+Mit dem universellen Feld
+
+```math
+\boxed{X_v(t)=e^{-t/4}K_tv}
+```
+
+ist jeder Prime-Power-Weilkanal nur eine Gitterabtastung:
+
+```math
+\boxed{
+\sqrt{w_{p,k}}K_{k\log p}v
+=\sqrt{\log p}\,X_v(k\log p).
+}
+```
+
+Somit sind die P11-Primaste verschiedene logarithmische Gitter desselben kontinuierlichen Critical-half-Jump-Feldes.
+
+Status:
+
+```text
+discrete stopped precision/boundary identity            ✓[M]
+discrete exterior-tail defect positive                  ✓[M]
+continuous stopped precision/boundary identity          ✓[M]
+continuous exterior-tail defect positive                ✓[M]
+normalized discrete/continuous boundary modes           ✓[M]
+all prime branches sample one universal X_v(t)           ✓[M]
+prime discrepancy as weighted lattice-count discrepancy ✓[M]
+Object-X boundary coupling                               ?[O]
+```
+
+---
+
+# 1. Discrete tail transform and its inverse orientation
+
+For a Hilbert-valued square-summable sequence `X=(X_k)_{k>=1}`, define
+
+```math
+\boxed{
+Y_m
+=(T_qX)_m
+=\sqrt{1-q^2}\sum_{k\ge m}q^{k-m}X_k.
+}
+```
+
+Then
+
+```math
+Y_m-qY_{m+1}
+=\sqrt{1-q^2}X_m,
+```
+
+so the correct inverse orientation is
+
+```math
+\boxed{
+X_m
+=\frac{Y_m-qY_{m+1}}{\sqrt{1-q^2}}.
+}
+```
+
+This orientation matters at a finite stopping boundary.
+
+---
+
+# 2. Natural finite precision and the boundary state
+
+The covariance of the visible vector `(Y_1,...,Y_m)` is the finite Kac--Murdock--Szego matrix
+
+```math
+R_q^{[m]}=(q^{|j-k|})_{1<=j,k<=m}.
+```
+
+Its inverse quadratic form is
+
+```math
+\boxed{
+\mathfrak p_m(Y)
+=
+\sum_{r=1}^{m-1}
+\frac{\|Y_r-qY_{r+1}\|^2}{1-q^2}
++\|Y_m\|^2.
+}
+```
+
+Indeed, expansion gives the standard tridiagonal inverse with endpoint diagonal `1/(1-q^2)` and interior diagonal `(1+q^2)/(1-q^2)`.
+
+Using the inverse tail relation,
+
+```math
+\boxed{
+\mathfrak p_m(Y)
+=
+\sum_{r=1}^{m-1}\|X_r\|^2+\|Y_m\|^2.
+}
+```
+
+Thus every visible interior bare channel is recovered exactly. Only the unobserved tail `X_m,X_{m+1},...` is replaced by one boundary state `Y_m`.
+
+---
+
+# 3. Boundary compression is contractive
+
+The coefficient vector of the boundary tail is
+
+```math
+b_m(k)
+=\sqrt{1-q^2}\,q^{k-m}\mathbf1_{k>=m}.
+```
+
+It has exact norm
+
+```math
+\sum_{k>=m}|b_m(k)|^2
+=(1-q^2)\sum_{r>=0}q^{2r}=1.
+```
+
+Therefore Cauchy--Schwarz gives
+
+```math
+\boxed{
+\|Y_m\|^2
+\le
+\sum_{k>=m}\|X_k\|^2.
+}
+```
+
+Equivalently the missing exterior energy
+
+```math
+\boxed{
+\mathcal D_m(X)
+:=
+\sum_{k>=m}\|X_k\|^2-\|Y_m\|^2
+\ge0
+}
+```
+
+is exactly the norm of the component of the tail orthogonal to the normalized exponential boundary mode `b_m`.
+
+This is the finite-dimensional/marginal Schur-shortening mechanism underlying the stopped AR(1) covariance.
+
+---
+
+# 4. Continuous tail transform
+
+Let `X in L^2((0,infty);H)` and define
+
+```math
+\boxed{
+Y(u)=\int_u^\infty e^{-(t-u)/2}X(t)dt.
+}
+```
+
+Then in the weak sense
+
+```math
+Y'(u)
+=-X(u)+\frac12Y(u),
+```
+
+hence
+
+```math
+\boxed{
+X(u)=\left(\frac12-\partial_u\right)Y(u).
+}
+```
+
+The covariance kernel of the transform of white input is
+
+```math
+\int_{\max(s,t)}^\infty
+ e^{-(u-s)/2}e^{-(u-t)/2}du
+=e^{-|s-t|/2}.
+```
+
+Thus this is the same full Critical-half OU kernel.
+
+---
+
+# 5. Continuous stopped precision and boundary shorting
+
+The inverse-covariance form of `e^{-|s-t|/2}` restricted to `[0,L]` is
+
+```math
+\int_0^L
+\left(\|Y'\|^2+\frac14\|Y\|^2\right)du
++\frac12\|Y(0)\|^2
++\frac12\|Y(L)\|^2.
+```
+
+Integration by parts rewrites it as
+
+```math
+\boxed{
+\mathfrak p_L(Y)
+=
+\int_0^L
+\left\|Y'(u)-\frac12Y(u)\right\|^2du
++\|Y(L)\|^2.
+}
+```
+
+Using `X=(1/2-partial)Y`,
+
+```math
+\boxed{
+\mathfrak p_L(Y)
+=
+\int_0^L\|X(u)\|^2du+\|Y(L)\|^2.
+}
+```
+
+The continuous boundary mode is
+
+```math
+g_L(t)=e^{-(t-L)/2}\mathbf1_{t>=L}
+```
+
+with
+
+```math
+\|g_L\|_{L^2(dt)}^2=\int_L^\infty e^{-(t-L)}dt=1.
+```
+
+Hence
+
+```math
+\boxed{
+\|Y(L)\|^2
+\le
+\int_L^\infty\|X(t)\|^2dt
+}
+```
+
+and the exterior defect is the positive orthogonal-complement energy.
+
+---
+
+# 6. Exact discrete/continuous boundary correspondence
+
+For a lattice step `h>0`, put
+
+```math
+q=e^{-h/2}.
+```
+
+The discrete boundary coefficients are
+
+```math
+\sqrt{1-e^{-h}}\,e^{-(k-m)h/2}.
+```
+
+Their `ell^2` norm is exactly one. The continuous boundary coefficient is
+
+```math
+e^{-(t-L)/2}\mathbf1_{t>=L},
+```
+
+whose `L^2` norm is exactly one.
+
+Thus the discrete P11 boundary state and the continuous Critical-half boundary state are normalized versions of the same exponential tail mode in their respective counting/Lebesgue geometries.
+
+---
+
+# 7. Universal Weil-Jump field sampled by every prime
+
+For `p` set
+
+```math
+h_p=\log p,
+\qquad
+q_p=e^{-h_p/2}.
+```
+
+The Weil weight is
+
+```math
+w_{p,k}
+=(\log p)p^{-k/2}
+=h_p e^{-kh_p/2}.
+```
+
+Define
+
+```math
+\boxed{
+X_v(t)=e^{-t/4}K_tv.
+}
+```
+
+Then
+
+```math
+\begin{aligned}
+\sqrt{w_{p,k}}K_{kh_p}v
+&=\sqrt{h_p}\,e^{-kh_p/4}K_{kh_p}v\\
+&=\boxed{\sqrt{h_p}\,X_v(kh_p)}.
+\end{aligned}
+```
+
+Hence all prime branches sample one and the same continuous channel field; only their logarithmic lattice spacing differs.
+
+Applying the P11 tail transform gives
+
+```math
+\boxed{
+Y_{p,m}
+=
+\sqrt{1-e^{-h_p}}
+\sum_{k>=m}
+ e^{-(k-m)h_p/2}\sqrt{h_p}\,X_v(kh_p).
+}
+```
+
+This is the exact normalized exponential tail quadrature used by the P11 branch.
+
+---
+
+# 8. Prime measure as a weighted logarithmic lattice comb
+
+Define the logarithmic prime-power counting measure
+
+```math
+\boxed{
+d\mathcal N_P(t)
+=
+\sum_{p}\sum_{k>=1}(\log p)\,\delta_{k\log p}(dt).
+}
+```
+
+Then
+
+```math
+\boxed{
+d\nu_P(t)=e^{-t/2}d\mathcal N_P(t).}
+```
+
+Since the pole compensator is `e^{t/2}dt`, the discrepancy is
+
+```math
+\boxed{
+d\Delta(t)
+=e^{-t/2}\left(d\mathcal N_P(t)-e^t dt\right).}
+```
+
+Thus `Delta` is exactly the weighted discrepancy between the union of logarithmic prime lattices and its smooth exponential main density.
+
+This formulation does not make the discrepancy easy: polynomial control is already RH-scale, as recorded in the preceding audit. Its value is geometric: the same depth coordinate `t` carries
+
+```text
+- the universal jump field X_v(t),
+- every P11 prime lattice,
+- the stopped OU boundary mode,
+- and the pole-subtracted prime-lattice counting discrepancy.
+```
+
+---
+
+# 9. Next gate
+
+The remaining nontrivial construction problem can now be stated without P11-specific index notation:
+
+```math
+\boxed{
+\text{Given one continuous field }X_v(t)\text{ on the overlap cone,}
+\atop
+\text{couple its prime-lattice stopped boundary quadratures and the continuous}
+\text{Critical-half boundary state in a forward positive system whose}
+\text{nonlocal boundary transfer reproduces the }\Delta\text{ pairing.}
+}
+```
+
+The signed measure itself cannot be a local positive energy density. Any success must occur through a boundary/transfer/spectral-shift mechanism and must pass the exact Prime-2 mixed-pair witness.
diff --git a/audits/P11_BULK_TRACE_GATE_CURRENT_2026-09-15.md b/audits/P11_BULK_TRACE_GATE_CURRENT_2026-09-15.md
new file mode 100644
index 00000000..9849039b
--- /dev/null
+++ b/audits/P11_BULK_TRACE_GATE_CURRENT_2026-09-15.md
@@ -0,0 +1,167 @@
+# P11 / Object X — Bulk–Trace gate after Euler-cell hard audit
+
+**Datum:** 15. September 2026  
+**Registry:** unveraendert.  
+**Nonclaim:** kein RH-/Object-X-Beweis, keine Publikationsneuheit.
+
+## 0. Current front
+
+The surviving local geometry is now fixed.
+
+Define
+
+```math
+X_v(t)=e^{-t/4}K_tv,
+\qquad
+\rho(h)=R'(e^h)e^h
+=\sum_{j\ge1}\frac{h^{j-1}}{j!\zeta(j+1)}.
+```
+
+For `m>=0`, `k>=1`, define the bulk field
+
+```math
+\boxed{
+(\mathcal G_m v)(h,k)
+=e^{-mkh}\sqrt{\frac{h}{e^{kh}-1}}X_v(kh).
+}
+```
+
+Then the Gamma mode `mu_m=2m+1/2` is exactly
+
+```math
+\boxed{
+\|\mathcal G_m v\|^2_{L^2(\rho(h)dh;\ell^2_k;L^2_x)}
+=\int_0^\infty e^{-\mu_m t}\|K_tv\|^2dt.
+}
+```
+
+At a prime scale `h=log p`, the ground bulk satisfies
+
+```math
+\boxed{
+(p^k-1)\|\mathcal G_0v(\log p,k)\|^2
+=\frac{\log p}{p^{k/2}}\|K_{k\log p}v\|^2.
+}
+```
+
+Hence:
+
+```text
+Gamma positive layer = continuum bulk norm;
+Prime positive layer = weighted atomic trace of the same ground bulk field.
+```
+
+The P11 multiplicity `p^k-1` is exactly the trace multiplicity.
+
+This is the current exact common geometry.
+
+---
+
+## 1. Equivalent OU interpretation
+
+For each prime, the P11 Hub+Rest form is the covariance/state norm of the AR(1) tail state, while the positive p-power Weil tower is the inverse-covariance/precision norm of that same state.
+
+At `h->0`, the discrete precision cells converge strongly to the archimedean Gamma resolvent modes. For mode `mu`,
+
+```math
+A_{h,\mu}=h\sum_{k\ge1}e^{-\mu kh}K_{kh}^*K_{kh}
+\to
+\frac2\mu D^2(D^2+\mu^2)^{-1}.
+```
+
+The convergence is never `L^2` operator-norm convergence because of aliasing, but after one source derivative it has the explicit graph-norm rate
+
+```math
+\|(A_{h,\mu}-A_{0,\mu})(1+D^2)^{-1/2}\|
+\le(4+2/\mu)h.
+```
+
+Thus the correct gluing topology is strong/source-local, with quantitative graph-norm stability on smooth sources.
+
+---
+
+## 2. What the Riemann-R continuum supplies
+
+The cell base density is the derivative of the classical Riemann R function. It satisfies
+
+```math
+\int_0^\infty\rho(h)\sum_{k\ge1}h\delta_{kh}(dt)dh
+=(e^t-1)dt.
+```
+
+Disintegrating the left side over `t=kh` gives the probability weights
+
+```math
+\boxed{
+\lambda_k(t)
+=\frac{t\rho(t/k)}{k^2(e^t-1)},
+\qquad \sum_{k\ge1}\lambda_k(t)=1.
+}
+```
+
+Thus the smooth archimedean background has a canonical exponent-fibre decomposition; a genuine prime power `p^r` selects one specific exponent coordinate `r`.
+
+However ordinary orthogonal shorting against this continuum is not the answer:
+- collapsing `(h,k)` to depth `t` makes the continuum total and kills the arithmetic;
+- pointwise exponent-fibre shorting reduces the exact Prime-2 mixed calibration and therefore fails if the Gamma baseline is left unchanged.
+
+---
+
+## 3. Exact local cell and archimedean boundary
+
+For every `h>0`,
+
+```math
+B_h
+=\sqrt{\frac{h q(1+q)}{1-q}}
+(T_h-I)(I-qT_h)^{-1},
+\qquad q=e^{-h/2},
+```
+
+compresses the complete logarithmic tower:
+
+```math
+B_h^*B_h
+=\sum_{k\ge1}h e^{-kh/2}K_{kh}^*K_{kh}.
+```
+
+At `h=log p` this is the full p-power positive Weil tower. At `h=0` its strong boundary is the Gamma ground operator
+
+```math
+4D^2(D^2+1/4)^{-1}.
+```
+
+The actual prime scales do not approach zero (`log p>=log2`); the `h->0` statement is a boundary theorem for the continuous generalized cell family, not a limit of actual finite places.
+
+---
+
+## 4. Closed natural couplings
+
+The following coefficient-free classes have now failed:
+
+```text
+root-only Prime/continuum coupling                     FAIL
+ordinary common-depth OU Gram shorting                 FAIL (continuum totality)
+pointwise exponent-fibre shorting with fixed Gamma     FAIL (Prime-2 calibration)
+orthogonal sum of local passive cells                  insufficient
+ordinary passive cascade of normalized Euler cells    FAIL (limit collapses to zero)
+ordinary positive Sobolev regularity in scale h        not supplied by current energy;
+                                                      high-frequency cell oscillation is too expensive
+```
+
+The last line is an architectural warning rather than a closed theorem class: any future scale-space norm must be derived explicitly and tested against high-frequency modulation before it is accepted.
+
+---
+
+## 5. Single remaining gate
+
+The missing object is now a **relative scale-storage structure** which:
+
+1. lives before the quotient `(h,k)->t=kh` so the base/factorization label is not lost;
+2. couples atomic scales `h=log p` to the Riemann-R bulk `rho(h)dh`;
+3. preserves the full exact Prime-2 mixed correction;
+4. uses the already fixed local OU covariance/precision metrics, not fitted coefficients;
+5. provides only the amount of scale regularity supported by the existing Gamma/P11 energy;
+6. remains compatible with the strong, non-norm-continuous `h->0` archimedean boundary.
+
+A successful construction would turn the prime layer into a genuine boundary/trace term of the positive Gamma/P11 bulk. Failure of every natural relative-storage kernel derived from the overlap cone would close this branch as an Object-X architecture while preserving the local identities as independent mathematics.
diff --git a/audits/P11_CAUSAL_TRANSFER_POLES_AND_RH_2026-09-15.md b/audits/P11_CAUSAL_TRANSFER_POLES_AND_RH_2026-09-15.md
new file mode 100644
index 00000000..6376f1af
--- /dev/null
+++ b/audits/P11_CAUSAL_TRANSFER_POLES_AND_RH_2026-09-15.md
@@ -0,0 +1,352 @@
+# P11 Audit — Causal transfer poles and the RH boundary
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Spektral-/Systeminterpretation des Volterra-Kerns `J_Delta`.  
+**Registry:** unveraendert.  
+**Nonclaim:** dies ist eine RH-aequivalente Reformulierung, kein RH-Beweis und kein Neuheitsclaim.
+
+---
+
+## 0. Kurzurteil
+
+Der im Critical-half-Finite-Part-Audit abgeleitete kausale Kern
+
+```math
+J_\Delta(L)
+=e^{L/2}
+\left(
+L-\gamma-
+\sum_{\log n<L}\frac{\Lambda(n)}{n}
+\right)
+```
+
+hat fuer `Re s>1/2` die Laplacetransformierte
+
+```math
+\boxed{
+\widehat J_\Delta(s)
+=
+\frac{
+\displaystyle
+\frac{d}{ds}\log\left((s-\tfrac12)\zeta(s+\tfrac12)\right)
+-\gamma
+}{s-\tfrac12}.
+}
+```
+
+Der scheinbare Punkt `s=1/2` ist entfernbar. Die nichttrivialen Pole der meromorphen Fortsetzung liegen genau bei
+
+```math
+\boxed{s=\rho-\frac12}
+```
+
+mit `rho` einer nichttrivialen Nullstelle von zeta, unter Beibehaltung der Multiplizitaet.
+
+Daher
+
+```math
+\boxed{
+RH
+\iff
+\widehat J_\Delta^{\rm mer}(s)
+\text{ hat keine Pole in }\Re s>0.
+}
+```
+
+Unter RH gilt sogar der Standardweg
+
+```math
+J_\Delta(L)=O(L^2)
+```
+
+und damit existiert die echte Laplacetransformierte fuer jedes `Re s>0`. Umgekehrt wuerde eine vorwaerts konstruierte passive/stabile Realisierung des arithmetischen Volterra-Transfers auf der offenen rechten Halbebene die unerwuenschten Pole ausschliessen und damit RH liefern.
+
+Das ist die praezise Systembedeutung des aktuellen Object-X-Gates:
+
+```text
+constructive positivity/passivity -> no unstable transfer poles -> RH.
+```
+
+Status:
+
+```text
+Laplace-transform formula                              ✓[M]
+removability at s=1/2                                  ✓[M]
+nontrivial poles = shifted zeta zeros                   ✓[M]
+RH <=> no poles in open right half-plane                ✓[M]
+RH => polynomial J_Delta via standard PNT error         ✓[M] (standard consequence)
+passive positive realization of V_J                     ?[O]
+```
+
+---
+
+# 1. Laplace transform from the Stieltjes inversion
+
+For `Re s>1/2`, the unconditional growth of `J_Delta` is harmless and
+
+```math
+\widehat J_\Delta(s)
+:=\int_0^\infty e^{-sL}J_\Delta(L)dL
+```
+
+converges.
+
+The discrepancy transform is
+
+```math
+\mathcal L\Delta(s)
+=-\frac{\zeta'}{\zeta}(s+1/2)
+-\frac1{s-1/2}.
+```
+
+The Stieltjes identity is
+
+```math
+d\Delta
+=\frac12J_\Delta dL-dJ_\Delta.
+```
+
+Taking the Laplace transform gives
+
+```math
+\begin{aligned}
+\mathcal L\Delta(s)
+&=\frac12\widehat J_\Delta(s)
+-\int_0^\infty e^{-sL}dJ_\Delta(L)\\
+&=\frac12\widehat J_\Delta(s)
++J_\Delta(0)-s\widehat J_\Delta(s)\\
+&=J_\Delta(0)-(s-\tfrac12)\widehat J_\Delta(s).
+\end{aligned}
+```
+
+Since
+
+```math
+J_\Delta(0)=-\gamma,
+```
+
+we obtain
+
+```math
+\boxed{
+\widehat J_\Delta(s)
+=\frac{-\gamma-\mathcal L\Delta(s)}{s-1/2}.
+}
+```
+
+Substitution yields
+
+```math
+\boxed{
+\widehat J_\Delta(s)
+=
+\frac{
+-\gamma
++\zeta'/\zeta(s+1/2)
++1/(s-1/2)
+}{s-1/2}.
+}
+```
+
+---
+
+# 2. Relative determinant notation
+
+Define the pole-removed relative factor
+
+```math
+\boxed{
+\mathcal D_{\rm rel}(s)
+=(s-\tfrac12)\zeta(s+\tfrac12).
+}
+```
+
+Then
+
+```math
+\frac{\mathcal D_{\rm rel}'(s)}{\mathcal D_{\rm rel}(s)}
+=
+\frac1{s-1/2}
++\frac{\zeta'}{\zeta}(s+1/2).
+```
+
+Hence
+
+```math
+\boxed{
+\widehat J_\Delta(s)
+=
+\frac{
+\mathcal D_{\rm rel}'(s)/\mathcal D_{\rm rel}(s)-\gamma
+}{s-1/2}.
+}
+```
+
+At `s=1/2`, the Laurent expansion
+
+```math
+\zeta(1+\varepsilon)
+=\frac1\varepsilon+\gamma+O(\varepsilon)
+```
+
+shows
+
+```math
+\mathcal D_{\rm rel}(1/2+\varepsilon)
+=1+\gamma\varepsilon+O(\varepsilon^2).
+```
+
+Thus
+
+```math
+\mathcal D_{\rm rel}'/\mathcal D_{\rm rel}-\gamma
+=O(s-1/2),
+```
+
+so the apparent singularity of `Jhat` at `s=1/2` is removable.
+
+---
+
+# 3. Pole set
+
+Away from `s=1/2`, poles of
+
+```math
+\mathcal D_{\rm rel}'/\mathcal D_{\rm rel}
+```
+
+occur exactly at zeros and poles of `D_rel`.
+
+The zeta pole at `1` has been removed by the factor `(s-1/2)`. The trivial zeta zeros produce poles only at
+
+```math
+s=-2m-\frac12,
+\qquad m>=1,
+```
+
+strictly in the left half-plane.
+
+Every nontrivial zero `rho` gives a pole at
+
+```math
+\boxed{s=rho-1/2.}
+```
+
+Its residue equals the zero multiplicity; division by `s-1/2` does not remove such a pole because no nontrivial zero is `rho=1`.
+
+---
+
+# 4. RH equivalence
+
+All nontrivial zeta zeros lie in the critical strip
+
+```math
+0<Re rho<1.
+```
+
+Therefore
+
+```math
+\widehat J_\Delta^{mer}
+\text{ has no pole in }Re s>0
+```
+
+if and only if zeta has no zero with
+
+```math
+Re rho>1/2.
+```
+
+By the functional equation / symmetry of nontrivial zeros under
+
+```math
+rho -> 1-rho,
+```
+
+absence of zeros to the right of `1/2` is equivalent to every nontrivial zero lying on `Re rho=1/2`.
+
+Hence
+
+```math
+\boxed{
+RH
+\iff
+\widehat J_\Delta^{mer}
+\text{ is pole-free in the open right half-plane}.
+}
+```
+
+---
+
+# 5. Actual Laplace convergence under RH
+
+The classical RH consequence
+
+```math
+\psi(x)=x+O(\sqrt x\log^2x)
+```
+
+implies by partial summation
+
+```math
+\sum_{n<x}\frac{\Lambda(n)}n
+=\log x-\gamma
++O(x^{-1/2}\log^2x).
+```
+
+With `x=e^L`, the closed formula for `J_Delta` gives
+
+```math
+\boxed{J_\Delta(L)=O(L^2).}
+```
+
+Thus, assuming RH, the defining Laplace integral for `Jhat_Delta` converges for every `Re s>0`, not merely by meromorphic continuation.
+
+No converse growth theorem is needed for the pole-set equivalence in Section 4.
+
+---
+
+# 6. System-theoretic meaning and firewall
+
+A causal passive or conservative transfer function produced by a well-posed positive state-space realization is analytic in its open stability half-plane. Therefore a **forward**, noncircular passive realization of the already-derived kernel `J_Delta` on `Re s>0` would force the pole-free property above and hence RH.
+
+This explains why the missing positive realization is exactly as hard as it should be.
+
+However it is forbidden to argue backwards:
+
+```text
+RH-equivalent analytic function
+-> invoke a generic realization theorem
+-> call the resulting storage space Object X.
+```
+
+Such a realization would merely encode the desired analyticity as an assumption.
+
+The admissible Object-X task remains:
+
+```text
+construct the passive/conservative state space directly from
+- the P11 stopped OU filtration,
+- the continuous pole compensator,
+- the higher Gamma filter bank,
+- and the overlap-cone boundary geometry,
+without assuming pole-freeness or Weil positivity.
+```
+
+---
+
+# 7. Strategic endpoint
+
+The research front can now be stated in one sentence:
+
+```math
+\boxed{
+\text{Can the concrete stopped-OU/Gamma geometry furnish a positive storage}
+\atop
+\text{realization of the causal arithmetic kernel }J_\Delta
+\text{ whose transfer is automatically analytic for }Re s>0?
+}
+```
+
+A PASS would imply RH. A natural-class FAIL would be a meaningful Object-X no-go.
diff --git a/audits/P11_CAUSAL_VOLterra_PASSIVITY_NORMAL_FORM_2026-09-15.md b/audits/P11_CAUSAL_VOLterra_PASSIVITY_NORMAL_FORM_2026-09-15.md
new file mode 100644
index 00000000..c87fd51f
--- /dev/null
+++ b/audits/P11_CAUSAL_VOLterra_PASSIVITY_NORMAL_FORM_2026-09-15.md
@@ -0,0 +1,374 @@
+# P11 Audit — Causal Volterra / passivity normal form
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Operatorverdichtung des Critical-half-Finite-Part-Transfers.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein Passivitaetsbeweis, kein Object-X-/RH-Abschluss, keine Publikationsneuheit.
+
+---
+
+## 0. Kurzurteil
+
+Der pole-subtrahierte Prime-Diskrepanzoperator besitzt eine exakte **kausale Volterra-Faktorisierung** mit dem festen Critical-half-Port
+
+```math
+D_+=\partial_x+\frac12.
+```
+
+Definiere den kausalen Volterra-Operator
+
+```math
+\boxed{
+(V_Jv)(x)
+=\int_{-\infty}^{x}J_\Delta(x-y)v(y)dy,
+}
+```
+
+wobei
+
+```math
+J_\Delta(L)
+=e^{L/2}
+\left(L-\gamma-\sum_{\log n<L}\frac{\Lambda(n)}{n}\right).
+```
+
+Mit
+
+```math
+D_-:=D_+^*=-\partial_x+\frac12
+```
+
+gilt auf glatten kompakt getragenen Funktionen exakt
+
+```math
+\boxed{
+D_-V_J+V_J^*D_+
+=2\gamma I+\mathcal C_\Delta,
+}
+```
+
+wobei `C_Delta` der selbstadjungierte Faltungsoperator ist, dessen Quadratform
+
+```math
+\langle v,\mathcal C_\Delta v\rangle
+=\int_0^\infty F_v(L)d\Delta(L)
+```
+
+ist.
+
+Die verbleibende positive Gamma-Schicht `m>=1` besitzt zugleich eine kausale stabile Filterbankrealisierung. Fuer
+
+```math
+\mu_m=2m+\frac12,
+\qquad
+y_m=(\partial_x+\mu_m)^{-1}v,
+```
+
+gilt
+
+```math
+\boxed{
+\int_0^\infty e^{-\mu_m t}\|K_tv\|_2^2dt
+=\frac{2}{\mu_m}\|y_m'\|_2^2.
+}
+```
+
+Damit wird die NULLPOL-Weilform zu
+
+```math
+\boxed{
+Q_W(v)
+=
+\sum_{m=1}^\infty\frac{2}{\mu_m}\|y_m'\|_2^2
+-c_*\|v\|_2^2
+-2\operatorname{Re}\langle D_+v,V_Jv\rangle,
+}
+```
+
+mit
+
+```math
+c_*
+=\log(8\pi)+\frac\pi2-4-\gamma
+=0.2177520894\ldots>0.
+```
+
+Dies ist eine radiusunabhaengige **Dissipation-minus-feedback**-Normalform. Der verbleibende Object-X-Gate kann daher als Passivitaets-/Storage-Problem eines kausalen Systems formuliert werden.
+
+Status:
+
+```text
+Volterra factorization of C_Delta                       ✓[M]
+fixed critical-half port D_+                            ✓[M]
+causal higher-Gamma filter bank                         ✓[M]
+causal dissipation-feedback normal form                 ✓[M]
+positive storage / bounded-real certificate             ?[O]
+RH / Object X                                           ?[O]
+```
+
+---
+
+# 1. One-sided kernel as a distribution
+
+Let
+
+```math
+j_+(t)=\mathbf1_{t\ge0}J_\Delta(t).
+```
+
+On `(0,infty)` the previous audit proved
+
+```math
+d\Delta
+=\frac12J_\Delta(t)dt-dJ_\Delta(t).
+```
+
+As a distribution on the whole line, differentiating the Heaviside extension adds the root atom:
+
+```math
+dj_+
+=\mathbf1_{t>0}dJ_\Delta+J_\Delta(0)\delta_0.
+```
+
+Since
+
+```math
+J_\Delta(0)=-\gamma,
+```
+
+we get
+
+```math
+\boxed{
+\left(\frac12-\partial_t\right)j_+
+=d\Delta+\gamma\delta_0.
+}
+```
+
+Here `dDelta` is understood on the positive half-line and extended by zero to negative `t`.
+
+---
+
+# 2. Volterra operator factorization
+
+Convolution by `j_+` is exactly
+
+```math
+(V_Jv)(x)
+=\int_{-\infty}^xJ_\Delta(x-y)v(y)dy.
+```
+
+Applying
+
+```math
+D_-=-\partial_x+\frac12
+```
+
+to the output gives convolution by
+
+```math
+\left(\frac12-\partial_t\right)j_+
+=d\Delta+\gamma\delta_0.
+```
+
+Hence `D_-V_J` has the positive-half discrepancy kernel plus `gamma I`. Its adjoint `V_J^*D_+` supplies the reflected negative-half kernel plus the second `gamma I`.
+
+Therefore
+
+```math
+\boxed{
+D_-V_J+V_J^*D_+
+=\mathcal C_\Delta+2\gamma I.
+}
+```
+
+In quadratic-form language,
+
+```math
+\boxed{
+2\operatorname{Re}\langle D_+v,V_Jv\rangle
+=2\gamma\|v\|_2^2
++\int_0^\infty F_v(t)d\Delta(t).
+}
+```
+
+This is equivalent to the midpoint-coordinate transfer identity of the preceding audit but makes causality explicit.
+
+---
+
+# 3. Causal realization of one higher Gamma mode
+
+For `mu>0` let
+
+```math
+y=(\partial_x+\mu)^{-1}v,
+```
+
+i.e. the stable causal first-order state satisfying
+
+```math
+y'+\mu y=v.
+```
+
+On the Fourier line,
+
+```math
+\widehat{y'}(z)
+=\frac{iz}{\mu+iz}\widehat v(z).
+```
+
+Thus
+
+```math
+\frac{2}{\mu}\|y'\|_2^2
+```
+
+has multiplier
+
+```math
+\frac{2}{\mu}\frac{z^2}{z^2+\mu^2}.
+```
+
+But directly
+
+```math
+\int_0^\infty e^{-\mu t}\|K_tv\|_2^2dt
+```
+
+has the same multiplier because
+
+```math
+2\int_0^\infty e^{-\mu t}(1-\cos zt)dt
+=\frac{2z^2}{\mu(z^2+\mu^2)}.
+```
+
+Therefore
+
+```math
+\boxed{
+\int_0^\infty e^{-\mu t}\|K_tv\|_2^2dt
+=\frac{2}{\mu}\|y'\|_2^2.
+}
+```
+
+---
+
+# 4. Higher-Gamma filter bank
+
+The post-ground Gamma density is
+
+```math
+h_1(t)
+=\sum_{m=1}^\infty e^{-\mu_m t},
+\qquad
+\mu_m=2m+\frac12.
+```
+
+By monotone convergence of the positive mode energies,
+
+```math
+\boxed{
+\int_0^\infty h_1(t)\|K_tv\|_2^2dt
+=\sum_{m=1}^\infty\frac{2}{\mu_m}\|y_m'\|_2^2,
+}
+```
+
+where
+
+```math
+y_m=(\partial_x+\mu_m)^{-1}v.
+```
+
+Thus the entire positive remainder after removing the Gamma ground mode is a parallel bank of stable first-order causal dissipative channels.
+
+---
+
+# 5. Causal NULLPOL normal form
+
+The Critical-half finite-part audit gave
+
+```math
+Q_W(v)
+=
+\int h_1(t)\|K_tv\|^2dt
+-c_*\|v\|^2
+-2\operatorname{Re}\langle D_+v,V_Jv\rangle.
+```
+
+Substituting the filter-bank realization gives
+
+```math
+\boxed{
+Q_W(v)
+=
+\sum_{m=1}^\infty\frac{2}{\mu_m}\|y_m'\|_2^2
+-c_*\|v\|_2^2
+-2\operatorname{Re}\langle D_+v,V_Jv\rangle.
+}
+```
+
+Every operator in this formula is now radius-independent. Compact support enters only through the input `v`; no prime cutoff or frequency truncation is present.
+
+---
+
+# 6. Passivity interpretation and firewall
+
+The form has the schematic structure
+
+```math
+Q_W
+=
+\text{positive causal dissipation}
+-
+\text{fixed scalar leakage}
+-
+\text{Hermitian arithmetic feedback}.
+```
+
+This suggests a bounded-real/KYP-type question: can one construct a positive storage state `S(v)` from the stopped OU boundary-tail geometry so that
+
+```text
+positive Gamma dissipation + storage drop
+>= arithmetic feedback + scalar leakage?
+```
+
+Such a theorem would be genuinely forward if the storage is fixed by the already-derived OU/overlap-cone system and not selected from the desired Weil inequality.
+
+However no such storage certificate is currently known. Simply applying a generic realization theorem to a transfer function already defined from `Q_W` would be circular and is forbidden as an Object-X claim.
+
+---
+
+# 7. Why the route is different from fixed-window prolate
+
+The prolate formulation encodes the arithmetic obstruction through a sign-changing Fourier multiplier whose dangerous frequency region grows doubly exponentially with the support radius under pointwise tail control.
+
+The Volterra formulation instead uses one fixed causal kernel `J_Delta` on the half-line and restricts it automatically to the source support. There is no radius-dependent prime cutoff in the operator definition.
+
+This does **not** prove that the Volterra route is easier; polynomial control of `J_Delta` is itself RH-scale. It does, however, remove the artificial pointwise-frequency-resolution barrier and exposes a different structural gate: causal passivity/storage rather than multiplier positivity.
+
+---
+
+# 8. Next gate
+
+The next destructive task is to search for a **canonical storage functional** generated by the stopped OU boundary state
+
+```math
+Y(L)=\int_L^\infty e^{-(t-L)/2}X(t)dt
+```
+
+and the higher-Gamma filter states `y_m`, and test it first on the exact `R=1` Prime-2 mixed witness.
+
+PASS criterion:
+
+```text
+a coefficient-free positive block/storage identity reproduces the exact
+Volterra cross term and scalar c_*.
+```
+
+FAIL criterion:
+
+```text
+the natural OU storage class cannot reach the Prime-2 mixed calibration;
+record the class no-go and do not fit free constants.
+```
diff --git a/audits/P11_COMMON_DEPTH_SHORTING_TOTALITY_NOGO_2026-09-15.md b/audits/P11_COMMON_DEPTH_SHORTING_TOTALITY_NOGO_2026-09-15.md
new file mode 100644
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--- /dev/null
+++ b/audits/P11_COMMON_DEPTH_SHORTING_TOTALITY_NOGO_2026-09-15.md
@@ -0,0 +1,54 @@
+# P11 Audit — Common-depth OU shorting totality no-go
+
+**Datum:** 15. September 2026
+
+## Result
+
+For `h>0`, `k>=1`, define the normalized OU innovation cell
+
+```math
+\eta_{h,k}(t)
+=\frac{\mathbf 1_{((k-1)h,kh]}(t)e^{-(kh-t)/2}}
+{\sqrt{1-e^{-h}}}.
+```
+
+For fixed `h`, the family in `k` is orthonormal in `L^2(R_+,dt)`. Hence the most direct nonorthogonal discrete/continuum coupling is to place all prime cells and all Riemann-R continuum cells in this common depth space and use their ordinary Gram overlaps.
+
+This architecture collapses: the continuum family is already total in the common depth space.
+
+Indeed its `k=1` vectors are
+
+```math
+\eta_{h,1}(t)=c(h)e^{t/2}\mathbf 1_{(0,h]}(t),
+\qquad c(h)\ne0.
+```
+
+Because the Riemann-R density `rho(h)` is strictly positive for every `h>0`, all these vectors occur in the continuum support. Their linear span is dense in `L^2(R_+)`: differences of the weighted initial-interval indicators generate weighted compactly supported step functions, and those are dense.
+
+Therefore every discrete prime cell `eta_(log p,k)` lies in the closed span of the continuum cells. Orthogonal shorting of the joint Gram against the full continuum block removes the discrete side completely.
+
+In geometric Schur-complement notation,
+
+```math
+\boxed{A_P-B A_C^{-1}B^*=0}
+```
+
+where this is understood as the shorted operator / projection onto the orthogonal complement of the closed continuum feature span (not as a claim that an unbounded literal matrix inverse exists).
+
+## Consequence
+
+```text
+common depth embedding + ordinary Gram overlap + orthogonal shorting   ×[M]
+```
+
+The common OU depth geometry is too complete to retain arithmetic information. A nontrivial relative storage mechanism must preserve additional structure not visible in the depth vector alone, for example:
+
+```text
+- the scale/base variable h itself,
+- the physical overlap-cone x-incidence,
+- or a relative/renormalized boundary form.
+```
+
+This does not invalidate the exact discrete/continuous precision identities. It only excludes the simplest canonical Schur complement on the shared `L^2(dt)` innovation space.
+
+No RH/Object-X claim.
diff --git a/audits/P11_COMMON_JUMP_DIAGONAL_FINITE_PART_2026-09-13.md b/audits/P11_COMMON_JUMP_DIAGONAL_FINITE_PART_2026-09-13.md
new file mode 100644
index 00000000..e36c8920
--- /dev/null
+++ b/audits/P11_COMMON_JUMP_DIAGONAL_FINITE_PART_2026-09-13.md
@@ -0,0 +1,413 @@
+# P11 Audit — COMMON-JUMP diagonal finite part
+
+**Datum:** 13. September 2026  
+**Basis:** NULLPOL-COMMON plus Critical-half Gamma-resolvent ladder.  
+**Rolle:** theorem-level Identifikation des bisher offenen Skalar-/Threshold-Charakters.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein NP-GAP-/RH-Beweis; keine Publikationsprioritaet behauptet.
+
+---
+
+## 0. Kurzurteil
+
+Die COMMON-JUMP-Schwelle
+
+```math
+\Gamma_a
+=2\sum_{n\in\mathcal P_a}w_n+\kappa_*,
+\qquad
+w_n=\frac{\Lambda(n)}{\sqrt n},
+```
+
+ist kein algebraisch fremder Skalar neben der positiven Featuregeometrie.
+
+- Der Prime-Anteil `2 w_n` ist exakt die Diagonalmasse des positiven Jump-Operators `w_n K_{log n}^*K_{log n}`.
+- Der archimedische Anteil `kappa_*` ist exakt die **renormalisierte Diagonal-/Hochfrequenzmasse** der positiven Gamma-Resolventenleiter.
+
+Mit
+
+```math
+\mu_m=2m+\frac12
+```
+
+gilt exakt
+
+```math
+\boxed{
+\kappa_*
+=
+\lim_{N\to\infty}
+\left(
+\sum_{m=0}^N\frac{2}{\mu_m}
+-
+\log\frac{N+1}{\pi}
+\right).
+}
+```
+
+Damit ist
+
+```math
+\boxed{
+\Gamma_a
+=
+\text{finite Prime diagonal mass}
++
+\text{renormalized archimedean diagonal mass}.
+}
+```
+
+Die bisherige `NP-SCALAR`-Frage ist damit strukturell weiter geschlossen: offen bleibt nicht mehr die Herkunft der Schwelle, sondern ausschliesslich, warum die **zentrierte/renormalisierte** Common-Jump-Form auf NULLPOL nichtnegativ sein sollte.
+
+Status:
+
+```text
+Prime diagonal threshold interpretation                 ✓[M]
+Gamma ladder finite-part identity for kappa_*           ✓[M]
+full COMMON-JUMP threshold = canonical diagonal mass    ✓[M]
+UV asymptotic normalization at scale 2pi                ✓[M]
+sharp centered NP-GAP                                   ?[O]
+full positive Object X / RH                             ?[O]
+publication novelty                                     ?[O]
+```
+
+---
+
+# 1. Prime-Atome: die Schwelle ist exakt die Diagonale
+
+Fuer
+
+```math
+K_t=T_{t/2}-T_{-t/2}
+```
+
+gilt
+
+```math
+\boxed{
+K_t^*K_t=2I-T_t-T_{-t}.
+}
+```
+
+Ein Prime-Power-Atom mit Gewicht `w_n` traegt daher
+
+```math
+w_nK_{\log n}^*K_{\log n}
+=
+2w_nI-w_n(T_{\log n}+T_{-\log n}).
+```
+
+Die zugehoerige COMMON-JUMP-Schwelle enthaelt exakt
+
+```math
+\boxed{2w_nI.}
+```
+
+Nach Zentrierung bleibt nur der off-diagonale/Translationsanteil
+
+```math
+-w_n(T_{\log n}+T_{-\log n}).
+```
+
+Summiert ueber die aktive endliche Prime-Power-Menge:
+
+```math
+\boxed{
+A_a^{\rm prime}
+=2\sum_{n\in\mathcal P_a}w_n
+}
+```
+
+ist daher exakt die endliche Diagonalmasse des positiven Prime-Jump-Operators.
+
+Dies erklaert zugleich die bereits bewiesene Cutoff-Gauge-Kovarianz: jedes neu eintretende, fuer den kleineren Traeger exterior liegende Prime-Atom fuegt `2w_n I` sowohl der positiven Gramform als auch der Schwelle hinzu.
+
+---
+
+# 2. Archimedes: positive Resolventenleiter
+
+Die archimedische positive Dichte lautet
+
+```math
+h(t)=\frac{e^{-t/2}}{1-e^{-2t}}
+=\sum_{m=0}^\infty e^{-\mu_m t},
+\qquad
+\mu_m=2m+\frac12.
+```
+
+Damit ist der positive archimedische COMMON-JUMP-Operator
+
+```math
+\boxed{
+\Phi_\infty(D)
+=
+\sum_{m=0}^\infty A_m(D),
+}
+```
+
+mit
+
+```math
+\boxed{
+A_m(D)
+:=
+\frac{2}{\mu_m}
+D^2(D^2+\mu_m^2)^{-1}
+\succeq0.
+}
+```
+
+Auf der reellen Fourierachse ist
+
+```math
+A_m(z)
+=\frac{2}{\mu_m}\frac{z^2}{z^2+\mu_m^2}.
+```
+
+Jeder einzelne Modus besitzt daher die endliche Hochfrequenz-/Diagonalmasse
+
+```math
+\boxed{
+\lim_{|z|\to\infty}A_m(z)
+=\frac{2}{\mu_m}.
+}
+```
+
+Die Summe dieser Diagonalmassen divergiert logarithmisch. Der volle Operator bleibt fuer jedes feste `z` wohldefiniert, weil die Resolventenfaktoren die hohen Moden daempfen. Die Schwelle `kappa_*` ist genau die kanonische finite-part dieser divergenten Diagonalsumme.
+
+---
+
+# 3. Exakte finite-part-Identitaet
+
+Da
+
+```math
+\mu_m=2\left(m+\frac14\right),
+```
+
+gilt
+
+```math
+\frac{2}{\mu_m}
+=\frac{1}{m+1/4}.
+```
+
+Die Digamma-Teleskopformel liefert fuer jedes `N>=0`
+
+```math
+\boxed{
+\sum_{m=0}^N\frac{2}{\mu_m}
+=
+\psi\left(N+\frac54\right)-\psi\left(\frac14\right).
+}
+```
+
+Mit der Standardasymptotik
+
+```math
+\psi(x)=\log x+O(x^{-1})
+\qquad(x\to+\infty)
+```
+
+folgt
+
+```math
+\lim_{N\to\infty}
+\left[
+\sum_{m=0}^N\frac{2}{\mu_m}-\log(N+1)
+\right]
+=-\psi(1/4).
+```
+
+Da aus NULLPOL-COMMON
+
+```math
+\kappa_*=\log\pi-\psi(1/4)
+```
+
+bekannt ist, erhalten wir exakt
+
+```math
+\boxed{
+\kappa_*
+=
+\log\pi+
+\lim_{N\to\infty}
+\left[
+\sum_{m=0}^N\frac{2}{\mu_m}-\log(N+1)
+\right].
+}
+```
+
+Aequivalent und kompakter:
+
+```math
+\boxed{
+\kappa_*
+=
+\lim_{N\to\infty}
+\left[
+\sum_{m=0}^N\frac{2}{\mu_m}
+-
+\log\frac{N+1}{\pi}
+\right].
+}
+```
+
+Dies ist keine numerische Beobachtung, sondern eine exakte Digamma-Identitaet.
+
+---
+
+# 4. UV-Normalisierung der zentrierten Gamma-Schicht
+
+Die volle archimedische positive Symbolfunktion ist
+
+```math
+\Phi_\infty(z)
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)
+-\psi(1/4).
+```
+
+Nach Abzug der finite-part-Diagonalmasse:
+
+```math
+\Phi_\infty(z)-\kappa_*
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)
+-\log\pi.
+```
+
+Aus der Digamma-Asymptotik folgt fuer reelles `|z|->infty`
+
+```math
+\boxed{
+\Phi_\infty(z)-\kappa_*
+=
+\log\frac{|z|}{2\pi}+O(|z|^{-2}).
+}
+```
+
+Die Schwelle ist somit genau die Konstante, welche die positive Resolventenleiter auf die natuerliche archimedische Hochfrequenzskala `2pi` renormiert.
+
+Dies erklaert auch, warum die fruehere Zerlegung
+
+```math
+\kappa_*=\log(4\pi)+\gamma+\log2+\frac\pi2
+```
+
+strukturell irrefuehrend sein kann: sie zerlegt eine intrinsische finite-part-Konstante in zwei rechnerische Stuecke. In der Resolventenbasis ist `kappa_*` ein einziges kanonisches Renormierungsdatum derselben positiven Gamma-Schicht.
+
+---
+
+# 5. Vollstaendige COMMON-JUMP-Diagonalmasse
+
+Fuer festes Fenster `a` ist die Prime-Power-Menge endlich. Deshalb ist keine Prime-Renormierung notwendig. Kombiniert mit §3:
+
+```math
+\boxed{
+\Gamma_a
+=
+2\sum_{n\in\mathcal P_a}w_n
++
+\operatorname{FP}_{\pi}
+\sum_{m\ge0}\frac{2}{\mu_m},
+}
+```
+
+wobei `FP_pi` hier exakt die in §3 definierte finite part
+
+```math
+\operatorname{FP}_{\pi}
+\sum_{m\ge0}\frac{2}{\mu_m}
+:=
+\lim_{N\to\infty}
+\left[
+\sum_{m=0}^N\frac{2}{\mu_m}
+-
+\log\frac{N+1}{\pi}
+\right]
+```
+
+bedeutet.
+
+Damit ist `Gamma_a` kanonisch aus den **Diagonalmassen der positiven COMMON-JUMP-Moden** rekonstruiert.
+
+---
+
+# 6. Reinterpretation der zentrierten Weilform
+
+Auf NULLPOL gilt exakt
+
+```math
+Q_W(v)
+=\langle v,(X_a^*X_a-\Gamma_aI)v\rangle.
+```
+
+Nach diesem Audit kann dies gelesen werden als
+
+> positive COMMON-JUMP-Energie minus ihre endliche/renormalisierte Diagonalmasse.
+
+Auf der Prime-Seite ist die Zentrierung algebraisch
+
+```math
+2w_nI-w_n(T_t+T_{-t})-2w_nI
+=
+-w_n(T_t+T_{-t}).
+```
+
+Auf der Gamma-Seite ist sie die finite-part-Zentrierung
+
+```math
+\Phi_\infty(D)-\kappa_*I.
+```
+
+Damit ist die verbleibende NP-GAP-Frage praeziser:
+
+```text
+Warum ist die renormalisierte off-diagonale/common-jump Paarung
+auf der critical-half Range-Klasse nichtnegativ?
+```
+
+Der Skalar selbst benoetigt keinen unabhaengigen Generator mehr.
+
+---
+
+# 7. Object-X-Bedeutung
+
+Dieser Befund schliesst eine der zuletzt verbliebenen konzeptionellen Luecken:
+
+```text
+Prime channels      -> positive modes + exact finite diagonal masses
+Gamma channels      -> positive modes + exact finite-part diagonal mass
+Threshold Gamma_a   -> Summe genau dieser beiden Diagonalbuchungen
+```
+
+Gemeinsam mit dem Critical-half Pole-Transfer-Audit ergibt sich nun:
+
+```text
+critical-half L_{1/2}
+   |
+   +-- boundary charges / pole transfer E_±
+   +-- logarithmic Prime OU/AR(1) geometry
+   +-- Gamma resolvent ground mode and ladder
+   +-- COMMON-JUMP diagonal finite-part threshold
+```
+
+Damit sind Pole, Prime-Korrelation, archimedische positive Schicht und der zuvor isolierte Threshold alle auf dieselbe dokumentierte Operatorfamilie zurueckgefuehrt.
+
+Was weiterhin fehlt, ist die **Positivitaet der zentrierten Form auf NULLPOL fuer alle Fenster**. Genau dies ist NP-GAP und global RH-aequivalent.
+
+---
+
+# 8. Firewalls
+
+Nicht behaupten:
+
+- finite-part-Zentrierung erhalte automatisch Positivitaet;
+- eine renormalisierte Diagonalmasse sei bereits eine positive Gramnorm;
+- NP-GAP oder RH seien bewiesen;
+- das gesamte Object X sei fertig konstruiert;
+- Publikationsneuheit sei geklaert.
+
+Die neue Aussage ist die exakte strukturelle Herkunft des gesamten Skalars `Gamma_a`, nicht seine Positivitaetswirkung nach Subtraktion.
diff --git a/audits/P11_CONTINUOUS_EULER_PRODUCT_AND_CRITICAL_ALLPASS_2026-09-15.md b/audits/P11_CONTINUOUS_EULER_PRODUCT_AND_CRITICAL_ALLPASS_2026-09-15.md
new file mode 100644
index 00000000..9275ebca
--- /dev/null
+++ b/audits/P11_CONTINUOUS_EULER_PRODUCT_AND_CRITICAL_ALLPASS_2026-09-15.md
@@ -0,0 +1,208 @@
+# P11 Audit — Continuous Euler product and Critical-half allpass
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Registry:** unveraendert.  
+**Nonclaim:** kein RH-/Object-X-Beweis; keine Publikationsneuheit behauptet.
+
+## 0. Result
+
+Let
+
+```math
+\rho(h)=\sum_{m\ge1}\frac{h^{m-1}}{m!\zeta(m+1)}
+=\frac{d}{dh}R(e^h),
+```
+
+where `R` is the classical Riemann prime-counting approximation (Gram series). The previous audit proved
+
+```math
+\int_0^\infty\rho(h)
+\sum_{k\ge1}h e^{-kh/2}\delta_{kh}(dt)\,dh
+=(e^{t/2}-e^{-t/2})dt.
+```
+
+Taking the shifted Laplace transform gives, for `Re S>1`,
+
+```math
+\boxed{
+\int_0^\infty \rho(h)\frac{h}{e^{Sh}-1}\,dh
+=\frac1{S(S-1)}.
+}
+```
+
+Define the positive continuous Euler product by
+
+```math
+\boxed{
+Z_0(S)
+:=
+\exp\left(
+\int_0^\infty\rho(h)
+\log\frac1{1-e^{-Sh}}\,dh
+\right),
+\qquad \Re S>1.
+}
+```
+
+Then
+
+```math
+\boxed{Z_0(S)=\frac{S}{S-1}.}
+```
+
+Consequently, after the critical shift `S=1/2+s`,
+
+```math
+\boxed{
+Z_0(1/2+s)^{-1}
+=\frac{s-1/2}{s+1/2}.
+}
+```
+
+This is the elementary right-half-plane Blaschke/allpass factor of the Critical-half first-order state.
+
+Status:
+
+```text
+continuous Euler product identity                 ✓[M]
+critical-half inverse allpass                     ✓[M]
+positive Riemann-R cell background                ✓[M]
+relative discrete/continuous determinant view     ✓[M]_part
+positive global cross-prime storage               ?[O]
+```
+
+---
+
+## 1. Laplace transform of the positive cell continuum
+
+For the cell
+
+```math
+\nu_h(dt)=\sum_{k\ge1}h e^{-kh/2}\delta_{kh}(dt),
+```
+
+and shifted parameter `s=S-1/2`,
+
+```math
+\int e^{-s t}d\nu_h(t)
+=\sum_{k\ge1}h e^{-S k h}
+=\frac{h}{e^{Sh}-1}.
+```
+
+The continuum-cell identity therefore gives
+
+```math
+\int_0^\infty \rho(h)\frac{h}{e^{Sh}-1}dh
+=\int_0^\infty e^{-s t}(e^{t/2}-e^{-t/2})dt.
+```
+
+Since `s=S-1/2`, the right side is
+
+```math
+\frac1{S-1}-\frac1S
+=\frac1{S(S-1)}.
+```
+
+All integrands on the cell side are positive for real `S>1`; complex extension follows by absolute convergence on `Re S>1`.
+
+---
+
+## 2. Integration to the continuous Euler product
+
+For `Re S>1`, differentiation under the integral gives
+
+```math
+\frac{d}{dS}\log Z_0(S)
+=-\int_0^\infty\rho(h)\frac{h}{e^{Sh}-1}dh
+=-\frac1{S(S-1)}.
+```
+
+But
+
+```math
+\frac{d}{dS}\log\frac{S}{S-1}
+=-\frac1{S(S-1)}.
+```
+
+Both logarithms tend to zero as real `S -> +infinity`; hence
+
+```math
+Z_0(S)=S/(S-1).
+```
+
+Near `h=0`, `rho(h)` is bounded and the logarithmic singularity in the integrand is integrable. For large `h`, the Gram/Riemann-R density has the expected exponential scale, while `Re S>1` supplies exponential decay; the differentiated identity also supplies a direct convergence control.
+
+---
+
+## 3. Relative Euler determinant
+
+For `Re S>1`, the ordinary Euler product is
+
+```math
+\zeta(S)=\prod_p(1-e^{-S\log p})^{-1}.
+```
+
+Thus
+
+```math
+\boxed{
+\frac{\zeta(S)}{Z_0(S)}
+=\frac{S-1}{S}\zeta(S).
+}
+```
+
+This can be read as a relative discrete-versus-continuum Euler product:
+
+```text
+discrete base spacings:   h=log p
+continuous background:    rho(h) dh = d R(e^h)
+local cell factor:         (1-e^{-S h})^{-1}.
+```
+
+The completed xi function can consequently be written
+
+```math
+\boxed{
+\xi(S)
+=\frac12 S^2\pi^{-S/2}\Gamma(S/2)
+\frac{\zeta(S)}{Z_0(S)}.
+}
+```
+
+This is only an exact reorganization of classical factors, not a new xi formula.
+
+---
+
+## 4. Critical-half allpass
+
+Put `S=1/2+s`. Then
+
+```math
+Z_0(1/2+s)^{-1}
+=\frac{s-1/2}{s+1/2}.
+```
+
+For `Re s>0`,
+
+```math
+|s-1/2|<|s+1/2|,
+```
+
+so the factor is Schur; for `s=i omega` its modulus is one. It is therefore the canonical one-pole lossless/allpass transfer associated with the Critical-half first-order state.
+
+This gives a positive-system meaning to the smooth pole compensator: division by the full positive Riemann-R continuum Euler background collapses to one lossless Critical-half boundary scatterer.
+
+---
+
+## 5. Firewall
+
+This does not make the discrete prime Euler product a Schur function on the whole shifted right half-plane. A normalized local cell
+
+```math
+F_h(s)=\frac{1-e^{-h/2}}{1-e^{-(s+1/2)h}}
+```
+
+is Schur for `Re s>0`, but the infinite orthogonal product over prime spacings carries a divergent normalization and its natural Euler/log-derivative realization is only directly convergent in the classical half-plane `Re s>1/2` (equivalently `Re S>1`).
+
+Hence a direct orthogonal sum/product of local passive cells cannot supply the needed all-right-half-plane Object-X realization. A nontrivial cross-prime / discrete-continuum coupling remains necessary.
diff --git a/audits/P11_CONTINUOUS_OU_MOMENT_QUOTIENT_2026-09-13.md b/audits/P11_CONTINUOUS_OU_MOMENT_QUOTIENT_2026-09-13.md
new file mode 100644
index 00000000..f895225d
--- /dev/null
+++ b/audits/P11_CONTINUOUS_OU_MOMENT_QUOTIENT_2026-09-13.md
@@ -0,0 +1,578 @@
+# P11 Audit — Continuous OU factorization and the 2D moment quotient at a=1/2
+
+**Datum:** 13. September 2026  
+**Basis:** Critical-half branch nach dem Prime-2 bad-channel PASS.  
+**Rolle:** theorem-level Strukturaudit fuer den verbleibenden `CRIT-HALF-COMPRESS-1`-Gate.  
+**Registry:** unveraendert.  
+**Nonclaim:** die volle `a=1/2`-Positivitaet ist noch nicht geschlossen.
+
+---
+
+## 0. Kurzurteil
+
+Zwei weitere exakte Reduktionen werden verfuegbar.
+
+Erstens besitzt **jede** Gamma-Resolventenstufe eine kausale OU-/Innovationsfaktorisierung
+
+```math
+\boxed{
+A_\mu
+=\frac2\mu V_\mu^*V_\mu,
+\qquad
+V_\mu=\partial_x(\partial_x+\mu)^{-1}.
+}
+```
+
+Der zugehoerige Zustand
+
+```math
+y_\mu=(\partial_x+\mu)^{-1}v
+```
+
+erfuellt die skalare Markov-Gleichung
+
+```math
+y_\mu'+\mu y_\mu=v.
+```
+
+Abtastung dieses Zustands in Schrittweite `h=log p` beim Gamma-Grundwert `mu=1/2` liefert exakt
+
+```math
+q_p=e^{-h/2}=p^{-1/2},
+```
+
+also den P11-AR(1)-Parameter. Der diskrete P11-Tail-/Innovationsmechanismus ist damit die logarithmische Abtastung des kontinuierlichen Gamma-Grund-OU-Kanals.
+
+Zweitens reduziert sich beim Spezialgate `a=1/2`, `p=2` nach Herausnahme der separat NULLPOL Rand- und Komplementunterraeume der verbleibende Momententransfer **exakt auf zwei komplexe Freiheitsgrade**:
+
+```math
+\boxed{
+\dim\left(
+\mathscr H_{NP,1/2}/(U\oplus W)
+\right)=2.
+}
+```
+
+Damit wird die volle Positivitaetsfrage, sobald `Q|_{U\oplus W}` coerciv geschlossen ist, zu einer `2 x 2` Hermite-Weyl-/Schur-Matrix auf den beiden Critical-half Randladungen.
+
+Status:
+
+```text
+continuous Gamma OU innovation factorization            ✓[M]
+P11 q_p as exact sampled Gamma-ground OU transition      ✓[M]
+NULLPOL closes both ground OU exterior states             ✓[M]
+ground Gamma channel Dirichlet estimate                  ✓[M]
+2D moment-transfer quotient at a=1/2                     ✓[M]
+bad Prime-2 channel U                                    ✓[M] positive (previous audit)
+coercivity on U direct-sum W                             ?[O]
+2x2 effective Weyl/Schur matrix                          ?[O]
+full a=1/2 NP-GAP                                        ?[O]
+```
+
+---
+
+# 1. Kausaler OU-Resolvent
+
+Fuer `mu>0` definiere auf `L^2(R)` den kausalen Resolventen
+
+```math
+(R_\mu^+f)(x)
+:=
+\int_{-\infty}^x e^{-\mu(x-y)}f(y)\,dy.
+```
+
+Dann
+
+```math
+R_\mu^+=(\partial_x+\mu)^{-1}
+```
+
+im ueblichen schwachen Sinn und sein Fouriermultiplikator ist
+
+```math
+\frac1{\mu+iz}.
+```
+
+Setze
+
+```math
+\boxed{
+V_\mu
+:=\partial_xR_\mu^+
+=I-\mu R_\mu^+.
+}
+```
+
+Der Fouriermultiplikator von `V_mu` ist
+
+```math
+\frac{iz}{\mu+iz}.
+```
+
+Daher
+
+```math
+V_\mu^*V_\mu
+\quad\widehat{}\quad
+\frac{z^2}{z^2+\mu^2}.
+```
+
+Fuer die Gamma-Stufe
+
+```math
+A_\mu(D)
+=\frac2\mu\frac{D^2}{D^2+\mu^2}
+```
+
+folgt exakt
+
+```math
+\boxed{
+A_\mu=\frac2\mu V_\mu^*V_\mu.
+}
+```
+
+Dies ist eine vorwaerts konstruierte positive Faktorisierung jeder einzelnen archimedischen Resolventenstufe.
+
+---
+
+# 2. Zustandsraum-/Markovform
+
+Sei
+
+```math
+y_\mu=R_\mu^+v.
+```
+
+Dann
+
+```math
+\boxed{
+y_\mu'+\mu y_\mu=v,}
+```
+
+und
+
+```math
+V_\mu v=y_\mu'.
+```
+
+Somit
+
+```math
+\boxed{
+\langle v,A_\mu v\rangle
+=\frac2\mu\|y_\mu'\|_2^2.
+}
+```
+
+Fuer `v` mit Traeger in `[-a,a]` ist `y_mu(x)=0` fuer `x<=-a`. Nach dem rechten Traegerrand gilt
+
+```math
+y_\mu(x)
+=e^{-\mu(x-a)}y_\mu(a),
+```
+
+mit
+
+```math
+\boxed{
+y_\mu(a)=e^{-\mu a}E_{\mu,+}(v),
+\qquad
+E_{\mu,+}(v)=\int e^{\mu x}v(x)dx.}
+```
+
+Daher kann die Energie auch auf dem endlichen Fenster geschrieben werden als
+
+```math
+\boxed{
+\langle v,A_\mu v\rangle
+=\frac2\mu\int_{-a}^a|y_\mu'(x)|^2dx
++|y_\mu(a)|^2.
+}
+```
+
+Der letzte Term ist exakt die Energie des exponentiellen rechten Aussenschwanzes.
+
+Die anti-kausale Faktorisierung mit `( -\partial_x+mu)^{-1}` liefert spiegelbildlich den linken Aussenzustand und `E_{mu,-}`.
+
+---
+
+# 3. NULLPOL schliesst den Gamma-Grundzustand an beiden Enden
+
+Fuer den Grundmodus
+
+```math
+\mu_0=\frac12
+```
+
+sind
+
+```math
+E_{\mu_0,+}(v)=E_+(v)=M(v)(1),
+```
+
+und anti-kausal
+
+```math
+E_{\mu_0,-}(v)=E_-(v)=M(v)(0).
+```
+
+Auf NULLPOL verschwinden beide. Fuer den kausalen Grundzustand
+
+```math
+y_0=(\partial_x+1/2)^{-1}v
+```
+
+gilt daher
+
+```math
+\boxed{
+y_0(-a)=y_0(a)=0.}
+```
+
+Insbesondere liegt `y_0` in `H_0^1(-a,a)` und
+
+```math
+\boxed{
+\langle v,A_{1/2}v\rangle
+=4\|y_0'\|_2^2.
+}
+```
+
+Da
+
+```math
+v=y_0'+\frac12y_0
+```
+
+und der Randterm verschwindet,
+
+```math
+\|v\|^2
+=\|y_0'\|^2+\frac14\|y_0\|^2.
+```
+
+Mit Dirichlet-Poincare
+
+```math
+\|y_0\|^2
+\le\left(\frac{2a}{\pi}\right)^2\|y_0'\|^2
+```
+
+folgt die explizite Ground-mode-Schranke
+
+```math
+\boxed{
+\langle v,A_{1/2}v\rangle
+\ge
+\frac{4}{1+a^2/\pi^2}\|v\|^2
+\qquad(v\in\mathscr D_{NP,a}).
+}
+```
+
+Diese benutzt beide Critical-half Randbedingungen in der Zustandsraumform und kein Weil-Positivitaetsinput.
+
+---
+
+# 4. P11-AR(1) ist die exakte logarithmische Abtastung dieses Zustands
+
+Die skalare Zustandsgleichung liefert fuer jedes `h>0`
+
+```math
+\boxed{
+y_\mu(x+h)
+=e^{-\mu h}y_\mu(x)
++
+\int_x^{x+h}e^{-\mu(x+h-s)}v(s)ds.}
+```
+
+Setze jetzt den **Gamma-Grundwert**
+
+```math
+\mu=\frac12
+```
+
+und einen Primschritt
+
+```math
+h_p=\log p.
+```
+
+Dann ist der Zustandsuebergang exakt
+
+```math
+\boxed{
+e^{-\mu h_p}=e^{-\log p/2}=p^{-1/2}=q_p.}
+```
+
+Bei iterierter Abtastung entsteht somit dieselbe AR(1)-Rekursion
+
+```math
+y_{k+1}=q_py_k+\text{segment innovation}
+```
+
+wie im P11-Kanalindex-Ledger.
+
+Die normierten, voneinander disjunkten Segmentinnovationen liefern die bekannte Zerlegung
+
+```math
+R_{q_p}=T_{q_p}^*T_{q_p}+u_pu_p^*.
+```
+
+Damit ist die fruehere diskrete Tail-/Hub-Faktorisierung nicht nur mit demselben Kovarianzkern kompatibel: sie ist die **diskrete Zustandsraumabtastung des kontinuierlichen Gamma-Grund-OU-Kanals**.
+
+---
+
+# 5. Prime-2-Faserdekomposition bei a=1/2
+
+Setze wie im Threshold-Audit
+
+```math
+h=\log2,
+\qquad
+b=\frac{1-h}{2},
+\qquad
+c=h-\frac12.
+```
+
+Dann zerfaellt
+
+```math
+H=L^2(-1/2,1/2)
+```
+
+orthogonal als
+
+```math
+\boxed{
+H=H_+\oplus H_-\oplus H_M,
+}
+```
+
+wobei
+
+```math
+H_+=\operatorname{Ran}P_+,
+\qquad
+H_-=\operatorname{Ran}P_-,
+```
+
+die symmetrischen/antisymmetrischen Zwei-Rand-Fasern sind und
+
+```math
+H_M=L^2(-c,c)
+```
+
+der mittlere Einblattbereich ist.
+
+Setze
+
+```math
+K:=H_-\oplus H_M.
+```
+
+Die Prime-2-Form ist in dieser Zerlegung diagonal:
+
+```math
+C_2=-w_2P_+ + w_2P_-,
+\qquad
+w_2=\frac{\log2}{\sqrt2}.
+```
+
+---
+
+# 6. Momentabbildungen auf beiden Seiten sind surjektiv
+
+Definiere die Critical-half Randladungsabbildung
+
+```math
+\mathcal E f=(E_+(f),E_-(f))\in\mathbb C^2.
+```
+
+Auf `H_+` gilt ueber die Isometrie `S_+ : L^2(-b,b)->H_+`
+
+```math
+\mathcal E(S_+g)
+=\sqrt2\cosh(h/4)\,\mathcal E_b(g).
+```
+
+Die beiden Funktionen `e^{x/2}` und `e^{-x/2}` sind auf jedem nichtleeren Intervall linear unabhaengig. Daher ist
+
+```math
+\mathcal E_b:L^2(-b,b)\to\mathbb C^2
+```
+
+surjektiv, also auch `E|_{H_+}`.
+
+Auf `K` ist bereits die Einschraenkung auf den Mittelraum `H_M=L^2(-c,c)` surjektiv. Somit
+
+```math
+\boxed{
+\mathcal E|_{H_+}:H_+\twoheadrightarrow\mathbb C^2,
+\qquad
+\mathcal E|_K:K\twoheadrightarrow\mathbb C^2.}
+```
+
+---
+
+# 7. Der separat NULLPOL-Unterraum und der 2D Quotient
+
+Definiere
+
+```math
+U:=H_+\cap\ker\mathcal E,
+```
+
+```math
+W:=K\cap\ker\mathcal E.
+```
+
+Dann ist
+
+```math
+U\oplus W\subset\mathscr H_{NP}:=\ker\mathcal E\subset H.
+```
+
+Fuer `v=p+k` mit `p in H_+`, `k in K` lautet die globale NULLPOL-Bedingung
+
+```math
+\mathcal E p+\mathcal E k=0.
+```
+
+Definiere auf `H_NP` die Quotientenkoordinate
+
+```math
+\boxed{
+\mathfrak m(v):=\mathcal E p=-\mathcal E k\in\mathbb C^2.}
+```
+
+Dann
+
+```math
+\ker\mathfrak m=U\oplus W.
+```
+
+Wegen der Surjektivitaet aus §6 ist `m` selbst surjektiv. Der erste Isomorphiesatz liefert daher
+
+```math
+\boxed{
+\mathscr H_{NP}/(U\oplus W)
+\cong\mathbb C^2.}
+```
+
+Insbesondere
+
+```math
+\boxed{
+\dim_\mathbb C\bigl(\mathscr H_{NP}/(U\oplus W)\bigr)=2.}
+```
+
+Dies ist eine exakte algebraische Aussage; keine Positivitaet wird dafuer vorausgesetzt.
+
+---
+
+# 8. Konsequenz fuer den verbleibenden Schur-Gate
+
+Der vorherige Audit hat bereits fuer den gesamten negativen Prime-Kanal
+
+```math
+\boxed{
+Q_{1/2}|_U\succeq\frac1{20}I.}
+```
+
+bewiesen.
+
+Angenommen, der naechste Gate zeigt zusaetzlich eine Coercivity
+
+```math
+Q_{1/2}|_{U\oplus W}\succeq\delta I
+```
+
+mit `delta>0`. Dann kann `U direct-sum W` in der vollen NULLPOL-Form geshortet werden.
+
+Waehle beliebige lineare Rechtsinverse der Momentabbildungen und damit fuer `m in C^2` einen Repraesentanten
+
+```math
+r(m)=p(m)+k(-m)\in\mathscr H_{NP}.
+```
+
+Definiere die effektive Quotientenform
+
+```math
+\boxed{
+q_{eff}(m)
+:=
+\inf_{h\in U\oplus W}
+Q_{1/2}(r(m)+h).
+}
+```
+
+Wegen der Coercivity ist das Infimum eindeutig durch das Schur-Minimum realisiert und `q_eff` ist eine Hermiteform auf `C^2`. Daher existiert eine eindeutige `2 x 2` Hermitematrix `M_eff` mit
+
+```math
+q_{eff}(m)=m^*M_{eff}m.
+```
+
+Dann gilt exakt
+
+```math
+\boxed{
+Q_{1/2}\succeq0\text{ auf }\mathscr H_{NP}
+\iff
+M_{eff}\succeq0,
+}
+```
+
+vorausgesetzt der coercive Gate auf `U direct-sum W` ist geschlossen.
+
+Somit besteht der verbleibende `a=1/2`-Angriff aus zwei klar getrennten Aufgaben:
+
+```text
+(A) coercivity on the separately-NULLPOL block U direct-sum W;
+(B) positivity of one 2x2 effective critical-half Weyl matrix.
+```
+
+---
+
+# 9. Warum die numerisch schwache Richtung nicht dem bad channel entspricht
+
+Der negative Prime-Kanal `U` besitzt bereits eine theorem-level positive Reserve. Daher kann eine fast-null Richtung der vollen Form nicht innerhalb von `U` liegen.
+
+Die Quotientenzerlegung zeigt den einzigen algebraisch neuen Mechanismus ausserhalb `U direct-sum W`: eine Funktion kann auf `H_+` **nichtzero** Critical-half Ladungen tragen und diese durch entgegengesetzte Ladungen im Komplement `K` exakt neutralisieren.
+
+Dieser Mechanismus wird hier als
+
+```text
+critical-half moment transfer
+```
+
+bezeichnet.
+
+Er besitzt genau zwei komplexe Koordinaten `E_+` und `E_-`. Das ist der natuerliche Kandidat fuer die verbleibende kleine Eigenrichtung.
+
+Dies ist eine Strukturdiagnose; ein numerischer Eigenvektor wird nicht als Beweisinput verwendet.
+
+---
+
+# 10. Naechster Gate
+
+Prioritaet ist jetzt nicht mehr eine globale `a=1/2`-Suche, sondern:
+
+1. `OU-SCHUR-A`: beweise Coercivity auf `U direct-sum W`, wobei der Ground-OU-Kanal und die Prime-2-Diagonalisierung exakt benutzt werden;
+2. `OU-SCHUR-B`: berechne/charakterisiere danach die `2x2` Weyl-Matrix `M_eff` ueber die beiden Randladungen;
+3. erst nach PASS von A+B Fortsetzung im ein-Prime-Intervall `log2/2<a<log3/2`.
+
+Die kausale OU-Faktorisierung macht fuer Schritt 1 jeden Gamma-Modus zu einem eindimensionalen Zustandskanal. Das ist die bevorzugte Schur-/Riccati-Darstellung.
+
+---
+
+# 11. Firewalls
+
+Nicht behauptet wird:
+
+- `Q|_{U direct-sum W}` sei bereits coerciv;
+- die `2x2` effektive Matrix sei bereits berechnet oder positiv;
+- die Quotientenreduktion allein beweise fixed-window Positivitaet;
+- die kontinuierliche OU-Faktorisierung allein liefere Object X;
+- globales NP-GAP oder RH seien geloest.
diff --git a/audits/P11_COVARIANCE_PRECISION_DUALITY_2026-09-15.md b/audits/P11_COVARIANCE_PRECISION_DUALITY_2026-09-15.md
new file mode 100644
index 00000000..d966a992
--- /dev/null
+++ b/audits/P11_COVARIANCE_PRECISION_DUALITY_2026-09-15.md
@@ -0,0 +1,151 @@
+# P11 Audit — Covariance/precision duality on the same OU state
+
+**Datum:** 15. September 2026
+
+## Result
+
+Fix a prime `p` and put
+
+```math
+q=p^{-1/2},
+\qquad s_q=\sqrt{1-q^2}.
+```
+
+For the Weil-normalized jump channels
+
+```math
+X_k
+=\sqrt{\log p}\,p^{-k/4}K_{k\log p}v,
+```
+
+define the existing P11 tail state
+
+```math
+Y_m
+=s_q\sum_{k\ge m}q^{k-m}X_k.
+```
+
+Then
+
+```math
+Y_m-qY_{m+1}=s_q X_m.
+```
+
+The P11 hub coordinate is
+
+```math
+\boxed{
+H_p:=\sum_{k\ge1}q^kX_k
+=\frac{q}{s_q}Y_1.
+}
+```
+
+Therefore Hub + Rest is exactly the state norm
+
+```math
+\boxed{
+\mathcal E_{P11}(Y)
+=\sum_{m\ge1}\|Y_m\|^2
++\frac{q^2}{1-q^2}\|Y_1\|^2.
+}
+```
+
+By the already established factorization `R_q=T_q^*T_q+uu^*`, this equals the P11 AR(1) covariance form on the channel vector `X`.
+
+On the **same state** the canonical inverse-covariance/precision form is
+
+```math
+\boxed{
+\mathcal P(Y)
+=\frac1{1-q^2}
+\sum_{m\ge1}\|Y_m-qY_{m+1}\|^2
+=\sum_{m\ge1}\|X_m\|^2.
+}
+```
+
+The final quantity is exactly the positive full p-power COMMON-JUMP/Weil channel energy.
+
+Thus the P11 prime geometry and the positive Weil prime tower are not two unrelated feature metrics: they are the covariance and precision/Dirichlet forms of one and the same AR(1)/OU tail state.
+
+**Status:** exact identity `✓[M]`; no lower-frame/RH conclusion.
+
+---
+
+## Continuous counterpart
+
+For the universal field
+
+```math
+X_v(t)=e^{-t/4}K_tv
+```
+
+set
+
+```math
+Y(u)=\int_u^\infty e^{-(t-u)/2}X_v(t)dt.
+```
+
+Then
+
+```math
+(1/2-\partial_u)Y=X_v,
+```
+
+and the continuous precision is
+
+```math
+\boxed{
+\int_0^\infty\|X_v(t)\|^2dt
+=\int_0^\infty\|Y'(u)-Y(u)/2\|^2du.
+}
+```
+
+The left side is exactly the Gamma ground energy.
+
+The stationary Critical-half OU kernel on the half-line has the natural Robin root condition. The corresponding precision quadratic form can equivalently be written as the Critical-half Dirichlet form with its root boundary mass.
+
+Hence:
+
+```text
+P11 Hub+Rest        = discrete OU covariance/state geometry
+Prime jump tower    = discrete OU precision geometry
+Gamma ground        = continuous OU precision geometry
+NULLPOL source      = Critical-half precision/range geometry in x
+```
+
+This is an exact duality statement, not a positivity proof for the centered Weil form.
+
+---
+
+## Why this does not contradict the primewise contraction no-go
+
+The earlier hard audit proved that the P11 tail transform `T_q` is neither contractive nor expansively ordered relative to the bare channel norm on the physical translation-generated range.
+
+No contraction is asserted here. Instead the state space carries two different canonical metrics:
+
+```math
+\mathcal E_{P11}
+\quad\text{and}\quad
+\mathcal P.
+```
+
+The second is forced by the inverse AR(1) covariance / local difference operator. Thus the bridge is metric duality, not feature shorting.
+
+---
+
+## Immediate spectral bound
+
+Since the AR(1) covariance spectrum lies in
+
+```math
+[(1-q)/(1+q),(1+q)/(1-q)],
+```
+
+one has the canonical comparison
+
+```math
+\sum_k\|X_k\|^2
+\ge\frac{1-q}{1+q}\langle X,R_qX\rangle.
+```
+
+This bound is exact at the spectral level but is not by itself sharp enough to close the global Weil threshold. Its role is only to record the canonical relation between the two metrics.
diff --git a/audits/P11_CRITICAL_HALF_FINITE_PART_TRANSFER_2026-09-15.md b/audits/P11_CRITICAL_HALF_FINITE_PART_TRANSFER_2026-09-15.md
new file mode 100644
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--- /dev/null
+++ b/audits/P11_CRITICAL_HALF_FINITE_PART_TRANSFER_2026-09-15.md
@@ -0,0 +1,503 @@
+# P11 Audit — Critical-half finite-part discrepancy transfer
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Abel-/Finite-Part-Transferform der Prime-Diskrepanz.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein Positivitaets-, Object-X-, RH- oder Publikationsneuheitsclaim.
+
+---
+
+## 0. Kurzurteil
+
+Die im vorherigen Audit eingefuehrte signierte Prime-Diskrepanz
+
+```math
+\Delta
+=
+\sum_{n=p^k}\frac{\Lambda(n)}{\sqrt n}\delta_{\log n}
+-e^{t/2}dt
+```
+
+besitzt neben dem absolut konvergenten OU-Hamiltonian-Tail mit Rate `1` einen kanonischen **kritischen Halb-Raten-Tail**. Prime- und Continuum-Anteil divergieren bei dieser Rate einzeln, aber ihre Abel-Differenz besitzt fuer jedes feste `L>=0` eine explizite endliche Grenze.
+
+Definiere fuer `epsilon>0`
+
+```math
+J_{\Delta,\varepsilon}(L)
+=
+\int_{[L,\infty)}
+ e^{-(1/2+\varepsilon)(t-L)}d\Delta(t).
+```
+
+Dann existiert
+
+```math
+\boxed{
+J_\Delta(L)
+:=\lim_{\varepsilon\downarrow0}J_{\Delta,\varepsilon}(L)
+}
+```
+
+und
+
+```math
+\boxed{
+J_\Delta(L)
+=e^{L/2}
+\left(
+L-\gamma-
+\sum_{\log n<L}\frac{\Lambda(n)}{n}
+\right).
+}
+```
+
+Insbesondere
+
+```math
+\boxed{J_\Delta(0)=-\gamma.}
+```
+
+Im Stieltjes-/Distributionssinn gilt exakt
+
+```math
+\boxed{
+d\Delta(L)
+=\frac12J_\Delta(L)dL-dJ_\Delta(L).
+}
+```
+
+Daher wird die Diskrepanzpaarung fuer die Autokorrelation `F_v=f_v+f_v(-.)` zu
+
+```math
+\boxed{
+-\int F_v\,d\Delta
+=
+2\gamma\|v\|_2^2
+-
+\int_0^\infty
+J_\Delta(L)
+\left(F_v'(L)+\frac12F_v(L)\right)dL.
+}
+```
+
+In Mittelpunktskoordinaten erscheint exakt der first-order Critical-half-Faktor:
+
+```math
+\boxed{
+F_v'(L)+\frac12F_v(L)
+=
+2\operatorname{Re}
+\int
+\left(\partial_x+\frac12\right)
+ v(x+L/2)
+\overline{v(x-L/2)}dx.
+}
+```
+
+Somit lautet der nichtlokale Transferterm
+
+```math
+\boxed{
+-\int F_v\,d\Delta
+=
+2\gamma\|v\|_2^2
+-2\operatorname{Re}
+\iint_{\mathcal C_R}
+J_\Delta(L)
+\left(\partial_x+\frac12\right)v(x+L/2)
+\overline{v(x-L/2)}\,dx\,dL.
+}
+```
+
+Dies ist die bislang schaerfste gemeinsame Form von Prime-Diskrepanz, overlap cone und Critical-half-Operator. Der Koeffizient `1/2` ist hier durch die kanonische pole-subtrahierte Abel-Grenze erzwungen, nicht angepasst.
+
+Status:
+
+```text
+Abel finite-part tail J_Delta exists                    ✓[M]
+closed formula for J_Delta                              ✓[M]
+Stieltjes inversion dDelta=(1/2)J dL-dJ                 ✓[M]
+critical-half first-order transfer identity              ✓[M]
+window/overlap-cone Volterra representation              ✓[M]
+positivity / positive parent realization                 ?[O]
+publication novelty                                      ?[O]
+```
+
+---
+
+# 1. Abel regularization at the critical rate
+
+For `epsilon>0` define
+
+```math
+J_{\Delta,\varepsilon}(L)
+=
+\int_{[L,\infty)}
+ e^{-(1/2+\varepsilon)(t-L)}d\Delta(t).
+```
+
+Both terms are absolutely convergent. The prime part is
+
+```math
+\begin{aligned}
+J_{P,\varepsilon}(L)
+&=
+\sum_{\log n\ge L}
+\frac{\Lambda(n)}{\sqrt n}
+ e^{-(1/2+\varepsilon)(\log n-L)}\\
+&=
+e^{(1/2+\varepsilon)L}
+\sum_{\log n\ge L}
+\frac{\Lambda(n)}{n^{1+\varepsilon}}.
+\end{aligned}
+```
+
+The continuous compensator is
+
+```math
+\begin{aligned}
+J_{0,\varepsilon}(L)
+&=
+\int_L^\infty
+ e^{-(1/2+\varepsilon)(t-L)}e^{t/2}dt\\
+&=
+\frac{e^{L/2}}{\varepsilon}.
+\end{aligned}
+```
+
+Hence
+
+```math
+J_{\Delta,\varepsilon}
+=J_{P,\varepsilon}-J_{0,\varepsilon}.
+```
+
+---
+
+# 2. Closed finite-part formula
+
+For `epsilon>0`,
+
+```math
+\sum_{n\ge2}\frac{\Lambda(n)}{n^{1+\varepsilon}}
+=-\frac{\zeta'}{\zeta}(1+\varepsilon).
+```
+
+The Laurent expansion of zeta at `1` gives
+
+```math
+-\frac{\zeta'}{\zeta}(1+\varepsilon)
+=\frac1\varepsilon-\gamma+O(\varepsilon).
+```
+
+Therefore, writing
+
+```math
+S_<(L)
+:=
+\sum_{\log n<L}\frac{\Lambda(n)}{n},
+```
+
+one has
+
+```math
+\begin{aligned}
+J_{P,\varepsilon}(L)
+&=e^{L/2}e^{\varepsilon L}
+\left(
+\frac1\varepsilon-\gamma-S_<(L)+O(\varepsilon)
+\right)\\
+&=e^{L/2}
+\left(
+\frac1\varepsilon
++L-\gamma-S_<(L)+O(\varepsilon)
+\right).
+\end{aligned}
+```
+
+Subtracting `e^{L/2}/epsilon` gives
+
+```math
+\boxed{
+J_\Delta(L)
+=e^{L/2}\left(L-\gamma-S_<(L)\right).
+}
+```
+
+No PNT or RH is needed for this fixed-`L` identity. It uses only the Euler product for `Re s>1` and the standard Laurent expansion of zeta at `s=1`.
+
+At `L=0`, the finite sum is empty, so
+
+```math
+\boxed{J_\Delta(0)=-\gamma.}
+```
+
+---
+
+# 3. Jumps and the continuous drift
+
+Between prime-power locations, `S_<(L)` is constant. Therefore
+
+```math
+J_\Delta'(L)
+=\frac12J_\Delta(L)+e^{L/2}.
+```
+
+Thus
+
+```math
+\frac12J_\Delta(L)-J_\Delta'(L)
+=-e^{L/2},
+```
+
+which is exactly the absolutely continuous density of `Delta`.
+
+At an atom `L_0=log n`, crossing `L_0` from below to above inserts `Lambda(n)/n` into `S_<`. Consequently
+
+```math
+J_\Delta(L_0+)-J_\Delta(L_0-)
+=-e^{L_0/2}\frac{\Lambda(n)}{n}
+=-\frac{\Lambda(n)}{\sqrt n}.
+```
+
+Therefore `-dJ_Delta` contains the positive atom of `Delta`. Both statements combine into
+
+```math
+\boxed{
+d\Delta
+=\frac12J_\Delta dL-dJ_\Delta.
+}
+```
+
+---
+
+# 4. Stieltjes integration by parts
+
+Let `v` be smooth and compactly supported and put
+
+```math
+F_v(L)=f_v(L)+f_v(-L),
+\qquad
+f_v=v*\widetilde v.
+```
+
+Then `F_v` is compactly supported on the positive half-line and
+
+```math
+F_v(0)=2\|v\|_2^2.
+```
+
+Using the inversion identity,
+
+```math
+\begin{aligned}
+-\int F_v\,d\Delta
+&=-\frac12\int F_vJ_\Delta dL
++\int F_v\,dJ_\Delta.
+\end{aligned}
+```
+
+Stieltjes integration by parts gives
+
+```math
+\int F_v\,dJ_\Delta
+=-F_v(0)J_\Delta(0)
+-\int J_\Delta F_v' dL.
+```
+
+Since `J_Delta(0)=-gamma`,
+
+```math
+\boxed{
+-\int F_v\,d\Delta
+=
+2\gamma\|v\|_2^2
+-
+\int_0^\infty
+J_\Delta(L)
+\left(F_v'(L)+\frac12F_v(L)\right)dL.
+}
+```
+
+---
+
+# 5. Midpoint-coordinate transfer identity
+
+With the convention
+
+```math
+f_v(L)
+=\int v(x+L/2)\overline{v(x-L/2)}dx,
+```
+
+differentiate:
+
+```math
+\begin{aligned}
+f_v'(L)
+&=\frac12\int v'(x+L/2)\overline{v(x-L/2)}dx\\
+&\quad-
+\frac12\int v(x+L/2)\overline{v'(x-L/2)}dx.
+\end{aligned}
+```
+
+Integration by parts in `x` turns the second integral into minus the first. Hence
+
+```math
+\boxed{
+f_v'(L)
+=\int v'(x+L/2)\overline{v(x-L/2)}dx.
+}
+```
+
+Since `F_v=2 Re f_v`,
+
+```math
+\boxed{
+F_v'(L)+\frac12F_v(L)
+=
+2\operatorname{Re}\int
+\left(\partial_x+\frac12\right)v(x+L/2)
+\overline{v(x-L/2)}dx.
+}
+```
+
+For `supp(v) subset (-R,R)`, the integrand is automatically supported in
+
+```math
+\mathcal C_R
+=\{(x,L):0<L<2(R-|x|)\}.
+```
+
+Thus
+
+```math
+\boxed{
+-\int F_v\,d\Delta
+=
+2\gamma\|v\|_2^2
+-2\operatorname{Re}
+\iint_{\mathcal C_R}
+J_\Delta(L)
+\left(\partial_x+\frac12\right)v(x+L/2)
+\overline{v(x-L/2)}dx\,dL.
+}
+```
+
+---
+
+# 6. Updated NULLPOL discrepancy normal form
+
+The previous theorem gives on NULLPOL
+
+```math
+Q_W(v)
+=
+\int_0^\infty h_1(L)\|K_Lv\|^2dL
+-(\kappa_*-4)\|v\|^2
+-\int F_v\,d\Delta.
+```
+
+Substituting the finite-part transfer identity yields
+
+```math
+\boxed{
+Q_W(v)
+=
+\int_0^\infty h_1(L)\|K_Lv\|^2dL
+-c_*\|v\|^2
+-2\operatorname{Re}
+\iint_{\mathcal C_R}
+J_\Delta(L)
+\left(\partial_x+\frac12\right)v(x+L/2)
+\overline{v(x-L/2)}dx\,dL,
+}
+```
+
+where
+
+```math
+\boxed{
+c_*
+=\kappa_*-4-2\gamma
+=\log(8\pi)+\frac\pi2-4-\gamma.
+}
+```
+
+Numerically only for orientation,
+
+```text
+c_* ~ 0.2177...
+```
+
+The large radius-dependent scalar has disappeared; the remaining scalar is fixed.
+
+---
+
+# 7. Why this is structurally stronger than the rate-1 Hamiltonian transform
+
+The absolutely convergent Hamiltonian transform `H_Delta` has inversion rate `1` and therefore couples to
+
+```math
+F_v'+F_v.
+```
+
+The canonical Abel finite part at the exact convergence boundary uses rate `1/2` and couples to
+
+```math
+F_v'+\frac12F_v.
+```
+
+The latter is precisely the midpoint correlation generated by the Critical-half first-order operator
+
+```math
+\partial_x+\frac12.
+```
+
+Together with
+
+```math
+L_{1/2}
+=\left(\frac12-\partial_x\right)
+ \left(\frac12+\partial_x\right),
+```
+
+this gives a sharper algebraic meeting point between
+
+```text
+NULLPOL source factorization,
+prime-pole discrepancy,
+and overlap-cone boundary transfer.
+```
+
+No positivity conclusion follows automatically because `J_Delta` is signed and RH-scale growth remains hidden in it.
+
+---
+
+# 8. Literature / novelty firewall
+
+The ingredients
+
+```text
+- Laurent expansion of zeta at s=1,
+- sums of Lambda(n)/n,
+- Abel/finite-part subtraction of the pole,
+- explicit-formula manipulations
+```
+
+are classical. No novelty is claimed for the scalar function `J_Delta` itself.
+
+The project-specific question is narrower: whether this exact critical-half Volterra transfer can be realized as the boundary term of the already present stopped P11/OU geometry in a forward positive system. That question remains open.
+
+---
+
+# 9. Next destructive gate
+
+The next candidate class should be tested at `R=1` before any global claim:
+
+1. use the stopped OU boundary-tail shorting as the positive bulk;
+2. use `J_Delta(L)` only as a **transfer coefficient**, never as a positive density;
+3. exploit the canonical factor `partial_x+1/2` forced by the finite-part inversion;
+4. check the exact Prime-2 mixed witness `B_W-c_Gamma=-log2/sqrt2`;
+5. if a natural coefficient-free transfer misses that target, record a class no-go rather than fitting a scalar.
diff --git a/audits/P11_CRITICAL_HALF_GREEN_TREE_BRIDGE_2026-09-13.md b/audits/P11_CRITICAL_HALF_GREEN_TREE_BRIDGE_2026-09-13.md
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,766 @@
+# P11 Audit — Critical-half Green/tree bridge
+
+**Datum:** 13. September 2026  
+**Rolle:** theorem-level Strukturaudit nach `NULLPOL-COMMON`; keine Registry-Promotion.  
+**Basis:** `main` nach Merge von PR #107 (`6e7b0c8229cc69a8d86d85a9db99d6646339ffda`).  
+**Scope:** exakte Operator-/Gram-Identitaeten; kein NP-GAP-, Object-X- oder RH-Abschluss.
+
+---
+
+## 0. Kurzurteil
+
+Die bisher getrennten Strukturen
+
+- NULLPOL-Bedingungen `M(v)(0)=M(v)(1)=0`,
+- Prime-Power-AR(1)/Kac--Murdock--Szego aus PR #98,
+- cross-prime Hub-Korrelation,
+- die POS-DIL-Amplitude `1-p^{-k/2}` aus PR #101,
+- der archimedische Root-Defekt `1-e^{-t/2}` aus NULLPOL-COMMON,
+- und der Grundmodus der archimedischen Digamma-/Gamma-Schicht
+
+werden durch **denselben critical-half Kernel** verbunden:
+
+```math
+\boxed{G_{1/2}(x,y)=e^{-|x-y|/2}.}
+```
+
+Er ist zugleich
+
+1. der Green-Kern von `L_{1/2}:=-\partial_x^2+1/4` auf `R`;
+2. der exakt auf `k log p` abgetastete AR(1)-Kernel `p^{-|j-k|/2}`;
+3. der Grundmodus `mu_0=1/2` der positiven Gamma-Resolventenleiter;
+4. der Root-Korrelationskern eines expliziten OU-/Sternbaum-Hilbertraums, dessen gewichtete Gram-Matrix den fensterlosen P11-Prime-Index-Ledger inklusive cross-prime Hub exakt reproduziert.
+
+Zusaetzlich gilt exakt
+
+```math
+\boxed{
+C_c^\infty(-a,a)\cap\ker M(0)\cap\ker M(1)
+=
+L_{1/2}C_c^\infty(-a,a).
+}
+```
+
+Damit wird der NULLPOL-Gate zu einer **Range-Bedingung desselben Operators**, dessen Green-Kern die Prime-Power-Korrelation erzeugt.
+
+**Status:**
+
+```text
+NULLPOL = range(L_{1/2}) on every compact window       ✓[M]
+critical-half Green kernel                             ✓[M]
+prime star-tree Gram realization                       ✓[M]
+P11 root+tail factorization from tree innovations      ✓[M]
+POS-DIL root-defect identity                           ✓[M]
+Gamma resolvent ladder with ground mass 1/2            ✓[M]
+archimedean J as same root-chord energy                 ✓[M]
+NP-GAP moment-free L_{1/2}-lift                        ✓[M]
+sharp NP-GAP lower frame bound                         ?[O]
+full positive Object-X realization / RH                ?[O]
+publication novelty                                    ?[O]
+```
+
+---
+
+# 1. Der critical-half Operator und sein Green-Kern
+
+Setze
+
+```math
+D=-i\partial_x,
+\qquad
+L_{1/2}:=D^2+\frac14=-\partial_x^2+\frac14.
+```
+
+Dann gilt distributionell auf `R`
+
+```math
+\boxed{
+L_{1/2}\,e^{-|x|/2}=\delta_0.
+}
+```
+
+Also besitzt `L_{1/2}^{-1}` den Faltungskern
+
+```math
+\boxed{
+G_{1/2}(x,y)=e^{-|x-y|/2}.
+}
+```
+
+Der Faktor vor dem Exponential ist hier genau `1`, weil der allgemeine Green-Kern von `D^2+mu^2` gleich
+
+```math
+\frac1{2\mu}e^{-\mu|x-y|}
+```
+
+ist und fuer `mu=1/2` gerade `2mu=1` gilt.
+
+---
+
+# 2. NULLPOL ist exakt die lokale Range von `L_{1/2}`
+
+Wir verwenden
+
+```math
+M(v)(s)=\int_{\mathbb R}v(x)e^{(s-1/2)x}\,dx.
+```
+
+Damit
+
+```math
+M(v)(0)=\int e^{-x/2}v(x)\,dx,
+\qquad
+M(v)(1)=\int e^{x/2}v(x)\,dx.
+```
+
+## Satz 2.1 — lokale Range-Charakterisierung
+
+Fuer jedes `a>0` gilt
+
+```math
+\boxed{
+\mathscr D_{NP,a}
+:=
+\{v\in C_c^\infty(-a,a):M(v)(0)=M(v)(1)=0\}
+=
+L_{1/2}C_c^\infty(-a,a).
+}
+```
+
+### Beweis: `Range subset NULLPOL`
+
+Sei `u in C_c^infty(-a,a)` und `v=L_{1/2}u`. Zweimalige partielle Integration liefert
+
+```math
+\int e^{\pm x/2}\left(-u''+\frac14u\right)dx=0,
+```
+
+denn
+
+```math
+\left(-\partial_x^2+\frac14\right)e^{\pm x/2}=0.
+```
+
+Also `M(v)(0)=M(v)(1)=0`.
+
+### Beweis: `NULLPOL subset Range`
+
+Sei umgekehrt `v in D_{NP,a}` und setze vorwaerts
+
+```math
+\boxed{
+u(x):=(L_{1/2}^{-1}v)(x)
+=\int_{\mathbb R}e^{-|x-y|/2}v(y)\,dy.
+}
+```
+
+Dann gilt `L_{1/2}u=v`. Liegt `x` rechts vom Traeger von `v`, so
+
+```math
+u(x)
+=e^{-x/2}\int e^{y/2}v(y)dy
+=e^{-x/2}M(v)(1)=0.
+```
+
+Liegt `x` links vom Traeger, so
+
+```math
+u(x)
+=e^{x/2}\int e^{-y/2}v(y)dy
+=e^{x/2}M(v)(0)=0.
+```
+
+Damit ist `u` wieder kompakt im selben Fenster getragen; aus `L_{1/2}u=v` und glattem `v` folgt `u in C_c^infty(-a,a)`. Also `v=L_{1/2}u`.
+
+Da `L_{1/2}>0` auf `L^2(R)` ist die Darstellung eindeutig.
+
+**Wichtig:** Hier wird kein Paley--Wiener-Satz benoetigt. Die beiden Mellin-Nullen sind exakt die Bedingungen, welche die beiden exponentiellen Aussenschwaenze des Green-Resolventen vernichten.
+
+---
+
+# 3. Der logarithmische Prim-Sternbaum
+
+Definiere einen verwurzelten metrischen Sternbaum `T_P` mit Root `o` und fuer jede Primzahl `p` einer eigenen Halbgeraden `R_p=[0,infty)`.
+
+Ein Punkt auf Ast `p` in Radius `x>=0` werde `(p,x)` genannt. Die Metrik ist
+
+```math
+d((p,x),(p,y))=|x-y|,
+```
+
+und fuer `p!=r`
+
+```math
+d((p,x),(r,y))=x+y.
+```
+
+Die Prime-Power-Knoten sind
+
+```math
+\boxed{x_{p,k}:=(p,k\log p),\qquad k>=1.}
+```
+
+Damit liegen Potenzen derselben Primzahl auf einer logarithmisch aequidistanten Halbgeraden, waehrend verschiedene Primzahlen nur den Root teilen.
+
+---
+
+# 4. Explizite positive Hilbert-Realisierung des Baumkerns
+
+Setze
+
+```math
+\mathscr K
+:=
+\mathbb C e_o
+\oplus
+\bigoplus_p L^2(\mathbb R_+,ds).
+```
+
+Fuer `x>=0` definiere
+
+```math
+\boxed{
+\Phi_{p,x}
+:=
+e^{-x/2}e_o
+\oplus
+\left[
+\mathbf1_{[0,x]}(s)e^{-(x-s)/2}
+\right]_p.
+}
+```
+
+Am Root sei `Phi_o=e_o`.
+
+Zunaechst
+
+```math
+\|\Phi_{p,x}\|^2
+=e^{-x}+\int_0^x e^{-(x-s)}ds
+=1.
+```
+
+Sind `0<=x<=y` auf demselben Ast, dann
+
+```math
+\begin{aligned}
+\langle\Phi_{p,x},\Phi_{p,y}\rangle
+&=e^{-(x+y)/2}
++e^{-(x+y)/2}\int_0^x e^s ds\\
+&=e^{-(y-x)/2}.
+\end{aligned}
+```
+
+Auf verschiedenen Aesten ueberlappt nur die Root-Koordinate:
+
+```math
+\langle\Phi_{p,x},\Phi_{r,y}\rangle
+=e^{-(x+y)/2}
+\qquad(p\ne r).
+```
+
+Somit allgemein
+
+```math
+\boxed{
+\langle\Phi_\alpha,\Phi_\beta\rangle
+=e^{-d(\alpha,\beta)/2}.
+}
+```
+
+Dies ist eine explizite Kolmogorov-/OU-Realisierung; Positivitaet des Baumkerns wird nicht aus einem abstrakten Existenzsatz importiert.
+
+---
+
+# 5. Der komplette fensterlose P11-Prime-Ledger ist dieser Baum-Gram
+
+Aus PR #98:
+
+```math
+w_{p,k}=(\log p)p^{-k/2},
+\qquad
+\alpha_{p,k}=\sqrt{\log p}\,p^{-3k/4},
+```
+
+und fuer denselben Primast
+
+```math
+C_{jk}^{(p)}
+=(\log p)p^{\min(j,k)}p^{-3(j+k)/4}
+=\sqrt{w_{p,j}w_{p,k}}\,p^{-|j-k|/2}.
+```
+
+Definiere gewichtete Baumfeatures
+
+```math
+\boxed{
+\Psi_{p,k}:=\sqrt{w_{p,k}}\,\Phi_{p,k\log p}.
+}
+```
+
+Dann gilt fuer `p=r`
+
+```math
+\begin{aligned}
+\langle\Psi_{p,j},\Psi_{p,k}\rangle
+&=\sqrt{w_{p,j}w_{p,k}}\,p^{-|j-k|/2}\\
+&=(\log p)p^{\min(j,k)}p^{-3(j+k)/4}\\
+&=C_{jk}^{(p)}.
+\end{aligned}
+```
+
+Fuer `p!=r` gilt
+
+```math
+\begin{aligned}
+\langle\Psi_{p,j},\Psi_{r,k}\rangle
+&=\sqrt{w_{p,j}w_{r,k}}
+ p^{-j/2}r^{-k/2}\\
+&=\sqrt{\log p\log r}\,
+ p^{-3j/4}r^{-3k/4}\\
+&=\alpha_{p,j}\alpha_{r,k},
+\end{aligned}
+```
+
+also exakt der gemeinsame Hub-Kreuzkoeffizient.
+
+Damit ist fuer jede endliche Prime-Power-Kanalmenge der **gesamte fensterlose Hub+Rest-Indexledger** ein einziger positiver Baum-Gram:
+
+```math
+\boxed{
+\mathcal C_{(p,j),(r,k)}
+=
+\langle\Psi_{p,j},\Psi_{r,k}\rangle.
+}
+```
+
+Insbesondere erhaelt der in PR #98 erzwungene Exponent `3/4` eine direkte geometrische Zerlegung:
+
+```math
+\boxed{
+\alpha_{p,k}
+=\sqrt{w_{p,k}}\,
+\langle\Phi_o,\Phi_{p,k\log p}\rangle
+=\sqrt{\log p}\,p^{-k/4}p^{-k/2}.
+}
+```
+
+Also `3/4 = 1/4 + 1/2`: halbes Weil-Diagonalgewicht plus critical-half Root-Korrelation.
+
+### Firewall
+
+Dies ist eine Aussage ueber den fensterlosen Kanalindex-Ledger. Fensterprojektionen und die raeumliche P11-Operatorgeometrie duerfen nicht stillschweigend entfernt werden.
+
+---
+
+# 6. Hub und Rest sind exakt Root plus OU-Innovationen
+
+Setze auf Ast `p`
+
+```math
+h_p:=\log p,
+\qquad
+q_p=e^{-h_p/2}=p^{-1/2},
+\qquad
+\Phi_{p,0}:=\Phi_o.
+```
+
+Definiere fuer `k>=1`
+
+```math
+\boxed{
+\eta_{p,k}
+:=
+\frac{\Phi_{p,kh_p}-q_p\Phi_{p,(k-1)h_p}}
+{\sqrt{1-q_p^2}}.
+}
+```
+
+Root-Anteil und alle frueheren Astsegmente heben sich exakt weg. `eta_{p,k}` lebt nur auf dem neuen Segment `((k-1)h_p,kh_p]`. Daher ist
+
+```math
+\boxed{
+\{\Phi_o\}\cup\{\eta_{p,k}:p\text{ prim},\ k>=1\}
+}
+```
+
+ein orthonormales System.
+
+Die Rekursion lautet
+
+```math
+\boxed{
+\Phi_{p,kh_p}
+=q_p^k\Phi_o
++\sqrt{1-q_p^2}
+\sum_{m=1}^k q_p^{k-m}\eta_{p,m}.
+}
+```
+
+Fuer eine endlich getragene Kanalfolge `X=(X_k)` folgt somit
+
+```math
+\left\|\sum_k X_k\Phi_{p,kh_p}\right\|^2
+=
+\left|\sum_k q_p^kX_k\right|^2
++
+\sum_m
+\left|
+\sqrt{1-q_p^2}\sum_{k>=m}q_p^{k-m}X_k
+\right|^2.
+```
+
+Mit
+
+```math
+(T_qX)_m
+=\sqrt{1-q^2}\sum_{k>=m}q^{k-m}X_k,
+\qquad
+u_k=q^k,
+```
+
+ist dies exakt
+
+```math
+\boxed{
+R_q=T_q^*T_q+uu^*,
+}
+```
+
+die bereits in PR #98 gefundene Tail-/Hub-Faktorisierung.
+
+Der Sternbaum liefert daher eine kanonische geometrische Bedeutung:
+
+- `Hub` = gemeinsame Root-Koordinate;
+- `Rest` = voneinander orthogonale Ast-Innovationen;
+- cross-prime Korrelation = ausschliesslich gemeinsam geerbte Root-Koordinate.
+
+Dies widerspricht nicht der frueheren Firewall, dass der Hub metrisch nicht irrelevant ist; er ist gerade die gemeinsame Root-Komponente.
+
+---
+
+# 7. POS-DIL-Amplitude ist die Root-Distanz desselben Kerns
+
+Aus PR #101 gilt fuer `n=p^k`
+
+```math
+\sqrt{w_{p,k}}D_{p^k}
+=\sqrt{\log p}(1-q_p^k)S.
+```
+
+Da alle `Phi` Einheitsvektoren sind,
+
+```math
+\boxed{
+1-q_p^k
+=1-\langle\Phi_o,\Phi_{p,k\log p}\rangle
+=\frac12\|\Phi_{p,k\log p}-\Phi_o\|^2.
+}
+```
+
+Also
+
+```math
+\boxed{
+\sqrt{w_{p,k}}D_{p^k}
+=\frac{\sqrt{\log p}}2
+\|\Phi_{p,k\log p}-\Phi_o\|^2 S.
+}
+```
+
+Der fruehere Faktor `flat minus AR(1)-root` ist damit kein isoliertes algebraisches Muster, sondern die quadrierte Chordaldistanz zum gemeinsamen Root im selben positiven Hilbertraum.
+
+---
+
+# 8. Die archimedische Gamma-Schicht ist eine positive Resolventenleiter
+
+Aus NULLPOL-COMMON ist
+
+```math
+h(t)=\frac{e^{-t/2}}{1-e^{-2t}}.
+```
+
+Fuer `t>0` gilt absolut positiv
+
+```math
+\boxed{
+h(t)=\sum_{m=0}^\infty e^{-\mu_m t},
+\qquad
+\mu_m:=2m+\frac12.}
+```
+
+Das archimedische positive Symbol ist
+
+```math
+\Phi_\infty(z)
+=2\int_0^\infty h(t)(1-\cos zt)dt.
+```
+
+Tonelli und
+
+```math
+\int_0^\infty e^{-\mu t}(1-\cos zt)dt
+=\frac{z^2}{\mu(\mu^2+z^2)}
+```
+
+liefern exakt
+
+```math
+\boxed{
+\Phi_\infty(z)
+=2\sum_{m=0}^\infty
+\frac{z^2}{\mu_m(\mu_m^2+z^2)}.
+}
+```
+
+Mit der klassischen Digamma-Partialbruchreihe ist dies identisch zu
+
+```math
+\boxed{
+\Phi_\infty(z)
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)
+-\psi(1/4).
+}
+```
+
+Operatorisch:
+
+```math
+\boxed{
+\Phi_\infty(D)
+=
+\sum_{m=0}^\infty
+\frac{2}{\mu_m}
+D^2(D^2+\mu_m^2)^{-1}
+}
+```
+
+als positive Form-/Spektralmultiplizierer.
+
+Jeder Summand ist ein positiver Resolventendefekt des massiven eindimensionalen Operators
+
+```math
+L_{\mu_m}=D^2+\mu_m^2.
+```
+
+Der Grundmodus ist
+
+```math
+\boxed{\mu_0=\frac12,}
+```
+
+und daher
+
+```math
+L_{\mu_0}=L_{1/2},
+\qquad
+L_{\mu_0}^{-1}(x,y)=e^{-|x-y|/2}.
+```
+
+**Dies ist exakt derselbe Kernel wie in §§1--7.**
+
+Zusaetzlich sind die Zahlen
+
+```math
+\mu_m=2m+\frac12
+```
+
+genau die reellen Abstaende der kritischen Mittellinie `Re(s)=1/2` von den Gammafaktor-Polstellen `s=-2m` von `Gamma(s/2)`. Dies ist die spektrale Herkunft der Leiter; es wird keine neue Aussage ueber Zeta-Nullstellen behauptet.
+
+Literaturcheck fuer die Digamma-Partialbruchreihe: NIST DLMF §5.7.6. Die hier relevante Prime-/NULLPOL-Verknuepfung wird nicht als publikationsneu beansprucht.
+
+---
+
+# 9. Der archimedische Skalar `J` ist dieselbe Root-Chord-Energie
+
+NULLPOL-COMMON hat exakt bewiesen
+
+```math
+J
+:=2\int_0^\infty h(t)(1-e^{-t/2})dt
+=\log2+\frac\pi2.
+```
+
+Fuege dem Sternbaum formal einen archimedischen kontinuierlichen Ast hinzu und verwende auf ihm denselben `mu_0=1/2`-Featurekernel. Dann
+
+```math
+\|\Phi_{\infty,t}-\Phi_o\|^2
+=2(1-e^{-t/2}).
+```
+
+Somit
+
+```math
+\boxed{
+J
+=\int_0^\infty
+h(t)\|\Phi_{\infty,t}-\Phi_o\|^2dt.
+}
+```
+
+Der nichttriviale Integralanteil der archimedischen Schwelle ist also die kontinuierliche `h(t)dt`-gemittelte Root-Distanz **derselben critical-half OU-Geometrie**, deren diskrete Root-Distanzen in §7 die Prime-moment-Amplituden erzeugen.
+
+Firewall: `kappa_* = log(4pi)+gamma+J` ist damit noch nicht vollstaendig als reine Baumenergie geometrisiert; `log(4pi)+gamma` bleibt eine separate Normalisierungskonstante.
+
+---
+
+# 10. Exakter momentfreier Lift von NP-GAP
+
+Die Range-Identitaet aus §2 entfernt die zwei Nebenbedingungen vollstaendig.
+
+Schreibe fuer beliebiges
+
+```math
+u\in C_c^\infty(-a,a)
+```
+
+```math
+v=L_{1/2}u.
+```
+
+Dann ist `v` automatisch Nullpol, und jedes Nullpol-`v` entsteht eindeutig so.
+
+Da `K_t` mit `L_{1/2}` kommutiert,
+
+```math
+K_t v=L_{1/2}K_tu.
+```
+
+Fuer die archimedische Resolventenleiter gilt daher exakt
+
+```math
+\begin{aligned}
+\langle v,\Phi_\infty(D)v\rangle
+&=
+\sum_{m=0}^\infty
+\frac{2}{\mu_m}
+\left\|
+D(D^2+\mu_m^2)^{-1/2}L_{1/2}u
+\right\|_2^2.
+\end{aligned}
+```
+
+Der Grundmodus wird **lokal**:
+
+```math
+\boxed{
+\frac{2}{\mu_0}
+\left\|D(D^2+\mu_0^2)^{-1/2}L_{1/2}u\right\|^2
+=4\|u''\|_2^2+\|u'\|_2^2.
+}
+```
+
+Ferner
+
+```math
+\boxed{
+\|L_{1/2}u\|_2^2
+=\|u''\|_2^2
++\frac12\|u'\|_2^2
++\frac1{16}\|u\|_2^2.
+}
+```
+
+Die Prime-Atome werden
+
+```math
+\boxed{
+\|K_{\log n}v\|_2^2
+=\|L_{1/2}K_{\log n}u\|_2^2.
+}
+```
+
+Damit ist NP-GAP exakt aequivalent zu der **momentfreien** Ungleichung
+
+```math
+\boxed{
+\begin{aligned}
+&\sum_{m=0}^\infty
+\frac{2}{\mu_m}
+\left\|
+D(D^2+\mu_m^2)^{-1/2}L_{1/2}u
+\right\|_2^2\\
+&\quad+
+\sum_{n\in\mathcal P_a}
+\frac{\Lambda(n)}{\sqrt n}
+\|L_{1/2}K_{\log n}u\|_2^2\\
+&\qquad\ge
+\Gamma_a\|L_{1/2}u\|_2^2,
+\qquad
+u\in C_c^\infty(-a,a).
+\end{aligned}
+}
+```
+
+Es gibt hier **keine Mellin-/Momentennebenbedingungen mehr**. Der Preis ist die feste positive Quellmetrik `||L_{1/2}u||^2`; dafuer ist der erste archimedische Modus ein rein lokaler coerciver `H^2/H^1`-Term.
+
+Status:
+
+```text
+NP-GAP-LIFT exact equivalence ✓[M]
+sharp inequality             ?[O]
+```
+
+Ein Beweis der Ungleichung fuer alle `a` bleibt RH-aequivalent und wird nicht behauptet.
+
+---
+
+# 11. Objekt-X-Bedeutung
+
+Die neue Struktur ist mehr als eine Umbenennung der bereits bekannten AR(1)-Formel:
+
+1. **Source-Seite:** NULLPOL ist exakt `Range(L_{1/2})` mit lokalem Green-Inversen.
+2. **Prime-Seite:** derselbe Green-Kern, logarithmisch auf Prime-Power-Aesten abgetastet, ist der P11-AR(1)-/Hub-Gramkern.
+3. **Hub/Rest:** die bestehende `T_q^*T_q+uu^*=R_q`-Faktorisierung ist die Root-/Innovationszerlegung dieses Baumes.
+4. **POS-DIL:** `1-u_k` ist exakt die Root-Chordaldistanz.
+5. **Archimedes:** die Digamma-Schicht ist eine positive Resolventenleiter, deren Grundoperator exakt `L_{1/2}` ist.
+6. **Archimedischer Skalar:** `J` ist das kontinuierliche Mittel derselben Root-Chordaldistanz.
+
+Damit existiert nun ein vorwaerts definierter, RH-unabhaengiger **critical-half Green/OU-Kern**, der mehrere bisher nur paarweise verbundene Objekt-X-Bausteine gleichzeitig erklaert.
+
+Der natuerliche naechste analytische Gate ist nicht, einen weiteren Generator zu erfinden, sondern den `NP-GAP-LIFT` aus §10 mit dieser Green-/Resolventenstruktur anzugreifen.
+
+---
+
+# 12. Firewalls / Nichtbehauptungen
+
+Nicht behauptet wird:
+
+- NP-GAP sei bewiesen;
+- `Q_W>=0` sei unkonditional bewiesen;
+- RH sei bewiesen;
+- der Sternbaum allein realisiere bereits die volle Weilform;
+- die archimedischen `K_t`-Kanäle seien bereits mit den Prime-Aesten zu einem einzigen vollstaendigen Object-X-Operator verklebt;
+- Fensterprojektionen duerften aus P11 entfernt werden;
+- `log(4pi)+gamma` sei bereits durch die Baumenergie erklaert;
+- Publikationsneuheit sei geklaert.
+
+Allgemeine OU-Prozesse auf Baeumen und KMS/AR(1)-Kovarianzen sind klassische Strukturen. Der neue projektinterne Befund ist die **exakte Gleichzeitigkeit** dieser Struktur mit dem NULLPOL-Green-Inversen, dem P11-Prime-Ledger, POS-DIL und dem Grundmodus der Gamma-Resolventenleiter.
+
+---
+
+# 13. Neuer Arbeitsauftrag
+
+Default-Hauptengpass bleibt
+
+```text
+NP-GAP / COMMON-JUMP FRAME BOUND.
+```
+
+Neue bevorzugte Route:
+
+```text
+CRITICAL-HALF GREEN BRIDGE
+    -> NULLPOL = Range(L_{1/2})
+    -> momentfreier NP-GAP-LIFT
+    -> Gamma-Resolventenleiter + Prime-Jumps in L_{1/2}-Metrik
+    -> echte Lower-Frame-Schranke oder enger Klassen-No-Go.
+```
+
+Insbesondere sollen als naechstes **vorwaerts** untersucht werden:
+
+1. coercive Schranken der ersten endlich vielen Gamma-Resolventenmodi in der `L_{1/2}`-Metrik;
+2. exakte Rolle der Prime-Jumps nach `L_{1/2}`-Lift;
+3. ob die OU-Tree-Innovationskoordinaten eine kanonische Vergleichsmetrik zwischen P11 und COMMON-JUMP liefern;
+4. ein No-Go gegen jede rein punktweise Kanal-Domination, falls diese die notwendige kleine-`t`-Geometrie nicht respektiert.
diff --git a/audits/P11_CRITICAL_HALF_POLE_TRANSFER_2026-09-13.md b/audits/P11_CRITICAL_HALF_POLE_TRANSFER_2026-09-13.md
new file mode 100644
index 00000000..0d8fe23a
--- /dev/null
+++ b/audits/P11_CRITICAL_HALF_POLE_TRANSFER_2026-09-13.md
@@ -0,0 +1,428 @@
+# P11 Audit — Critical-half pole transfer
+
+**Datum:** 13. September 2026  
+**Basis:** Critical-half Green/tree branch nach COMMON-JUMP.  
+**Rolle:** theorem-level Strukturidentitaet.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein globaler NP-GAP-/RH-Beweis; keine Publikationsprioritaet behauptet.
+
+---
+
+## 0. Kurzurteil
+
+Der Operator
+
+```math
+L_{1/2}=-\partial_x^2+\frac14
+```
+
+und sein Green-Kern
+
+```math
+G_{1/2}(x,y)=e^{-|x-y|/2}
+```
+
+verbinden nicht nur NULLPOL, P11-AR(1) und den Gamma-Grundmodus. **Auch der bisher separat behandelte Rang-2-Polblock ist exakt die orientierte Außen-Transferform desselben Green-Kerns.**
+
+Sind zwei kompakte Traeger geordnet und disjunkt, faktorisiert der Green-Kern rank-one ueber die beiden exponentiellen Randmoden. Fuer die im Projekt verwendeten Funktionale
+
+```math
+E_+(f)=\int e^{x/2}f(x)dx,
+\qquad
+E_-(f)=\int e^{-x/2}f(x)dx
+```
+
+gilt exakt:
+
+```math
+\boxed{
+\operatorname{supp}f<\operatorname{supp}g
+\Longrightarrow
+\langle f,G_{1/2}g\rangle
+=E_+(f)\overline{E_-(g)},
+}
+```
+
+und in umgekehrter Orientierung
+
+```math
+\boxed{
+\operatorname{supp}g<\operatorname{supp}f
+\Longrightarrow
+\langle f,G_{1/2}g\rangle
+=E_-(f)\overline{E_+(g)}.
+}
+```
+
+Die beiden Terme sind genau die zwei Eintraege des Pole-pairings
+
+```math
+\boxed{
+\langle\mathcal Ef,P\mathcal Eg\rangle
+=E_+(f)\overline{E_-(g)}
++E_-(f)\overline{E_+(g)},
+\qquad
+P=\begin{pmatrix}0&1\\1&0\end{pmatrix}.
+}
+```
+
+Damit erhaelt die bisherige OX-GEN-A/Pole-layer eine direkte Green-Transfer-Bedeutung innerhalb derselben Critical-half-Geometrie.
+
+Status:
+
+```text
+oriented separated-support Green factorization          ✓[M]
+pole amplitudes = critical-half exterior transfer       ✓[M]
+NULLPOL = zero exterior Green charge                     ✓[M]
+cutoff exterior decoupling in Green metric               ✓[M]
+Prime AR(1) decay = same transfer law at log p           ✓[M]
+full positive gluing of pole+COMMON-JUMP                 ?[O]
+sharp NP-GAP                                              ?[O]
+RH / full Object X                                       ?[O]
+publication novelty                                      ?[O]
+```
+
+---
+
+# 1. Konventionen
+
+Wir verwenden die projektweite lineare-erste Hermitesche Konvention und
+
+```math
+\langle f,g\rangle=\int f(x)\overline{g(x)}dx.
+```
+
+Setze
+
+```math
+G:=L_{1/2}^{-1},
+\qquad
+(Gg)(x)=\int_{\mathbb R}e^{-|x-y|/2}g(y)dy.
+```
+
+Ferner
+
+```math
+E_+(f)=\int e^{x/2}f(x)dx,
+\qquad
+E_-(f)=\int e^{-x/2}f(x)dx.
+```
+
+---
+
+# 2. Orientierter Green-Transfer
+
+## Satz 2.1 — links nach rechts
+
+Seien `f,g in C_c^infty(R)` und
+
+```math
+\sup\operatorname{supp}f
+<
+\inf\operatorname{supp}g.
+```
+
+Dann ist fuer jedes `x in supp f` und `y in supp g` stets `x<y`, also
+
+```math
+e^{-|x-y|/2}
+=e^{-(y-x)/2}
+=e^{x/2}e^{-y/2}.
+```
+
+Daher
+
+```math
+\begin{aligned}
+\langle f,Gg\rangle
+&=
+\int f(x)
+\overline{\int e^{-(y-x)/2}g(y)dy}
+\,dx\\
+&=
+\left(\int e^{x/2}f(x)dx\right)
+\overline{\left(\int e^{-y/2}g(y)dy\right)}.
+\end{aligned}
+```
+
+Somit
+
+```math
+\boxed{
+\langle f,Gg\rangle
+=E_+(f)\overline{E_-(g)}.
+}
+```
+
+## Satz 2.2 — rechts nach links
+
+Ist umgekehrt
+
+```math
+\sup\operatorname{supp}g
+<
+\inf\operatorname{supp}f,
+```
+
+so `x>y` und
+
+```math
+e^{-|x-y|/2}
+=e^{-x/2}e^{y/2}.
+```
+
+Daher
+
+```math
+\boxed{
+\langle f,Gg\rangle
+=E_-(f)\overline{E_+(g)}.
+}
+```
+
+Diese beiden Formeln sind exakte algebraische Folgen des Green-Kerns; keine explizite Formel, keine Zeta-Nullstelle und keine Positivitaetsannahme wird verwendet.
+
+---
+
+# 3. Pole-pairing = Summe der beiden Orientierungen
+
+Der COMMON-JUMP-Audit schreibt den Polblock als
+
+```math
+\langle\mathcal Ef,P\mathcal Eg\rangle
+=E_+(f)\overline{E_-(g)}
++E_-(f)\overline{E_+(g)}.
+```
+
+Die Saetze 2.1--2.2 zeigen daher:
+
+> Die beiden algebraischen Eintraege des Pole-pairings sind exakt die zwei moeglichen orientierten rank-one Außen-Faktorisierungen des Critical-half-Greenkerns auf getrennten Traegern.
+
+Wichtig ist die Firewall: Das bedeutet **nicht**, dass der indefinite `P`-Block selbst bereits eine positive Green-Norm ist. Es bedeutet, dass seine beiden Off-Diagonal-Amplituden aus demselben positiven Green-Propagator stammen.
+
+---
+
+# 4. Translationen erzeugen exakt den Faktor `e^{-t/2}`
+
+Mit der Projekttranslation
+
+```math
+(T_tg)(x)=g(x-t)
+```
+
+gilt
+
+```math
+E_+(T_tg)=e^{t/2}E_+(g),
+\qquad
+E_-(T_tg)=e^{-t/2}E_-(g).
+```
+
+Sei `f` links von `T_tg` und die unverschobene Geometrie so fixiert, dass der Abstand durch `t` vergroessert wird. Dann traegt der Green-Transfer den Faktor
+
+```math
+\boxed{e^{-t/2}.}
+```
+
+Genau derselbe Zerfall erscheint
+
+```text
+bei t = k log p:  e^{-t/2}=p^{-k/2},
+in P11:           q_p^k=p^{-k/2},
+im OU-Tree:       <Phi_o,Phi_{p,k}>=p^{-k/2},
+in Gamma-Ground:  mu_0=1/2.
+```
+
+Der Prime-Power-AR(1)-Abfall ist damit die logarithmische Diskretisierung desselben Außen-Transfergesetzes.
+
+---
+
+# 5. NULLPOL = keine Außenladung
+
+Ist
+
+```math
+E_+(v)=E_-(v)=0,
+```
+
+so verschwinden beide orientierten Außen-Transferamplituden identisch.
+
+Dies hat zwei exakt aequivalente Manifestationen:
+
+### 5.1 Green-Inverser bleibt lokal
+
+Wie im Green/tree Audit bewiesen,
+
+```math
+u=Gv
+```
+
+hat rechts vom Traeger die Form
+
+```math
+u(x)=e^{-x/2}E_+(v),
+```
+
+und links die Form
+
+```math
+u(x)=e^{x/2}E_-(v).
+```
+
+NULLPOL bedeutet also
+
+```math
+\boxed{Gv\text{ besitzt keine exponentiellen Aussenschwaenze}.}
+```
+
+### 5.2 Getrennte Kopien sind Green-orthogonal
+
+Sind `T_s v` und `T_t v` disjunkt getragen, dann folgt aus Satz 2.1 oder 2.2
+
+```math
+\boxed{
+\langle T_sv,G T_tv\rangle=0.
+}
+```
+
+Translationskovarianz erhaelt die Nullpolbedingungen, weil `E_±(T_tv)` nur mit einem nichtverschwindenden Skalar multipliziert werden.
+
+Somit werden getrennte Nullpol-Zellen im `G`-Hilbertraum exakt orthogonal.
+
+---
+
+# 6. Exterior-shell-Kovarianz in zwei Metriken
+
+Sei `v` in `[-a,a]` getragen und `t>2a`.
+
+Dann sind `T_{t/2}v` und `T_{-t/2}v` disjunkt.
+
+In `L^2` gilt bereits aus COMMON-JUMP
+
+```math
+\|K_tv\|_2^2=2\|v\|_2^2.
+```
+
+Ist `v` zusaetzlich NULLPOL, dann sind die beiden Kopien nach §5 auch in der Green-Metrik orthogonal. Mit
+
+```math
+\|f\|_G^2:=\langle f,Gf\rangle
+```
+
+folgt daher
+
+```math
+\boxed{
+\|K_tv\|_G^2
+=2\|v\|_G^2
+\qquad(t>2a,\ v\in\mathscr D_{NP}).
+}
+```
+
+Die exterior-shell-Diagonalisierung ist somit nicht nur eine `L^2`-Traegertatsache: auf NULLPOL tritt dieselbe exakte Entkopplung auch in der Critical-half-Green-Metrik auf.
+
+---
+
+# 7. Der Pole-Block als Root-Transfer der Tree-Geometrie
+
+Im OU-Sternbaum besitzt ein Knoten im Abstand `x` vom Root die Root-Korrelation
+
+```math
+\langle\Phi_o,\Phi_x\rangle=e^{-x/2}.
+```
+
+Der Green-Transfer zwischen getrennten Zellen besitzt nach §4 exakt dasselbe Fortpflanzungsgesetz. Deshalb kann die Tree-Root-Koordinate nun konkret gelesen werden als abstrakte Version der exponentiellen Green-Randladung.
+
+Fuer Prime-Power-Knoten `x=k log p` ist
+
+```math
+\langle\Phi_o,\Phi_{p,k}\rangle=p^{-k/2}.
+```
+
+Cross-prime Root-Korrelationen
+
+```math
+p^{-j/2}r^{-k/2}
+```
+
+sind folglich genau das Produkt zweier Transfers zum gemeinsamen Root.
+
+Damit erhalten drei frueher getrennte Begriffe dieselbe Bedeutung:
+
+```text
+OX-GEN-A pole amplitudes E_±
+        = Green boundary charges,
+P11 common Hub
+        = common Tree root,
+AR(1) q_p^k
+        = critical-half propagation to/from that root.
+```
+
+Dies ist eine strukturelle Identifikation; die konkrete raeumliche P11-Fenstergeometrie wird nicht entfernt.
+
+---
+
+# 8. Konsequenz fuer die Object-X-Architektur
+
+Vor dieser Identitaet war die konservative Architektur:
+
+```text
+Pole layer (OX-GEN-A)
+    +
+COMMON-JUMP positive nonpole layer
+    +
+scalar threshold / NP-GAP.
+```
+
+Jetzt existiert ein gemeinsamer darunterliegender Propagator:
+
+```text
+                    L_{1/2} = -d^2 + 1/4
+                         |
+        -------------------------------------------
+        |                    |                    |
+   exterior pole       Prime log-grid       Gamma ground
+   transfer E_±        e^{-d/2}/AR(1)       resolvent mu_0
+        |                    |                    |
+      NULLPOL             P11 Tree          COMMON-JUMP
+   = zero charge       root/innovations      ladder
+```
+
+Damit ist die Pole-Schicht nicht mehr konzeptionell fremd zur Prime-/Gamma-Geometrie. Alle drei besitzen denselben Critical-half-Transferkern.
+
+### Was noch fehlt
+
+Der Pole-block hat Signatur `(1,1)` und COMMON-JUMP besitzt den negativen skalaren Threshold. Die jetzige Identifikation liefert daher noch **keine** vollstaendige positive Object-X-Norm.
+
+Der naechste harte Gate bleibt:
+
+```math
+X_a^*X_a\succeq\Gamma_aI
+\quad\text{auf NULLPOL fuer alle }a.
+```
+
+Aber die Suche nach einer Verklebung darf nun auf **einen einzigen Propagator `L_{1/2}`** beschraenkt werden, statt unabhaengige Prime-, Pole- und Archimedes-Generatoren zu erfinden.
+
+---
+
+# 9. Neuer bevorzugter Forschungsauftrag
+
+1. Formuliere COMMON-JUMP nach dem Source-Lift `v=L_{1/2}u` vollstaendig in der Green-/Energie-Metrik von `L_{1/2}`.
+2. Suche einen Schur-/Shorting-Mechanismus, in dem die Tree-root boundary charges `E_±` und die Gamma-Resolventenleiter als Rand- und Bulk-Anteile desselben positiven Blocks auftreten.
+3. Nutze die exakte Green-Orthogonalitaet disjunkter NULLPOL-Zellen fuer die Prime exterior-shells.
+4. Teste, ob der noch offene Skalar `log(4pi)+gamma` als renormierter Root-/boundary trace desselben Propagators auftreten kann.
+5. Wenn dies scheitert, liefere ein Klassen-No-Go fuer positive Blockverklebungen, die nur `L_{1/2}` und seine Tree-Abtastungen verwenden.
+
+---
+
+# 10. Firewalls
+
+Nicht behaupten:
+
+- der Pole-block sei selbst positiv;
+- der Green-Transfer beweise NP-GAP;
+- P11-Fensterprojektionen seien entbehrlich;
+- der abstrakte Sternbaum sei bereits der volle raeumliche Operator;
+- `log(4pi)+gamma` sei bereits erklaert;
+- Object X oder RH seien bewiesen;
+- Publikationsneuheit sei geklaert.
diff --git a/audits/P11_CRITICAL_HALF_RIGIDITY_AND_SCHUR_GAP_2026-09-13.md b/audits/P11_CRITICAL_HALF_RIGIDITY_AND_SCHUR_GAP_2026-09-13.md
new file mode 100644
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--- /dev/null
+++ b/audits/P11_CRITICAL_HALF_RIGIDITY_AND_SCHUR_GAP_2026-09-13.md
@@ -0,0 +1,549 @@
+# P11 Audit — Critical-half rigidity and analytic Schur-gap
+
+**Datum:** 13. September 2026  
+**Basis:** Branch `research/critical-half-green-tree-bridge-2026-09-13`, nach dem Critical-half Green/tree Audit.  
+**Rolle:** theorem-level Folgerungen aus der neuen Green-/OU-Brücke.  
+**Registry:** unverändert.  
+**Nonclaim:** kein globaler NP-GAP-, Object-X- oder RH-Abschluss; keine Publikationsneuheit behauptet.
+
+---
+
+## 0. Kurzurteil
+
+Die neue Green-/OU-Brücke besitzt eine **rigide Massenskala**. In der natürlichen Einparameterfamilie
+
+```math
+kappa_sigma(alpha,beta)=e^{-sigma d(alpha,beta)},
+\qquad sigma>0,
+```
+
+wird `sigma=1/2` gleichzeitig und eindeutig durch drei bereits unabhängig vorhandene Projektstrukturen erzwungen:
+
+1. den P11-Prime-Power-AR(1)-Kern `p^{-|j-k|/2}` und den erzwungenen P11-Hubexponenten `3/4`;
+2. die beiden Nullpol-Moden `e^{±x/2}`, also die Pole `s=0,1` um den Mittelpunkt `1/2`;
+3. den Grundmodus `mu_0=1/2` der archimedischen Gamma-Resolventenleiter `mu_m=2m+1/2`.
+
+Damit ist die im Green/tree Audit gefundene `1/2` **nicht frei gewaehlt**: innerhalb dieser natuerlichen OU-/Green-Familie ist sie die einzige gemeinsame Skala.
+
+Zusaetzlich liefert dieselbe Resolventenleiter eine rein analytische, unkonditionale Lower-Frame-Schranke fuer kleine Fenster. Der saubere elementare Bereich
+
+```math
+0<a<=1/16
+```
+
+ist vollständig zertifiziert. Diese kleine-Fenster-Positivitaet ist **nicht als literarisch neu** zu verstehen; bekannte Arbeiten reichen wesentlich weiter. Ihr Wert hier ist, dass sie direkt aus dem neuen Critical-half-Mechanismus folgt und damit dessen analytische Tragfaehigkeit testet.
+
+Status:
+
+```text
+critical-half rigidity sigma=1/2                     ✓[M]
+P11 exponent identity beta=1/4+sigma -> sigma=1/2   ✓[M]
+NULLPOL-pole rigidity                                ✓[M]
+Gamma-ground-mode rigidity                           ✓[M]
+archimedean Schur frame bound S(a)                   ✓[M]
+strict NP-GAP for every 0<a<=1/16                    ✓[M]
+new global NP-GAP mechanism beyond known local range ?[O]
+publication novelty                                  ?[O]
+RH / full Object X                                   ?[O]
+```
+
+---
+
+# 1. Die normierte OU-Familie auf dem Prim-Sternbaum
+
+Fuer `sigma>0` definiere im selben Sternbaum-Hilbertraum
+
+```math
+\boxed{
+\Phi^{(\sigma)}_{p,x}
+:=
+e^{-\sigma x}e_o
+\oplus
+\left[
+\sqrt{2\sigma}\,\mathbf1_{[0,x]}(s)e^{-\sigma(x-s)}
+\right]_p.
+}
+```
+
+Dann
+
+```math
+\|\Phi^{(\sigma)}_{p,x}\|^2
+=e^{-2\sigma x}
++2\sigma\int_0^xe^{-2\sigma(x-s)}ds
+=1.
+```
+
+Direkte Integration liefert fuer beliebige Baumknoten
+
+```math
+\boxed{
+\langle\Phi^{(\sigma)}_\alpha,
+       \Phi^{(\sigma)}_\beta\rangle
+=e^{-\sigma d(\alpha,\beta)}.
+}
+```
+
+Die zuvor gefundene Critical-half-Geometrie ist der Spezialfall `sigma=1/2`; dort ist `sqrt(2 sigma)=1` und zugleich der normierte OU-Kern identisch mit dem unskalierten Green-Kern von `D^2+1/4`.
+
+---
+
+# 2. P11 erzwingt `sigma=1/2`
+
+Die Weil-Diagonalgewichte sind
+
+```math
+w_{p,k}=(\log p)p^{-k/2},
+```
+
+also
+
+```math
+\sqrt{w_{p,k}}
+=\sqrt{\log p}\,p^{-k/4}.
+```
+
+Der Root-Korrelationsfaktor der `sigma`-Familie am Prime-Power-Knoten `x=k log p` ist
+
+```math
+\langle\Phi_o,\Phi^{(\sigma)}_{p,k\log p}\rangle
+=p^{-\sigma k}.
+```
+
+Damit besitzt die gewichtete Rootamplitude notwendigerweise den Exponenten
+
+```math
+\boxed{
+\beta(\sigma)=\frac14+\sigma,
+}
+```
+
+denn
+
+```math
+\sqrt{w_{p,k}}p^{-\sigma k}
+=\sqrt{\log p}\,p^{-(1/4+\sigma)k}.
+```
+
+P11 hat unabhaengig aus der Martingalmultiplizitaet und der geforderten Weil-Diagonale bereits exakt
+
+```math
+\beta=\frac34
+```
+
+erzwungen. Daher
+
+```math
+\boxed{
+\frac14+\sigma=\frac34
+\iff
+\sigma=\frac12.
+}
+```
+
+Dasselbe folgt direkt aus dem normalisierten same-prime Kern:
+
+```math
+p^{-\sigma|j-k|}
+=p^{-|j-k|/2}
+\quad\text{fuer alle }j,k
+\iff
+\sigma=\frac12.
+```
+
+Somit ist die Critical-half-Skala bereits rein aus dem P11-Prime-Ledger eindeutig bestimmt.
+
+---
+
+# 3. NULLPOL erzwingt dieselbe Skala
+
+Fuer die allgemeine massive Operatorfamilie
+
+```math
+L_\sigma=D^2+\sigma^2=-\partial_x^2+\sigma^2
+```
+
+sind die homogenen exponentiellen Moden
+
+```math
+e^{\pm\sigma x}.
+```
+
+Genau wie im Green/tree Audit gilt auf einem kompakten Fenster:
+
+```math
+L_\sigma C_c^\infty(-a,a)
+=
+\left\{
+v\in C_c^\infty(-a,a):
+\int e^{\sigma x}v(x)dx
+=
+\int e^{-\sigma x}v(x)dx
+=0
+\right\}.
+```
+
+Die reale Nullpolklasse verwendet aber exakt
+
+```math
+M(v)(0)=\int e^{-x/2}v(x)dx=0,
+\qquad
+M(v)(1)=\int e^{x/2}v(x)dx=0.
+```
+
+Daher stimmt eine `L_sigma`-Range genau mit NULLPOL ueberein nur fuer
+
+```math
+\boxed{\sigma=1/2.}
+```
+
+Aequivalent: die beiden Pole `s=0,1` liegen relativ zum Mellin-Mittelpunkt `1/2` bei den Abstaenden `±1/2`.
+
+---
+
+# 4. Die Gamma-Schicht erzwingt dieselbe Skala
+
+Die positive archimedische Dichte aus NULLPOL-COMMON ist
+
+```math
+h(t)=\frac{e^{-t/2}}{1-e^{-2t}}
+=\sum_{m=0}^\infty e^{-\mu_m t},
+\qquad
+\mu_m=2m+\frac12.
+```
+
+Daraus folgt die positive Resolventenleiter
+
+```math
+\Phi_\infty(D)
+=
+\sum_{m=0}^\infty
+\frac{2}{\mu_m}D^2(D^2+\mu_m^2)^{-1}.
+```
+
+Ihr Grundoperator ist
+
+```math
+D^2+\mu_0^2
+=D^2+\frac14
+=L_{1/2}.
+```
+
+Damit wird aus der Gamma-Schicht selbst
+
+```math
+\boxed{\sigma=\mu_0=1/2.}
+```
+
+erzwungen.
+
+Die drei Mechanismen — P11 Prime-Ledger, NULLPOL und Gamma-Grundmodus — treffen daher auf **derselben eindeutig bestimmten Massenskala** zusammen.
+
+---
+
+# 5. Critical-half rigidity theorem
+
+## Satz 5.1
+
+Betrachte die natuerliche OU-/Green-Einparameterfamilie `e^{-sigma d}` auf dem logarithmischen Prim-Sternbaum und den zugehoerigen massiven Quellenoperator `L_sigma=D^2+sigma^2`. Fordere gleichzeitig:
+
+1. die P11-Weil-Diagonalnormalisierung und den bereits erzwungenen Hubexponenten `beta=3/4`;
+2. die exakte lokale Range-Identifikation mit den Nullpolbedingungen bei `s=0,1`;
+3. Uebereinstimmung des Grundoperators mit der positiven Gamma-Resolventenleiter.
+
+Dann existiert genau ein Parameter:
+
+```math
+\boxed{\sigma=\frac12.}
+```
+
+Jede der drei Bedingungen fixiert `sigma` bereits separat; ihre Gleichzeitigkeit ist daher kein Parameterfit.
+
+### Bedeutung
+
+Die Zahl `1/2` erscheint hier in vier algebraisch verschiedenen Rollen:
+
+```text
+critical Mellin midpoint / pole distance,
+OU decay mass,
+Prime-power AR(1) exponent,
+Gamma resolvent ground mass.
+```
+
+Innerhalb der dokumentierten Familie sind diese Rollen exakt dieselbe Geometrie.
+
+---
+
+# 6. Schur-Untergrenze fuer jeden Gamma-Resolventenmodus
+
+Setze fuer `mu>0`
+
+```math
+R_\mu=(D^2+\mu^2)^{-1}.
+```
+
+Der volle Linienkern ist
+
+```math
+R_\mu(x,y)=\frac1{2\mu}e^{-\mu|x-y|}.
+```
+
+Sei `v` in `[-a,a]` getragen. Dann ist fuer `x in [-a,a]`
+
+```math
+\int_{-a}^aR_\mu(x,y)dy
+=
+\frac{1-e^{-\mu a}\cosh(\mu x)}{\mu^2}
+\le
+\frac{1-e^{-\mu a}}{\mu^2}.
+```
+
+Der Schur-Test gibt daher
+
+```math
+\boxed{
+\langle v,R_\mu v\rangle
+\le
+\frac{1-e^{-\mu a}}{\mu^2}\|v\|_2^2.
+}
+```
+
+Da
+
+```math
+D^2(D^2+\mu^2)^{-1}
+=I-\mu^2R_\mu,
+```
+
+folgt
+
+```math
+\boxed{
+\left\langle v,
+D^2(D^2+\mu^2)^{-1}v
+\right\rangle
+\ge e^{-\mu a}\|v\|_2^2.
+}
+```
+
+Diese Ungleichung benutzt weder RH noch Weil-Positivitaet noch Nullpol.
+
+---
+
+# 7. Geschlossene archimedische Frame-Untergrenze
+
+Mit `mu_m=2m+1/2` folgt durch Summation der positiven Moden
+
+```math
+\langle v,\Phi_\infty(D)v\rangle
+\ge
+S(a)\|v\|_2^2,
+```
+
+wobei
+
+```math
+\boxed{
+S(a)
+:=2\sum_{m=0}^\infty\frac{e^{-\mu_m a}}{\mu_m}.
+}
+```
+
+Setze
+
+```math
+x=e^{-a/2}.
+```
+
+Dann
+
+```math
+S(a)
+=4\sum_{m=0}^\infty\frac{x^{4m+1}}{4m+1}
+=4\int_0^x\frac{dt}{1-t^4}.
+```
+
+Die elementare Stammfunktion liefert
+
+```math
+\boxed{
+S(a)
+=
+\log\frac{1+e^{-a/2}}{1-e^{-a/2}}
++2\arctan(e^{-a/2}).
+}
+```
+
+`S(a)` ist streng fallend in `a` und divergiert fuer `a downarrow0`.
+
+---
+
+# 8. Rein elementarer Zertifikatbereich `0<a<=1/16`
+
+Fuer `a<=1/16` gilt
+
+```math
+2a<=1/8<\log2,
+```
+
+also ist noch kein Prime-Power-Atom aktiv und
+
+```math
+\Gamma_a=\kappa_*
+=\log(8\pi)+\gamma+\frac\pi2.
+```
+
+Wegen der Monotonie genuegt `a=1/16`. Setze
+
+```math
+x=e^{-1/32}>1-\frac1{32}=\frac{31}{32}.
+```
+
+Dann
+
+```math
+\log\frac{1+x}{1-x}>\log63.
+```
+
+Ferner
+
+```math
+\frac\pi4-\arctan x
+=\int_x^1\frac{dt}{1+t^2}
+<1-x<\frac1{32},
+```
+
+also
+
+```math
+2\arctan x>\frac\pi2-\frac1{16}.
+```
+
+Daher
+
+```math
+S(1/16)
+>
+\log63+\frac\pi2-\frac1{16}.
+```
+
+Um dies mit `kappa_*` zu vergleichen, benutzen wir nur die elementaren Schranken
+
+```math
+\pi<\frac{22}{7},
+\qquad
+\gamma<\frac23.
+```
+
+Dann
+
+```math
+\frac{63}{8\pi}
+>
+\frac{441}{176}
+>
+\frac52.
+```
+
+Aus der positiven `artanh`-Reihe folgt
+
+```math
+\begin{aligned}
+\log\frac52
+&=2\operatorname{artanh}\frac37\\
+&>2\left(\frac37+\frac1{3}\left(\frac37\right)^3\right)
+=\frac{312}{343}.
+\end{aligned}
+```
+
+Und exakt
+
+```math
+\frac{312}{343}-\frac23-\frac1{16}
+=\frac{2971}{16464}>0.
+```
+
+Somit
+
+```math
+\boxed{
+S(1/16)>\kappa_*.
+}
+```
+
+Also fuer jedes `0<a<=1/16` und jedes nichttriviale `v in C_c^infty(-a,a)`:
+
+```math
+\boxed{
+\|\mathcal X_av\|^2
+\ge
+\langle v,\Phi_\infty(D)v\rangle
+\ge
+S(a)\|v\|_2^2
+>
+\Gamma_a\|v\|_2^2.
+}
+```
+
+Insbesondere gilt auf NULLPOL strikt
+
+```math
+\boxed{
+Q_W(v)>0
+\qquad(0<a<=1/16,\ v\ne0).
+}
+```
+
+Status:
+
+```text
+CRIT-HALF-SCHUR local NP-GAP on 0<a<=1/16  ✓[M]
+```
+
+### Literatur-/Neuheitsfirewall
+
+Dieser Radius wird **nicht** als neuer Rekord oder Publikationsneuheit beansprucht. Unkonditionale lokale Weil-Positivitaet ist aus der Literatur fuer wesentlich groessere Fenster bekannt. Der Satz dient hier als forward analytic certificate dafuer, dass die neue Critical-half-Resolventenstruktur tatsaechlich einen Lower-Frame-Bound erzeugt, ohne bekannte Weil-Positivitaet rueckwaerts zu importieren.
+
+---
+
+# 9. Was dieser Durchlauf fuer Objekt X aendert
+
+Vor diesem Durchlauf war der Common-Jump-Engpass ein zweifach momentbeschraenktes Frameproblem. Jetzt stehen zusaetzlich zur exakten Common-Jump-Geometrie drei neue harte Tatsachen zur Verfuegung:
+
+1. `NULLPOL = Range(L_{1/2})` beseitigt die zwei Nebenbedingungen durch einen lokalen positiven Quellenoperator.
+2. Der P11 Prime-Power-/Hub-Ledger ist die logarithmische Abtastung des zugehoerigen Critical-half-OU-Kerns.
+3. Die Gamma-Schicht beginnt exakt mit demselben Resolventenoperator und liefert per Schur-Test bereits eigenstaendig eine echte Lower-Frame-Schranke.
+
+Die Route ist daher nicht mehr nur
+
+```text
+COMMON-JUMP -> abstract NP-GAP,
+```
+
+sondern
+
+```text
+COMMON-JUMP
+ -> critical-half rigidity
+ -> NULLPOL source lift L_{1/2}
+ -> OU/tree Prime geometry
+ -> Gamma resolvent ladder
+ -> analytic frame estimates.
+```
+
+Das schliesst NP-GAP global nicht, liefert aber erstmals innerhalb dieser Route einen expliziten coerciven Mechanismus und ein rigoroses Positivitaetszertifikat.
+
+---
+
+# 10. Naechster Gate
+
+Die kleine-Fenster-Schur-Schranke ist absichtlich grob. Der naechste sinnvolle Angriff ist nicht, ihren Radius numerisch zu optimieren, sondern die **Nullpol-/Range-Struktur** zu nutzen, die in §6--8 noch gar nicht verwendet wurde.
+
+Prioritaet:
+
+1. Schur-/Resolventenabschätzung nach `v=L_{1/2}u` statt fuer beliebiges `v`;
+2. endliche erste Gamma-Moden lokal in der `L_{1/2}`-Metrik behandeln, Rest per Schur;
+3. Prime-Jump-Energie ab dem ersten Cutoff nicht wegwerfen, sondern mit der Tree-/Root-Geometrie koppeln;
+4. suchen, ob daraus ein radiusstabiler Lower-Frame-Mechanismus entsteht.
+
+Ein Beweis fuer alle `a` bleibt RH-aequivalent und offen.
diff --git a/audits/P11_CRITICAL_RATE_LOCAL_RENORMALIZATION_NOGO_2026-09-15.md b/audits/P11_CRITICAL_RATE_LOCAL_RENORMALIZATION_NOGO_2026-09-15.md
new file mode 100644
index 00000000..e6106f3b
--- /dev/null
+++ b/audits/P11_CRITICAL_RATE_LOCAL_RENORMALIZATION_NOGO_2026-09-15.md
@@ -0,0 +1,258 @@
+# P11 Audit — Critical-rate local-renormalization no-go
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** enger Klassen-No-Go fuer die lokale Positivierung des Critical-half-Finite-Part-Transfers.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein No-Go gegen relative Determinanten, Streuung, Spektralverschiebung oder groessere nichtlokale positive Systeme.
+
+---
+
+## 0. Kurzurteil
+
+Der Critical-half-Finite-Part-Tail
+
+```math
+J_\Delta(L)
+=e^{L/2}
+\left(L-\gamma-\sum_{\log n<L}\frac{\Lambda(n)}{n}\right)
+```
+
+ist die endliche Differenz zweier bei der kritischen Rate einzeln divergenter positiver Tail-Amplituden. Der naheliegende Versuch,
+
+1. Prime- und Continuum-Amplitude in einem positiven `2x2`-Hamiltonian zu mischen und
+2. danach nur die gemeinsame skalare Divergenz von der Diagonale abzuziehen,
+
+kann **keinen positiven endlichen lokalen Hamiltonian** liefern.
+
+Schon am Root `L=0` ist der verbleibende nichttriviale Eigenwert
+
+```math
+2J_\Delta(0)=-2\gamma<0.
+```
+
+Damit ist die Klasse
+
+```text
+positive Prime+Continuum 2-channel Hamiltonian
++ common scalar counterterm
++ local finite positive limit
+```
+
+rigoros ausgeschlossen.
+
+Status:
+
+```text
+cutoff Prime amplitude positive                         ✓[M]
+cutoff continuum amplitude positive                     ✓[M]
+Hadamard 2-channel parent positive                      ✓[M]
+common-divergence subtraction finite                    ✓[M]
+positive local finite-part limit                        ×[M]
+relative / transfer / spectral-shift realization        ?[O]
+```
+
+---
+
+# 1. Positive cutoff amplitudes
+
+Fix `L>=0` and an upper depth cutoff `T>L`. At the critical amplitude rate define
+
+```math
+\boxed{
+J_P^{(T)}(L)
+=
+\sum_{L\le\log n\le T}
+\frac{\Lambda(n)}{\sqrt n}
+ e^{-(\log n-L)/2}
+=
+e^{L/2}
+\sum_{L\le\log n\le T}
+\frac{\Lambda(n)}{n}.
+}
+```
+
+This is positive.
+
+For the continuous pole compensator,
+
+```math
+\boxed{
+J_0^{(T)}(L)
+=
+\int_L^T
+ e^{-(t-L)/2}e^{t/2}dt
+=e^{L/2}(T-L),
+}
+```
+
+also positive.
+
+Their difference converges, in the canonical Abel/finite-part sense and equivalently under the usual prime-number-theorem asymptotics, to `J_Delta(L)`.
+
+---
+
+# 2. Canonical positive two-channel mixing
+
+Take the direct sum of the two positive scalar reservoirs and apply the fixed Hadamard rotation. The resulting local Hamiltonian is
+
+```math
+\boxed{
+\mathbb H_T(L)
+=
+\begin{pmatrix}
+J_P^{(T)}+J_0^{(T)} & J_P^{(T)}-J_0^{(T)}\\
+J_P^{(T)}-J_0^{(T)} & J_P^{(T)}+J_0^{(T)}
+\end{pmatrix}.
+}
+```
+
+Its eigenvalues are exactly
+
+```math
+2J_P^{(T)}(L),
+\qquad
+2J_0^{(T)}(L),
+```
+
+so
+
+```math
+\boxed{\mathbb H_T(L)\succeq0.}
+```
+
+The off-diagonal entry tends to the desired signed finite-part transfer coefficient `J_Delta(L)`.
+
+---
+
+# 3. The common diagonal divergence
+
+Both positive amplitudes have the same leading divergence
+
+```math
+J_P^{(T)}(L)
+=
+e^{L/2}(T-L)+J_\Delta(L)+o(1),
+```
+
+```math
+J_0^{(T)}(L)
+=
+e^{L/2}(T-L).
+```
+
+Hence the diagonal of `H_T` diverges like
+
+```math
+2e^{L/2}(T-L).
+```
+
+The canonical scalar counterterm that removes this common divergence is
+
+```math
+2J_0^{(T)}(L)I.
+```
+
+Subtracting it gives
+
+```math
+\mathbb H_T-2J_0^{(T)}I
+=
+\begin{pmatrix}
+J_P^{(T)}-J_0^{(T)} & J_P^{(T)}-J_0^{(T)}\\
+J_P^{(T)}-J_0^{(T)} & J_P^{(T)}-J_0^{(T)}
+\end{pmatrix}.
+```
+
+Its limit is
+
+```math
+\boxed{
+\mathbb H_{\rm fp}(L)
+=
+J_\Delta(L)
+\begin{pmatrix}1&1\\1&1\end{pmatrix}.
+}
+```
+
+The eigenvalues are
+
+```math
+0,
+\qquad
+2J_\Delta(L).
+```
+
+---
+
+# 4. Root obstruction
+
+From the exact finite-part formula,
+
+```math
+J_\Delta(0)=-\gamma.
+```
+
+Therefore
+
+```math
+\operatorname{spec}\mathbb H_{\rm fp}(0)
+=\{0,-2\gamma\}.
+```
+
+Since `gamma>0`,
+
+```math
+\boxed{
+\mathbb H_{\rm fp}(0)\not\succeq0.
+}
+```
+
+Thus positivity is destroyed already at the root by the canonical common-counterterm subtraction.
+
+More generally, whenever `J_Delta(L)<0`, the same local obstruction occurs.
+
+---
+
+# 5. Scope of the no-go
+
+The argument excludes only a very specific but natural class:
+
+```text
+- two local positive reservoirs (Prime and continuum),
+- fixed unitary/Hadamard mixing,
+- a common scalar diagonal counterterm removing the shared divergence,
+- positivity demanded after taking the finite local limit.
+```
+
+It does not exclude:
+
+```text
+- keeping the positive cutoff parent unrenormalized and extracting only a relative observable;
+- a relative Fredholm determinant;
+- Krein spectral shift / scattering phase;
+- a boundary transfer obtained before the cutoff is removed;
+- nonlocal coupling in L;
+- a larger conservative colligation;
+- an indefinite intermediate observable inside an overall positive Hilbert dilation.
+```
+
+---
+
+# 6. Strategic consequence
+
+The signed coefficient `J_Delta` is **not** to be treated as a renormalized positive Hamiltonian density.
+
+The only surviving use consistent with the hard audit is as a relative quantity:
+
+```math
+\boxed{
+J_\Delta
+=
+\text{Prime boundary response}
+-
+\text{free-continuum boundary response}.
+}
+```
+
+The next Object-X candidate therefore has to keep the positive Prime and continuum reservoirs themselves and recover the Weil term through a relative boundary observable (transfer matrix, scattering phase, determinant ratio, spectral shift, etc.), rather than by subtracting their energies pointwise.
diff --git a/audits/P11_CRITICAL_TRANSFER_AS_STOPPED_ROOT_TRACE_DEFECT_2026-09-16.md b/audits/P11_CRITICAL_TRANSFER_AS_STOPPED_ROOT_TRACE_DEFECT_2026-09-16.md
new file mode 100644
index 00000000..b32bbf65
--- /dev/null
+++ b/audits/P11_CRITICAL_TRANSFER_AS_STOPPED_ROOT_TRACE_DEFECT_2026-09-16.md
@@ -0,0 +1,295 @@
+# P11 Audit — Critical transfer as stopped Root-trace defect
+
+**Datum:** 16. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Registry:** unveraendert.  
+**Nonclaim:** kein Positivitaets-/RH-/Object-X-Abschluss.
+
+---
+
+## 0. Result
+
+The critical finite-part kernel
+
+```math
+J_\Delta(L)
+=e^{L/2}\left(
+L-\gamma-
+\sum_{\log n<L}\frac{\Lambda(n)}n
+\right)
+```
+
+has an exact P11 stopped-root interpretation.
+
+For the normalized OU/P11 branch at prime `p`, with `h_p=log p` and
+
+```math
+q_p=p^{-1/2},
+```
+
+the normalized root correlation of channel `k` is
+
+```math
+\langle\Phi_o,\Phi_{p,k}\rangle=q_p^k=p^{-k/2}.
+```
+
+Therefore the squared root correlation is `p^{-k}`, and the stopped visible root trace
+
+```math
+\boxed{
+M_P(L)
+:=\sum_{p,k:\,k\log p<L}
+(\log p)\,
+|\langle\Phi_o,\Phi_{p,k}\rangle|^2
+}
+```
+
+is exactly
+
+```math
+\boxed{
+M_P(L)
+=\sum_{\log n<L}\frac{\Lambda(n)}n.
+}
+```
+
+Hence
+
+```math
+\boxed{
+J_\Delta(L)=e^{L/2}\bigl(L-\gamma-M_P(L)\bigr).
+}
+```
+
+At a spatial P11 point `x` in window radius `R`, the stopped depth is
+
+```math
+L_R(x)=2(R-|x|).
+```
+
+Thus `J_Delta(L_R(x))` is determined directly by the root trace of precisely the P11 channels already visible at that boundary depth.
+
+The smooth reference `L` itself also splits into the already existing positive Riemann-R continuum root layer plus the Gamma-ground root layer. Consequently the whole finite-part coefficient is a relative stopped-root balance of the same geometry.
+
+---
+
+# 1. Prime stopped-root trace
+
+The windowless normalized prime-branch covariance is
+
+```math
+R_q(j,k)=q^{|j-k|}.
+```
+
+Its root component is `u_k=q^k`. Therefore the diagonal trace of the root projection, with the natural base-spacing weight `h_p=log p`, is
+
+```math
+\sum_k h_p|u_k|^2
+=\sum_k(\log p)p^{-k}.
+```
+
+Stopping at depth `L` retains exactly the indices satisfying
+
+```math
+k\log p<L.
+```
+
+Thus
+
+```math
+\begin{aligned}
+M_P(L)
+&=\sum_p\sum_{k\log p<L}(\log p)p^{-k}\\
+&=\sum_{n=p^k:\log n<L}\frac{\Lambda(n)}n.
+\end{aligned}
+```
+
+No prime-number theorem or RH is used.
+
+---
+
+# 2. Critical scaling converts root-square mass into Weil mass
+
+When `L` crosses a node
+
+```math
+L_0=k\log p,
+```
+
+the visible root trace jumps upward by
+
+```math
+\Delta M_P(L_0)
+=(\log p)p^{-k}.
+```
+
+Since
+
+```math
+J_\Delta(L)=e^{L/2}(L-\gamma-M_P(L)),
+```
+
+the corresponding jump of `J_Delta` is
+
+```math
+\begin{aligned}
+\Delta J_\Delta(L_0)
+&=-e^{L_0/2}(\log p)p^{-k}\\
+&=-\frac{\log p}{p^{k/2}}.
+\end{aligned}
+```
+
+Thus the Critical-half rescaling converts
+
+```text
+squared normalized root correlation:  p^{-k}
+                    into
+Weil prime-power weight:              p^{-k/2}.
+```
+
+This is exactly the atomic jump already required by
+
+```math
+d\Delta=\frac12J_\Delta dL-dJ_\Delta.
+```
+
+---
+
+# 3. The smooth root trace is the R-continuum plus Gamma ground
+
+The positive Riemann-R cell density satisfies
+
+```math
+\int_0^\infty\rho(h)
+\sum_{k\ge1}h\,\delta_{kh}(dt)dh
+=(e^t-1)dt.
+```
+
+The normalized OU root correlation at depth `t` has squared modulus `e^{-t}`. Therefore the stopped R-continuum root mass is
+
+```math
+\begin{aligned}
+M_R(L)
+&=\int_0^Le^{-t}(e^t-1)dt\\
+&=\int_0^L(1-e^{-t})dt\\
+&=L-1+e^{-L}.
+\end{aligned}
+```
+
+The Gamma-ground root contribution is
+
+```math
+\boxed{
+M_{\Gamma,0}(L)
+=\int_0^Le^{-t}dt
+=1-e^{-L}.
+}
+```
+
+Hence
+
+```math
+\boxed{M_R(L)+M_{\Gamma,0}(L)=L.}
+```
+
+The critical finite-part coefficient therefore has the exact relative-storage bookkeeping
+
+```math
+\boxed{
+e^{-L/2}J_\Delta(L)
+=M_R(L)+M_{\Gamma,0}(L)-M_P(L)-\gamma.
+}
+```
+
+Every nonconstant term on the right is a stopped positive root trace from geometry already present in the branch.
+
+---
+
+# 4. Relation to the spatial stopped filtration
+
+For the actual P11 masks,
+
+```math
+L_R(x)=2(R-|x|),
+\qquad
+m_p(x)=\left\lfloor\frac{L_R(x)}{\log p}\right\rfloor
+```
+
+away from threshold equalities. Hence the locally visible normalized root trace is exactly
+
+```math
+M_P(L_R(x)).
+```
+
+The overlap-cone Volterra coefficient from the critical finite-part audit can therefore be rewritten as
+
+```math
+\boxed{
+J_\Delta(L_R(x))
+=e^{R-|x|}
+\left(
+M_R(L_R(x))
++M_{\Gamma,0}(L_R(x))
+-M_P(L_R(x))
+-\gamma
+\right).
+}
+```
+
+Thus the scalar arithmetic transfer coefficient is not an external fitted function when restricted to the P11 overlap cone: it is the Critical-half rescaling of the stopped root-trace defect between the discrete prime tree and its positive smooth reference.
+
+---
+
+# 5. First-level versus higher prime powers
+
+Split
+
+```math
+M_P(L)
+=\sum_{p<e^L}\frac{\log p}{p}
++
+\sum_{\substack{k\ge2\\k\log p<L}}\frac{\log p}{p^k}.
+```
+
+The higher-power total converges absolutely:
+
+```math
+\boxed{
+\sum_p\sum_{k\ge2}\frac{\log p}{p^k}
+=\sum_p\frac{\log p}{p(p-1)}<\infty.
+}
+```
+
+Hence all critical large-depth renormalization lies in the first prime/root level `k=1`. The higher root levels are an absolutely summable correction.
+
+This is the time-domain counterpart of
+
+```math
+\log\zeta(S)
+=\sum_pp^{-S}
++\sum_{m\ge2}\frac1m\sum_pp^{-mS},
+```
+
+where the `m>=2` part converges absolutely for `Re S>1/2`.
+
+Consequently the unresolved global storage gate can be narrowed further:
+
+```text
+hard relative channel: first prime/root level versus smooth root continuum;
+higher prime powers: absolutely summable correction;
+local Euler cells: already passively subordinate to Gamma ground.
+```
+
+---
+
+# 6. Firewall
+
+The relative root trace
+
+```math
+M_R+M_{Gamma,0}-M_P-\gamma
+```
+
+changes sign. It is not a positive local storage density. The earlier root-only coupling no-go also remains valid: the scalar root trace cannot by itself repair the full higher-rank physical channel defect.
+
+What is new here is narrower: the **coefficient** of the already exact nonlocal Volterra transfer is now reconstructed canonically from the stopped P11 root filtration and its positive smooth reference. A successful Object-X mechanism would still have to place this signed relative root balance as a boundary/transfer term of a larger positive system.
diff --git a/audits/P11_CRIT_HALF_COMPRESS1_BAD_CHANNEL_POSITIVITY_2026-09-13.md b/audits/P11_CRIT_HALF_COMPRESS1_BAD_CHANNEL_POSITIVITY_2026-09-13.md
new file mode 100644
index 00000000..57e197a9
--- /dev/null
+++ b/audits/P11_CRIT_HALF_COMPRESS1_BAD_CHANNEL_POSITIVITY_2026-09-13.md
@@ -0,0 +1,677 @@
+# P11 Audit — CRIT-HALF-COMPRESS-1: bad-channel positivity at a=1/2
+
+**Datum:** 13. September 2026  
+**Basis:** Critical-half branch nach dem Prime-threshold bad-channel Audit.  
+**Rolle:** theorem-level Teilabschluss des `a=1/2`, `p=2` Kompressionsgates.  
+**Registry:** unveraendert.  
+**Nonclaim:** noch kein Beweis der vollen NULLPOL-Positivitaet bei `a=1/2`; kein globaler NP-GAP-/RH-Abschluss.
+
+---
+
+## 0. Kurzurteil
+
+Der exakt negative Prime-2-Kanal aus dem Threshold-Audit ist bei `a=1/2` **nicht** die verbleibende Obstruktion.
+
+Sei
+
+```math
+h=\log2,
+\qquad
+w_2=\frac{\log2}{\sqrt2},
+\qquad
+b=\frac12-\frac h2=\frac{1-\log2}{2}.
+```
+
+Der negative Kanal ist die isometrische symmetrische Randkopie
+
+```math
+S_+:\mathscr D_{NP,b}\to
+\mathscr D_{NP,1/2}\cap\operatorname{Ran}P_+.
+```
+
+Fuer jedes `g in D_{NP,b}` gilt auf diesem Kanal exakt
+
+```math
+Q_{1/2}(S_+g)
+=Q_\infty(g)
+-w_2\|g\|^2
+-\sum_{m\ge1}e^{-\mu_mh}
+\operatorname{Re}\bigl(E_{\mu_m,+}(g)\overline{E_{\mu_m,-}(g)}\bigr),
+```
+
+mit
+
+```math
+\mu_m=2m+\frac12,
+\qquad
+E_{\mu,+}(g)=\int e^{\mu x}g(x)dx,
+\qquad
+E_{\mu,-}(g)=\int e^{-\mu x}g(x)dx.
+```
+
+Der `m=0`-Cross-Term verschwindet **exakt** wegen NULLPOL.
+
+Eine rein vorwaerts gerichtete elementare Abschaetzung liefert
+
+```math
+\boxed{
+Q_{1/2}(S_+g)>\frac1{20}\|g\|^2.
+}
+```
+
+Damit ist der komplette isolierte negative Prime-2-Eigenkanal strikt positiv kontrolliert.
+
+Status:
+
+```text
+prime-2 bad-channel exact recursion                    ✓[M]
+critical gamma-ground cross term killed by NULLPOL     ✓[M]
+small-window archimedean source bound > 0.64 centered  ✓[M]
+remaining higher-gamma cross defect < 0.09             ✓[M]
+prime-2 bad-channel Q > 0.05                           ✓[M]
+full a=1/2 NULLPOL Schur complement                    ?[O]
+full NP-GAP / Object X / RH                            ?[O]
+```
+
+---
+
+# 1. Der schlechte Prime-2-Kanal
+
+Aus dem Threshold-Audit gilt fuer
+
+```math
+a=\frac12,
+\qquad
+h=\log2,
+```
+
+wegen
+
+```math
+\frac12<h<1
+```
+
+die nilpotente Randkanalzerlegung
+
+```math
+C_{2,1/2}
+=-w_2P_+ + w_2P_-.
+```
+
+Der negative Kanal `Ran P_+` ist isometrisch zu
+
+```math
+I_b=(-b,b),
+\qquad
+b=\frac{1-h}{2}.
+```
+
+Explizit
+
+```math
+(S_+g)(y-h/2)=\frac{g(y)}{\sqrt2},
+\qquad
+(S_+g)(y+h/2)=\frac{g(y)}{\sqrt2}.
+```
+
+Die beiden Kopien sind disjunkt und
+
+```math
+\|S_+g\|=\|g\|.
+```
+
+Ferner
+
+```math
+E_\pm(S_+g)
+=\sqrt2\cosh(h/4)E_\pm(g),
+```
+
+also
+
+```math
+S_+g\in\mathscr D_{NP,1/2}
+\iff
+g\in\mathscr D_{NP,b}.
+```
+
+Der Prime-2-Term ist auf diesem Kanal exakt
+
+```math
+\boxed{
+\langle S_+g,C_{2,1/2}S_+g\rangle
+=-w_2\|g\|^2.
+}
+```
+
+---
+
+# 2. Gamma-Moden in Realraumform
+
+Die positive archimedische Featureform besitzt die Resolventenzerlegung
+
+```math
+\Phi_\infty(D)
+=\sum_{m\ge0}A_m(D),
+\qquad
+A_m(D)=\frac2{\mu_m}\frac{D^2}{D^2+\mu_m^2},
+\qquad
+\mu_m=2m+\frac12.
+```
+
+Mit der Fourierkonvention des Projekts ist
+
+```math
+A_m
+=\frac2{\mu_m}I-H_{\mu_m},
+```
+
+wobei `H_mu` der positive Faltungsoperator mit Kern
+
+```math
+e^{-\mu|x-y|}
+```
+
+ist.
+
+Fuer zwei disjunkte Kopien von `g`, deren Mittelpunkte um `h` getrennt sind, verschwindet der Identitaets-Cross-Term. Der Green-Cross-Term faktorisiert dagegen exakt:
+
+```math
+\langle T_{-h/2}g,H_\mu T_{h/2}g\rangle
+=e^{-\mu h}
+E_{\mu,+}(g)\overline{E_{\mu,-}(g)}
+```
+
+(bis zur harmlosen Wahl, welcher Faktor konjugiert wird; in der quadratischen Form steht dessen Realteil).
+
+Daher
+
+```math
+\boxed{
+\langle S_+g,A_mS_+g\rangle
+=
+\langle g,A_mg\rangle
+-e^{-\mu_mh}
+\operatorname{Re}
+\bigl(E_{\mu_m,+}(g)\overline{E_{\mu_m,-}(g)}\bigr).
+}
+```
+
+Fuer den Grundmodus `mu_0=1/2` sind `E_{mu_0,±}` genau die NULLPOL-Momente. Also
+
+```math
+\boxed{
+\text{der gesamte }m=0\text{-Cross-Term verschwindet auf }\mathscr D_{NP,b}.
+}
+```
+
+Nach Summation und da die Norm unter `S_+` erhalten bleibt:
+
+```math
+\boxed{
+Q_\infty(S_+g)
+=Q_\infty(g)
+-\sum_{m\ge1}e^{-\mu_mh}
+\operatorname{Re}
+\bigl(E_{\mu_m,+}(g)\overline{E_{\mu_m,-}(g)}\bigr).
+}
+```
+
+Da auf `I_b` noch kein Prime-Power-Kanal aktiv ist, ist `Q_b(g)=Q_\infty(g)`.
+
+---
+
+# 3. Source-Lift auf dem kleineren Fenster
+
+Da `g in D_{NP,b}`, liefert die Critical-half Range-Identitaet eindeutig
+
+```math
+\boxed{
+g=L_{1/2}u,
+\qquad
+L_{1/2}=-\partial_x^2+\frac14,
+\qquad
+u\in C_c^\infty(-b,b).
+}
+```
+
+Schreibe
+
+```math
+A=\|u''\|_2^2,
+\qquad
+B=\|u'\|_2^2,
+\qquad
+C=\|u\|_2^2.
+```
+
+Dann
+
+```math
+\boxed{
+\|g\|^2=A+\frac12B+\frac1{16}C.
+}
+```
+
+---
+
+# 4. Scharfe Ableitungs-Poincare-Ungleichung
+
+Weil `u in C_c^infty(-b,b)`, liegt `w=u'` in `H_0^1(-b,b)` und erfuellt zusaetzlich
+
+```math
+\int_{-b}^{b}w(x)dx=u(b)-u(-b)=0.
+```
+
+## Lemma 4.1
+
+Fuer jedes `w in H_0^1(-b,b)` mit Integral null gilt
+
+```math
+\boxed{
+\|w'\|_2^2\ge\left(\frac\pi b\right)^2\|w\|_2^2.
+}
+```
+
+### Beweis-Skizze
+
+Das variationale Problem zerfaellt in Paritaet. Im ungeraden Sektor ist der erste Dirichlet-Modus auf `(0,b)` bereits
+
+```math
+\sin(\pi x/b)
+```
+
+mit Eigenwert `(pi/b)^2` und Integral null. Im geraden Sektor fuehrt die Lagrange-Multiplikator-Gleichung fuer die zusaetzliche Integralbedingung auf
+
+```math
+\tan(kb)=kb;
+```
+
+der erste positive Root liegt strikt oberhalb `pi`, also ist der gerade constrained Eigenwert groesser. Damit ist `(pi/b)^2` die scharfe Unterkante.
+
+Auf `u` angewandt:
+
+```math
+\boxed{
+A\ge\Lambda B,
+\qquad
+\Lambda=\left(\frac\pi b\right)^2.
+}
+```
+
+Aus der gewoehnlichen Dirichlet-Poincare-Ungleichung folgt ausserdem
+
+```math
+\boxed{
+B\ge\lambda C,
+\qquad
+\lambda=\left(\frac\pi{2b}\right)^2=\frac\Lambda4.
+}
+```
+
+---
+
+# 5. Die ersten neun Gamma-Moden liefern bereits 6.02
+
+Fuer `mu>0` setze
+
+```math
+R_\mu=(D^2+\mu^2)^{-1}.
+```
+
+Wie im vorherigen Schur-Audit gilt auf `I_b`
+
+```math
+\langle u,R_\mu u\rangle
+\le\frac{1-e^{-\mu b}}{\mu^2}\|u\|^2.
+```
+
+Setze `c=1/2` und `d_mu=mu^2-c^2`. Polynomdivision liefert exakt
+
+```math
+\begin{aligned}
+\langle L_cu,A_\mu L_cu\rangle
+=\frac2\mu\Bigl[
+&A+(2c^2-\mu^2)B+d_\mu^2C\\
+&-\mu^2d_\mu^2\langle u,R_\mu u\rangle
+\Bigr].
+\end{aligned}
+```
+
+Daher insbesondere
+
+```math
+\boxed{
+\langle g,A_\mu g\rangle
+\ge
+\frac2\mu\left[A+\left(\frac12-\mu^2\right)B\right],
+}
+```
+
+weil der verbleibende `C`-Term nach dem Schur-Test gleich
+
+```math
+\frac2\mu d_\mu^2e^{-\mu b}C\ge0
+```
+
+ist und fuer eine Untergrenze verworfen werden darf.
+
+Summiere nur die ersten neun Modi `m=0,...,8`. Setze
+
+```math
+\alpha
+:=\sum_{m=0}^8\frac2{\mu_m}
+=\sum_{m=0}^8\frac4{4m+1}
+=\frac{710302388}{111035925},
+```
+
+```math
+\beta
+:=\sum_{m=0}^8\frac2{\mu_m}
+\left(\frac12-\mu_m^2\right)
+=-\frac{16633345331}{111035925}.
+```
+
+Dann
+
+```math
+\langle g,\Phi_\infty g\rangle
+\ge\alpha A+\beta B.
+```
+
+Wir verwenden nur die sehr groben elementaren Schranken
+
+```math
+\log2>0.69,
+\qquad
+\pi>3.14.
+```
+
+Daraus
+
+```math
+b=\frac{1-\log2}{2}<\frac{31}{200},
+```
+
+und somit
+
+```math
+\Lambda
+=\left(\frac\pi b\right)^2
+>
+\left(\frac{628}{31}\right)^2
+=:\Lambda_0.
+```
+
+Mit
+
+```math
+r:=\frac{301}{50}=6.02
+```
+
+rechnet man rein rational nach:
+
+```math
+(\alpha-r)\Lambda_0+\beta-\frac r2>0,
+```
+
+und nach nochmaliger Verwendung `B>=Lambda_0 C/4` sogar
+
+```math
+\left[(\alpha-r)\Lambda_0+\beta-\frac r2\right]
+\frac{\Lambda_0}{4}
+-\frac r{16}>0.
+```
+
+Damit
+
+```math
+\alpha A+\beta B
+-r\left(A+\frac12B+\frac1{16}C\right)>0
+```
+
+fuer jedes nichttriviale `u`. Also
+
+```math
+\boxed{
+\langle g,\Phi_\infty g\rangle
+>6.02\|g\|^2.
+}
+```
+
+Alle Gamma-Moden `m>=9` wurden dabei einfach als positive Formen verworfen.
+
+---
+
+# 6. Zentrierte archimedische Reserve > 0.64
+
+Die archimedische Schwelle ist
+
+```math
+\kappa_*
+=\log(8\pi)+\gamma+\frac\pi2.
+```
+
+Mit den klassischen elementaren Schranken
+
+```math
+\pi<3.142,
+\qquad
+\gamma<0.578,
+\qquad
+\log(8\pi)<3.225
+```
+
+folgt
+
+```math
+\boxed{\kappa_*<5.38.}
+```
+
+(Die Log-Schranke folgt z.B. aus `8pi<25.136` und einer positiven Taylor-Untergrenze fuer `exp(3.225)>25.136`; die angezeigten Dezimalzahlen koennen durch die entsprechenden rationalen Zahlen `1571/500`, `289/500`, `129/40` ersetzt werden.)
+
+Somit
+
+```math
+\boxed{
+Q_\infty(g)
+=\langle g,\Phi_\infty g\rangle-\kappa_*\|g\|^2
+>0.64\|g\|^2.
+}
+```
+
+---
+
+# 7. Der hoehere Gamma-Cross-Defekt ist < 0.09
+
+Cauchy-Schwarz auf `(-b,b)` liefert
+
+```math
+|E_{\mu,+}(g)|^2
+\le\frac{\sinh(2\mu b)}{\mu}\|g\|^2,
+```
+
+und denselben Bound fuer `E_{mu,-}`. Also
+
+```math
+\left|
+\operatorname{Re}
+(E_{\mu,+}(g)\overline{E_{\mu,-}(g)})
+\right|
+\le
+\frac{\sinh(2\mu b)}{\mu}\|g\|^2.
+```
+
+Daher ist der gesamte `m>=1`-Cross-Defekt hoechstens
+
+```math
+R
+:=\sum_{m\ge1}
+\frac{e^{-\mu_mh}\sinh(2\mu_mb)}{\mu_m}.
+```
+
+Im Spezialfall
+
+```math
+h=\log2,
+\qquad
+2b=1-\log2
+```
+
+wird
+
+```math
+\boxed{
+R
+=\frac12\sum_{m\ge1}
+\frac{(e/4)^{\mu_m}-e^{-\mu_m}}{\mu_m}.
+}
+```
+
+Wir brauchen nur eine sehr grobe rationale Majorante. Aus `e<2.72` folgen
+
+```math
+\frac e4<\frac{17}{25}=0.68,
+\qquad
+e^{-1}>\frac{367}{1000}=0.367.
+```
+
+Ferner
+
+```math
+\sqrt{17/25}<33/40,
+\qquad
+\sqrt{367/1000}>121/200.
+```
+
+Die ersten drei Summanden werden mit diesen rationalen Schranken direkt majorisiert; fuer den Rest verwendet man `1/(4m+1)<=1/17` und die geometrische Reihe in `(17/25)^2`. Der rein rationale Wert der so erhaltenen Majorante ist kleiner als `9/100`.
+
+Somit
+
+```math
+\boxed{R<0.09.}
+```
+
+---
+
+# 8. Der Prime-2-Verlust ist < 0.5
+
+Mit
+
+```math
+\log2<0.7,
+\qquad
+\sqrt2>1.4
+```
+
+folgt sofort
+
+```math
+\boxed{
+w_2=\frac{\log2}{\sqrt2}<\frac12.}
+```
+
+---
+
+# 9. Bad-channel theorem
+
+Kombiniere §§2, 6, 7, 8. Fuer jedes nichttriviale `g in D_{NP,b}` gilt
+
+```math
+\begin{aligned}
+Q_{1/2}(S_+g)
+&\ge
+Q_\infty(g)-w_2\|g\|^2-R\|g\|^2\\
+&>
+(0.64-0.50-0.09)\|g\|^2.
+\end{aligned}
+```
+
+Daher
+
+```math
+\boxed{
+Q_{1/2}(S_+g)>\frac1{20}\|g\|^2.
+}
+```
+
+Da `S_+` isometrisch ist:
+
+```math
+\boxed{
+Q_{1/2}(v)>\frac1{20}\|v\|^2
+\qquad
+\forall\,0\ne v\in
+\mathscr D_{NP,1/2}\cap\operatorname{Ran}P_+.
+}
+```
+
+Dies ist ein echter theorem-level PASS fuer den **gesamten isolierten negativen Prime-2-Eigenkanal**.
+
+---
+
+# 10. Was noch offen bleibt
+
+Die volle Form zerfaellt relativ zu
+
+```math
+\mathscr D_{NP,1/2}
+=(P_+\mathscr D_{NP})\oplus
+(P_+\mathscr D_{NP})^\perp
+```
+
+nicht orthogonal fuer die archimedische Form. Obwohl
+
+- der Prime-Term auf `P_+` strikt negativ, aber jetzt kontrolliert ist,
+- der Prime-Term auf `P_-` positiv ist,
+- der Prime-Term auf dem mittleren Kanal null ist,
+
+besitzt `Q_\infty` archimedische Kreuzblöcke zwischen diesen Unterraeumen.
+
+Daher folgt aus der positiven Diagonalrestriktion auf `P_+` **noch nicht** die Positivitaet der vollen Matrix.
+
+Der verbleibende Gate ist jetzt exakt ein Schur-/Shorting-Problem:
+
+```math
+\boxed{
+\operatorname{Short}_{P_+}
+\bigl(Q_\infty+w_2P_-\bigr)
+\stackrel?\succeq
+w_2P_+.
+}
+```
+
+Dies ist wesentlich enger als der vorherige abstrakte `a=1/2`-Gate.
+
+---
+
+# 11. Strategische Konsequenz
+
+Der bisher naheliegendste Verdacht war, dass der direkte negative Prime-2-Kanal selbst den `a=1/2`-Gate blockiert. Das ist nun ausgeschlossen.
+
+Die verbleibende Schwierigkeit ist **nicht**
+
+```text
+negative prime diagonal mass,
+```
+
+sondern ausschliesslich
+
+```text
+archimedean off-diagonal coupling
+between the localized bad channel and its complement.
+```
+
+Damit wird `CRIT-HALF-COMPRESS-1` zu einem echten, lokalisierten Schur-Komplementproblem. Der naechste Angriff soll genau den archimedischen Cross-Block mit der Gamma-Resolventen-/Green-Markov-Struktur kontrollieren.
+
+---
+
+# 12. Firewalls
+
+Nicht behauptet wird:
+
+- volle Positivitaet auf `D_{NP,1/2}` sei bewiesen;
+- die archimedischen Kreuzblöcke seien klein oder positiv;
+- externe bekannte Weil-Positivitaet bei `a=1/2` werde als Beweisinput verwendet;
+- die angezeigte `1/20`-Reserve sei optimal;
+- NP-GAP fuer alle Fenster, Object X oder RH seien geloest.
diff --git a/audits/P11_DISCRETE_CONTINUOUS_PRECISION_UNIFICATION_2026-09-15.md b/audits/P11_DISCRETE_CONTINUOUS_PRECISION_UNIFICATION_2026-09-15.md
new file mode 100644
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--- /dev/null
+++ b/audits/P11_DISCRETE_CONTINUOUS_PRECISION_UNIFICATION_2026-09-15.md
@@ -0,0 +1,176 @@
+# P11 Audit — Discrete/continuous Critical-half precision unification
+
+**Datum:** 15. September 2026
+
+## 0. Result
+
+Define the universal Hilbert-valued jump field
+
+```math
+\boxed{X_v(t)=e^{-t/4}K_tv.}
+```
+
+The Gamma ground-mode energy is
+
+```math
+\boxed{
+E_{\Gamma,0}(v)
+=\int_0^\infty\|X_v(t)\|^2dt.
+}
+```
+
+Let
+
+```math
+Y(u)=\int_u^\infty e^{-(t-u)/2}X_v(t)dt.
+```
+
+Then
+
+```math
+(1/2-\partial_u)Y=X_v
+```
+
+and therefore
+
+```math
+\boxed{
+E_{\Gamma,0}(v)
+=\int_0^\infty
+\|Y'(u)-Y(u)/2\|^2du.
+}
+```
+
+For a prime `p`, put `h=log p`, `q=e^{-h/2}=p^{-1/2}` and sample
+
+```math
+\boxed{
+X_{p,k}=\sqrt h\,X_v(kh)
+=\sqrt{\frac{\log p}{p^{k/2}}}\,K_{k\log p}v.
+}
+```
+
+Define the P11/AR(1) tail state
+
+```math
+Y_{p,m}
+=\sqrt{1-q^2}\sum_{k\ge m}q^{k-m}X_{p,k}.
+```
+
+Then
+
+```math
+Y_{p,m}-qY_{p,m+1}=\sqrt{1-q^2}X_{p,m},
+```
+
+hence
+
+```math
+\boxed{
+\sum_{k\ge1}\|X_{p,k}\|^2
+=\frac1{1-q^2}
+\sum_{m\ge1}\|Y_{p,m}-qY_{p,m+1}\|^2.
+}
+```
+
+The left side is exactly the full positive Weil-weighted p-power jump tower.
+
+Thus the Gamma ground continuum and every prime-power tower are the same Critical-half precision geometry: continuous in `t` on the archimedean side, sampled at `t=k log p` on the prime side.
+
+**Status:** `✓[M]` identity; no positivity/RH conclusion beyond the already positive individual energies.
+
+---
+
+## 1. Continuum limit
+
+As `h downarrow 0`,
+
+```math
+q=e^{-h/2}=1-h/2+O(h^2),
+\qquad
+1-q^2=h+O(h^2).
+```
+
+Moreover
+
+```math
+X_{p,k}=\sqrt h X_v(kh).
+```
+
+The discrete tail state is a Riemann-sum approximation to
+
+```math
+Y(u)=\int_u^\infty e^{-(t-u)/2}X_v(t)dt,
+```
+
+and
+
+```math
+\frac{Y_{p,m}-qY_{p,m+1}}{\sqrt{1-q^2}}
+=X_{p,m}
+```
+
+is the exact discrete counterpart of
+
+```math
+(1/2-\partial_u)Y=X_v.
+```
+
+Consequently the discrete precision sum converges to the continuous Dirichlet/precision energy on smooth compact data.
+
+---
+
+## 2. Relation to P11
+
+The operator
+
+```math
+(T_qX)_m
+=\sqrt{1-q^2}\sum_{k\ge m}q^{k-m}X_k
+```
+
+is exactly the P11 AR(1) tail/innovation factor already present in the project. Its inverse is the local precision step
+
+```math
+D_q=(I-qS)/\sqrt{1-q^2}.
+```
+
+Hence the diagonal positive COMMON-JUMP p-tower is not an unrelated metric placed on top of P11: it is the **canonical precision norm of the P11 OU tail state**.
+
+This is different from the previously excluded claim that `T_q` itself should be a contraction from one feature metric to another. No contraction is used here. The metric is the intrinsic inverse-covariance/precision metric of the OU state.
+
+---
+
+## 3. Scale-quadrature viewpoint
+
+Writing
+
+```math
+g_v(t)=\|X_v(t)\|^2
+=e^{-t/2}\|K_tv\|^2,
+```
+
+one has
+
+```math
+E_{\Gamma,0}(v)=\int_0^\infty g_v(t)dt,
+```
+
+whereas for a prime `p`
+
+```math
+E_p(v)=\sum_{k\ge1}\|X_{p,k}\|^2
+=(\log p)\sum_{k\ge1}g_v(k\log p).
+```
+
+So every prime tower is literally a logarithmic-grid quadrature of the same nonnegative energy density whose continuum integral is the Gamma ground mode.
+
+The positive Riemann-R cell density from the companion audit mixes these grid quadratures over all spacings `h`; genuine primes select the discrete spacings `h=log p`.
+
+---
+
+## 4. Firewall
+
+This identity does not supply the required global inequality. A nonnegative function can have Riemann sums above or below its continuum integral, and the source-induced densities `g_v` are not monotone in `t` in general.
+
+The remaining Object-X problem is therefore a **structured scale-sampling/storage problem**, not a local positivity problem for any one prime or gamma mode.
diff --git a/audits/P11_EULERIZED_AR1_PRIME_TOWER_COMPRESSION_2026-09-15.md b/audits/P11_EULERIZED_AR1_PRIME_TOWER_COMPRESSION_2026-09-15.md
new file mode 100644
index 00000000..d4a58107
--- /dev/null
+++ b/audits/P11_EULERIZED_AR1_PRIME_TOWER_COMPRESSION_2026-09-15.md
@@ -0,0 +1,382 @@
+# P11 Audit — Eulerized AR(1) prime-tower compression
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Prime-Kompression nach dem Hard Audit.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein NP-GAP-/Object-X-/RH-Beweis; keine Publikationsneuheit behauptet.
+
+---
+
+## 0. Kurzurteil
+
+Der gesamte Weil-gewichtete Prime-Power-Jump-Turm einer festen Primzahl `p`
+laesst sich exakt auf **einen** stabilen AR(1)-/Euler-Zustand komprimieren.
+
+Setze
+
+```math
+h=\log p,
+\qquad q=p^{-1/2},
+\qquad U=T_h.
+```
+
+Definiere
+
+```math
+\boxed{
+B_p
+:=
+\sqrt{\frac{h q(1+q)}{1-q}}
+\,(U-I)(I-qU)^{-1}.
+}
+```
+
+Dann gilt auf `L^2(R)` exakt
+
+```math
+\boxed{
+B_p^*B_p
+=
+\sum_{k\ge1}h q^k K_{kh}^*K_{kh},
+}
+```
+
+also
+
+```math
+\boxed{
+\|B_pv\|_2^2
+=
+\sum_{k\ge1}
+\frac{\log p}{p^{k/2}}
+\|K_{k\log p}v\|_2^2.
+}
+```
+
+Der Nenner `(I-qU)^(-1)` ist derselbe AR(1)-Resolvent, der im P11-Cholesky-/Innovationsfaktor auftritt. Die neue Operation ist die kanonische Eulerisierung des Kanalindex-Shifts durch den physischen Log-p-Shift `U=T_logp`.
+
+Noch staerker: `B_p` stammt aus einer exakten positiven-real Storage-Identitaet mit demselben Zustandspol `q`.
+
+Status:
+
+```text
+prime tower one-state compression                         ✓[M]
+AR(1)/Euler denominator q=p^-1/2                         ✓[M]
+passive KYP/storage identity                              ✓[M]
+centered local-finiteness without prime-power cutoff      ✓[M]
+exact PR91 Prime-2 mixed calibration                      ✓[M]
+full lower-frame / Object X                               ?[O]
+```
+
+---
+
+# 1. Operator identity
+
+Recall
+
+```math
+K_t=T_{t/2}-T_{-t/2},
+\qquad
+K_t^*K_t=2I-T_t-T_{-t}.
+```
+
+For the fixed prime `p`, let `U=T_h`, `h=log p`. Then
+
+```math
+\sum_{k\ge1}h q^k K_{kh}^*K_{kh}
+=
+\sum_{k\ge1}h q^k(2I-U^k-U^{-k}).
+```
+
+Since `0<q<1`, the geometric series converge in operator norm:
+
+```math
+\sum_{k\ge1}q^k U^k
+=qU(I-qU)^{-1},
+```
+
+and similarly for `U^*`.
+
+On the other hand,
+
+```math
+B_p
+=
+\sqrt{\frac{hq(1+q)}{1-q}}
+(U-I)(I-qU)^{-1}.
+```
+
+Because `U` is unitary, functional calculus on the unit circle reduces the equality to
+
+```math
+\frac{hq(1+q)}{1-q}
+\frac{|z-1|^2}{|1-qz|^2}
+=
+\sum_{k\ge1}hq^k(2-z^k-z^{-k}),
+\qquad |z|=1.
+```
+
+The right side is the elementary Poisson/geometric identity
+
+```math
+2h\sum_{k\ge1}q^k(1-\cos k\theta)
+=
+\frac{hq(1+q)}{1-q}
+\frac{|e^{i\theta}-1|^2}{|1-qe^{i\theta}|^2}.
+```
+
+Hence the operator identity follows.
+
+---
+
+# 2. One-state form
+
+Set
+
+```math
+s_q:=\sqrt{1-q^2},
+\qquad
+\alpha_q:=\sqrt{\frac{1+q}{1-q}}
+=\frac{1+q}{s_q}.
+```
+
+Define the causal/logarithmic AR(1) state
+
+```math
+\boxed{
+\xi_p
+:=s_q U(I-qU)^{-1}v.
+}
+```
+
+Equivalently,
+
+```math
+\boxed{
+\xi_p=qU\xi_p+s_qUv.
+}
+```
+
+Then
+
+```math
+\xi_p-\alpha_qv
+=
+\alpha_q(U-I)(I-qU)^{-1}v,
+```
+
+and therefore
+
+```math
+\boxed{
+B_pv
+=\sqrt{hq}\,(\xi_p-\alpha_qv).
+}
+```
+
+Thus the full infinite prime-power positive Gram tower is represented by a single scalar AR(1) state in one copy of `L^2(R)`.
+
+This is not the native P11 Rest map applied directly to the jump-channel sequence. It is a **mediated Eulerization** of the same AR(1) recurrence under the intrinsic identification `k -> k log p`. This distinction is essential because the native P11 Rest/Feshbach pairing is already known to miss the exact Prime-2 mixed calibration.
+
+---
+
+# 3. Positive-real realization
+
+On a fibre modulo `h`, use a discrete index `n` and define
+
+```math
+x_{n+1}=q x_n+s_q u_n,
+```
+
+```math
+y_n
+=-\frac{hq}{s_q}x_n
++\frac{hq}{1-q}u_n.
+```
+
+The transfer function, with the standard one-step state delay, is
+
+```math
+\boxed{
+H_p(z)
+=\frac{hq(1-z)}{(1-q)(1-qz)}
+=h\sum_{k\ge1}q^k(1-z^k).
+}
+```
+
+Set the storage coefficient
+
+```math
+\boxed{
+P_p:=\frac{hq}{1-q^2}>0.
+}
+```
+
+A direct expansion gives the exact KYP/storage identity
+
+```math
+\boxed{
+2\operatorname{Re}(\overline{u_n}y_n)
+-P_p(|x_{n+1}|^2-|x_n|^2)
+=hq\left|
+x_n-\sqrt{\frac{1+q}{1-q}}u_n
+\right|^2
+\ge0.
+}
+```
+
+The associated `2x2` KYP matrix has determinant zero and is positive semidefinite of rank one. Hence this is a lossless/passive one-state realization of the full p-power positive-real jump response.
+
+Summing over a fibre for compactly supported input telescopes the storage term, and the total dissipation is exactly the prime-tower Gram energy from Section 1.
+
+---
+
+# 4. Centered window form is cutoff-free per prime
+
+Let `v` be supported in `[-a,a]`. Define
+
+```math
+c_p:=\sum_{k\ge1}hq^k=\frac{hq}{1-q}.
+```
+
+The centered p-block is
+
+```math
+Q_p(v)
+:=
+\sum_{k\ge1}hq^k
+\left(\|K_{kh}v\|^2-2\|v\|^2\right).
+```
+
+For every `kh>2a`, the shifted supports are disjoint and
+
+```math
+\|K_{kh}v\|^2=2\|v\|^2.
+```
+
+Therefore the centered sum is actually finite on every compact support and
+
+```math
+\boxed{
+Q_p(v)
+=\|B_pv\|^2-2c_p\|v\|^2.
+}
+```
+
+If `h=log p>2a`, every term is exterior and
+
+```math
+Q_p(v)=0.
+```
+
+Thus the Prime-Power cutoff is unnecessary at the level of the centered one-state p-block: local finiteness is automatic.
+
+---
+
+# 5. Exact Prime-2 firewall PASS
+
+Use the exact PR #91 mixed witness `a,b` with
+
+```math
+\langle a,b\rangle=0,
+```
+
+and whose correlation support meets the Prime-Power lattice only at `log 2`, with
+
+```math
+g_{a,b}(\log2)=1/2.
+```
+
+The exact Weil Prime-2 mixed target is
+
+```math
+-\frac{\log2}{\sqrt2}.
+```
+
+By the operator identity,
+
+```math
+\langle B_2a,B_2b\rangle
+=
+\sum_{k\ge1}\frac{\log2}{2^{k/2}}
+\langle K_{k\log2}a,K_{k\log2}b\rangle.
+```
+
+All `k>=2` terms vanish by support. For `k=1`,
+
+```math
+\langle K_{\log2}a,K_{\log2}b\rangle=-1.
+```
+
+Hence
+
+```math
+\boxed{
+\langle B_2a,B_2b\rangle
+=-\frac{\log2}{\sqrt2},
+}
+```
+
+exactly the required mixed Weil calibration.
+
+This is important relative to the earlier native P11 Rest/Feshbach firewalls: the Eulerized one-state mediator is outside those failed native classes and passes their calibration witness with no fitted coefficient.
+
+---
+
+# 6. Relation to the local Euler factor
+
+The internal state denominator is
+
+```math
+(1-qz)^{-1}.
+```
+
+With
+
+```math
+q=p^{-1/2},
+\qquad z=e^{-s\log p},
+```
+
+this becomes
+
+```math
+\boxed{
+\frac1{1-p^{-(s+1/2)}},
+}
+```
+
+the local Euler factor. Thus the same one-state recurrence simultaneously encodes:
+
+```text
+P11 AR(1) transition q
+Prime-power geometric tail
+local Euler denominator
+positive prime jump tower
+```
+
+No novelty claim is made for the individual state-space or Euler identities. The project-specific point is that the already-derived P11 AR(1) parameter yields an exact forward mediator for the complete prime-tower positive Gram and passes the pre-existing exact mixed-pair firewall.
+
+---
+
+# 7. What remains open
+
+The theorem compresses the positive Prime-Power feature family, but it does not remove the centered scalar
+
+```math
+2c_p I.
+```
+
+Summed over active primes and combined with the archimedean finite-part mass, these scalars are exactly the sharp NP-GAP threshold problem in another form.
+
+Therefore this result does **not** prove positivity. It improves the candidate state architecture:
+
+```math
+Prime powers for p
+   -> one canonical AR(1)/Euler state B_p
+Gamma modes
+   -> one canonical stable first-order state per mu_m.
+```
+
+The next gate is whether these already-passive local states admit a canonical nonorthogonal/global storage coupling (forced by the overlap-cone/stopped-OU geometry) that absorbs the centered feedthrough masses without importing Lagarias positivity or RH.
diff --git a/audits/P11_EULER_AR1_RESOLVENT_BRIDGE_2026-09-15.md b/audits/P11_EULER_AR1_RESOLVENT_BRIDGE_2026-09-15.md
new file mode 100644
index 00000000..2975a12b
--- /dev/null
+++ b/audits/P11_EULER_AR1_RESOLVENT_BRIDGE_2026-09-15.md
@@ -0,0 +1,329 @@
+# P11 Audit — Euler factors as adjoint AR(1) resolvents
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level lokale Transferidentitaet zwischen P11 und Eulerfaktoren.  
+**Registry:** unveraendert.  
+**Nonclaim:** Eulerprodukt/Determinantdarstellung ist klassisch; keine Publikationsneuheit.
+
+---
+
+## 0. Kurzurteil
+
+Der lokale Eulerfaktor einer Primzahl ist exakt die `z`-Transferfunktion des **adjungierten P11-AR(1)-Choleskyfaktors**.
+
+Fuer
+
+```math
+q=p^{-1/2}
+```
+
+und den einseitigen Shift `S` auf `ell^2(N_0)` gilt
+
+```math
+T_q
+=\sqrt{1-q^2}(I-qS^*)^{-1},
+```
+
+also
+
+```math
+\boxed{
+T_q^*
+=\sqrt{1-q^2}(I-qS)^{-1}.
+}
+```
+
+Am Root-Impuls `e_0`:
+
+```math
+\boxed{
+\frac1{\sqrt{1-q^2}}T_q^*e_0
+=(1,q,q^2,\ldots).
+}
+```
+
+Dessen `z`-Transformierte ist
+
+```math
+\boxed{
+\frac1{1-qz}.
+}
+```
+
+Mit dem physikalischen Delay
+
+```math
+z=e^{-s\log p}
+```
+
+wird dies exakt
+
+```math
+\boxed{
+\frac1{1-p^{-(s+1/2)}}.
+}
+```
+
+Der Prime-Anteil der logarithmischen Ableitung ist entsprechend
+
+```math
+\boxed{
+-\log p\,
+\frac{p^{-(s+1/2)}}{1-p^{-(s+1/2)}}
+=
+-h_p\left[
+\frac1{1-q_pe^{-sh_p}}-1
+\right].
+}
+```
+
+Damit sind P11-AR(1)-Tail und lokaler Euler-Feedbackloop nicht nur durch dieselbe geometrische Reihe verwandt: sie besitzen denselben Resolventenoperator, in adjungierter Kausalitaetsorientierung.
+
+Status:
+
+```text
+P11 tail = anti-causal AR1 resolvent                 ✓[M]
+adjoint P11 tail = causal AR1 resolvent              ✓[M]
+local Euler factor = z-transfer of adjoint resolvent ✓[M]
+prime log derivative = resolvent excess              ✓[M]
+Euler product determinant over primes                 classical / ✓[M]
+constructive global passive coupling                  ?[O]
+```
+
+---
+
+# 1. Shift orientation
+
+Use the unilateral right shift
+
+```math
+Se_k=e_{k+1}.
+```
+
+Then
+
+```math
+(I-qS^*)^{-1}
+=\sum_{r>=0}q^r(S^*)^r.
+```
+
+Hence
+
+```math
+\left[(I-qS^*)^{-1}X\right]_m
+=\sum_{k>=m}q^{k-m}X_k.
+```
+
+This is exactly the P11 tail orientation, so
+
+```math
+\boxed{
+T_q=\sqrt{1-q^2}(I-qS^*)^{-1}.
+}
+```
+
+Taking adjoints gives
+
+```math
+\boxed{
+T_q^*=\sqrt{1-q^2}(I-qS)^{-1}.
+}
+```
+
+The Gram identity remains
+
+```math
+R_q=T_q^*T_q+uu^*.
+```
+
+---
+
+# 2. Root impulse response
+
+Since
+
+```math
+(I-qS)^{-1}e_0
+=\sum_{r>=0}q^re_r,
+```
+
+we have
+
+```math
+\boxed{
+\frac1{\sqrt{1-q^2}}T_q^*e_0
+=(1,q,q^2,\ldots).
+}
+```
+
+This is the canonical causal impulse response of the AR(1) feedback loop.
+
+---
+
+# 3. Transfer function
+
+For the sequence `z`-transform,
+
+```math
+\sum_{r>=0}q^rz^r
+=\frac1{1-qz}
+```
+
+whenever `|qz|<1`.
+
+For a physical delay step
+
+```math
+h_p=\log p,
+```
+
+the Laplace delay is
+
+```math
+z=e^{-sh_p}.
+```
+
+Since
+
+```math
+q_p=e^{-h_p/2},
+```
+
+we obtain
+
+```math
+q_pz
+=e^{-(s+1/2)h_p}
+=p^{-(s+1/2)}.
+```
+
+Therefore
+
+```math
+\boxed{
+\mathcal Z
+\left[
+\frac1{\sqrt{1-q_p^2}}T_{q_p}^*e_0
+\right](e^{-sh_p})
+=
+\frac1{1-p^{-(s+1/2)}}.
+}
+```
+
+---
+
+# 4. Logarithmic derivative
+
+The local Euler factor is
+
+```math
+\zeta_p(s+1/2)
+=(1-p^{-(s+1/2)})^{-1}.
+```
+
+Its logarithmic derivative with respect to `s` is
+
+```math
+\frac{d}{ds}\log\zeta_p(s+1/2)
+=-\log p\,
+\frac{p^{-(s+1/2)}}{1-p^{-(s+1/2)}}.
+```
+
+Writing `q=p^{-1/2}`, `h=log p` gives
+
+```math
+\boxed{
+\frac{d}{ds}\log\zeta_p(s+1/2)
+=-h\frac{qe^{-sh}}{1-qe^{-sh}}
+=-h\left[
+\frac1{1-qe^{-sh}}-1
+\right].
+}
+```
+
+Thus the arithmetic transfer in the Lagarias decomposition is the weighted resolvent excess of the adjoint P11-AR(1) loop.
+
+---
+
+# 5. Finite-depth / boundary remainder
+
+For every integer `m>=0`,
+
+```math
+\boxed{
+\frac1{1-qz}
+=
+\sum_{r=0}^{m-1}(qz)^r
++
+\frac{(qz)^m}{1-qz}.
+}
+```
+
+The first term is the finite resolved impulse response; the second is one boundary state carrying the unresolved geometric future.
+
+This algebra is the scalar-transfer counterpart of the stopped-OU filtration and its boundary-tail shorting.
+
+Firewall: for a general physical channel sequence `X_k(v)`, the exterior tail is not restricted to this single geometric mode. The positive exterior defect proved in the boundary-shortening audit measures precisely the lost orthogonal directions. Therefore the exact Euler impulse-response truncation does not undo the existing Prime-2 calibration no-go for arbitrary sources.
+
+---
+
+# 6. Euler determinant over the prime loops
+
+For `Re s>1/2`, the diagonal prime operator
+
+```math
+Q(s)
+=\operatorname{diag}_{p}
+\left(p^{-(s+1/2)}\right)
+```
+
+is trace class because
+
+```math
+\sum_p p^{-Re(s)-1/2}<\infty.
+```
+
+Its Fredholm determinant is
+
+```math
+\boxed{
+\det(I-Q(s))
+=\prod_p(1-p^{-(s+1/2)})
+=\zeta(s+1/2)^{-1}.
+}
+```
+
+This determinant formula is simply the Euler product in operator notation and is not a novelty claim.
+
+What is project-specific is that the same scalar resolvent denominator had already appeared from the P11 Cholesky/innovation geometry independently of this determinant notation.
+
+---
+
+# 7. Spatial stopping interpretation
+
+The P11 masks set the visible depth
+
+```math
+m_p(x)=\left\lfloor\frac{2(R-|x|)}{\log p}\right\rfloor.
+```
+
+Thus, locally in the overlap cone, increasing the radius exposes more coefficients of the same AR(1)/Euler impulse response while retaining a boundary state for the unresolved future.
+
+At `m=0` all information sits in the conditional/root boundary state. Each increase `m -> m+1` releases one new positive OU innovation, matching the finite-depth resolvent expansion.
+
+Again, this is a structural interpretation only: the full Weil form contains source-dependent tail directions beyond the one-dimensional scalar Euler impulse mode.
+
+---
+
+# 8. Next gate
+
+The sharpened constructive problem is now:
+
+```text
+Can the locally finite stopped P11/Euler feedback network be coupled to the
+continuous pole compensator and higher Gamma filter bank so that the direct
+limit has a positive storage realization on Re s>0?
+```
+
+The infinite product itself is only trace-class for `Re s>1/2`; extending the positive realization to the larger half-plane is precisely where the RH difficulty remains.
diff --git a/audits/P11_EULER_CELL_DOMINATED_BY_GAMMA_GROUND_2026-09-16.md b/audits/P11_EULER_CELL_DOMINATED_BY_GAMMA_GROUND_2026-09-16.md
new file mode 100644
index 00000000..fba63a22
--- /dev/null
+++ b/audits/P11_EULER_CELL_DOMINATED_BY_GAMMA_GROUND_2026-09-16.md
@@ -0,0 +1,412 @@
+# P11 Audit — Every Euler cell is a contraction of the Gamma ground
+
+**Datum:** 16. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Registry:** unveraendert.  
+**Nonclaim:** kein RH-/Object-X-Beweis; keine globale Prime/Continuum-Kopplung.
+
+---
+
+## 0. Result
+
+For `h>0`, `q=e^{-h/2}`, let
+
+```math
+A_h
+:=\sum_{k\ge1}h e^{-kh/2}K_{kh}^*K_{kh}
+=B_h^*B_h,
+```
+
+where
+
+```math
+B_h
+=\sqrt{\frac{h q(1+q)}{1-q}}
+(T_h-I)(I-qT_h)^{-1}.
+```
+
+Let the Gamma ground operator be
+
+```math
+A_0
+:=\int_0^\infty e^{-t/2}K_t^*K_tdt
+=4D^2(D^2+1/4)^{-1}.
+```
+
+Then for every `h>0`,
+
+```math
+\boxed{0\preceq A_h\preceq A_0.}
+```
+
+Thus the strong `h->0` limit from the earlier audit is actually accompanied by a global Loewner domination valid at every positive scale.
+
+Equivalently there is a canonical commuting contraction `C_h` such that
+
+```math
+\boxed{A_h^{1/2}=C_hA_0^{1/2},\qquad \|C_h\|\le1.}
+```
+
+At the transfer-function level one may choose
+
+```math
+b_h(s)
+=\sqrt{\frac{h q(1+q)}{1-q}}
+\frac{1-e^{-hs}}{1-qe^{-hs}},
+\qquad
+b_0(s)=\frac{2s}{s+1/2},
+```
+
+and
+
+```math
+\boxed{\Phi_h(s):=\frac{b_h(s)}{b_0(s)}.}
+```
+
+The singularity at `s=0` is removable, all poles of `Phi_h` lie in `Re s<0`, and the boundary inequality proved below gives
+
+```math
+|\Phi_h(i\omega)|\le1.
+```
+
+Hence `Phi_h` is a Schur transfer on the right half-plane. Each complete Euler/AR(1) cell therefore has a coefficient-free passive relative realization with respect to the Gamma ground port.
+
+This does **not** solve the global storage problem: the orthogonal row of all prime contractions is unbounded. Thus the remaining difficulty is genuinely the non-diagonal relative scale coupling.
+
+---
+
+# 1. Exact multipliers
+
+The earlier `h=0` audit gives
+
+```math
+A_h=a_h(D),
+\qquad
+a_h(z)
+=h\left[\frac{1+q}{1-q}-P_q(hz)\right],
+```
+
+where
+
+```math
+P_q(\theta)
+=\frac{1-q^2}{1-2q\cos\theta+q^2}.
+```
+
+Also
+
+```math
+a_0(z)=\frac{4z^2}{z^2+1/4}.
+```
+
+Set
+
+```math
+x=\frac h4,
+\qquad
+y=\frac{hz}{2}.
+```
+
+Since `q=e^{-2x}`,
+
+```math
+\frac{1+q}{1-q}=\coth x,
+```
+
+and
+
+```math
+P_q(2y)
+=\frac{\sinh(2x)}{\cosh(2x)-\cos(2y)}.
+```
+
+Using
+
+```math
+\cosh(2x)-\cos(2y)
+=2(\sinh^2x+\sin^2y),
+```
+
+one obtains
+
+```math
+\boxed{
+a_h(z)
+=4x\coth x\,
+\frac{\sin^2y}{\sinh^2x+\sin^2y},
+}
+```
+
+while
+
+```math
+\boxed{
+a_0(z)=\frac{4y^2}{x^2+y^2}.}
+```
+
+Thus the desired operator order is the scalar inequality `a_h(z)<=a_0(z)`.
+
+---
+
+# 2. Circular auxiliary inequality
+
+For every real `y`,
+
+```math
+\boxed{\frac{y^2}{\sin^2y}\ge1+\frac{y^2}{3}}
+```
+
+with the usual interpretation at zeros of `sin y`.
+
+Equivalently
+
+```math
+\sin^2y\le\frac{3y^2}{3+y^2}.
+```
+
+For `y^2>=3/2` the right side is at least one, so the estimate is immediate.
+
+For `0<=y^2<=3/2`, alternating Taylor gives
+
+```math
+\frac{\sin y}{y}
+\le1-\frac{y^2}{6}+\frac{y^4}{120}.
+```
+
+Put `t=y^2`. Direct algebra gives
+
+```math
+1-
+\left(1-\frac t6+\frac{t^2}{120}\right)^2
+\left(1+\frac t3\right)
+=
+\frac{t^2(2880-520t+37t^2-t^3)}{43200}>0
+```
+
+on `0<=t<=3/2`. This proves the auxiliary inequality globally.
+
+---
+
+# 3. Two hyperbolic inequalities
+
+Put
+
+```math
+a(x):=\frac{\tanh x}{x}.
+```
+
+We need two elementary estimates.
+
+## 3.1 Lazarevic-type inequality
+
+```math
+\boxed{a(x)\sinh^2x\ge x^2.}
+```
+
+This is equivalent to
+
+```math
+\left(\frac{\sinh x}{x}\right)^3\ge\cosh x.
+```
+
+For a self-contained proof define
+
+```math
+F(x)=3\log\frac{\sinh x}{x}-\log\cosh x.
+```
+
+Then
+
+```math
+F'(x)=3\coth x-\frac3x-\tanh x.
+```
+
+Multiplying by `x sinh x cosh x>0` reduces the sign to
+
+```math
+G(x)=x(3+2\sinh^2x)-3\sinh x\cosh x.
+```
+
+Now
+
+```math
+G'(x)=4\sinh x\,(x\cosh x-\sinh x)\ge0,
+```
+
+because `(x cosh x-sinh x)'=x sinh x>=0`. Since `G(0)=0`, `F'>=0`; and `F(0)=0`, proving the claim.
+
+## 3.2 Coefficient inequality
+
+```math
+\boxed{a(x)-1+\frac{a(x)\sinh^2x}{3}\ge0.}
+```
+
+This is equivalent to
+
+```math
+\tanh x\,(3+\sinh^2x)\ge3x.
+```
+
+For
+
+```math
+H(x)=\tanh x\,(3+\sinh^2x)-3x
+```
+
+one gets exactly
+
+```math
+H'(x)=\frac{2\sinh^4x}{\cosh^2x}\ge0,
+```
+
+and `H(0)=0`.
+
+---
+
+# 4. Loewner domination
+
+For `sin y !=0`, the inequality `a_h<=a_0` is equivalent, after taking reciprocals, to
+
+```math
+\frac{\tanh x}{x}
+\left(1+\frac{\sinh^2x}{\sin^2y}\right)
+\ge
+1+\frac{x^2}{y^2}.
+```
+
+Multiplying by `y^2` gives the left-minus-right expression
+
+```math
+y^2(a-1)
++a\sinh^2x\frac{y^2}{\sin^2y}
+-x^2.
+```
+
+By Section 2 this is at least
+
+```math
+\bigl(a\sinh^2x-x^2\bigr)
++y^2\left(a-1+\frac{a\sinh^2x}{3}\right),
+```
+
+which is nonnegative by Sections 3.1 and 3.2.
+
+At zeros of `sin y`, `a_h=0`; at `y=0` the same conclusion follows by continuity. Hence
+
+```math
+\boxed{0\le a_h(z)\le a_0(z)\quad\text{for all real }z.}
+```
+
+Functional calculus yields
+
+```math
+\boxed{0\preceq A_h\preceq A_0.}
+```
+
+The previous aliasing result remains sharp: at `z=2pi n/h`, `a_h(z)=0` while `a_0(z)->4`, hence
+
+```math
+\|A_0-A_h\|=4.
+```
+
+---
+
+# 5. Canonical local Schur factor
+
+On the imaginary boundary,
+
+```math
+|b_h(i\omega)|^2=a_h(\omega),
+\qquad
+|b_0(i\omega)|^2=a_0(\omega).
+```
+
+Thus
+
+```math
+|\Phi_h(i\omega)|\le1.
+```
+
+The zeros at `s=0` cancel. The poles from `1-qe^{-hs}` satisfy
+
+```math
+s=-\frac12+\frac{2\pi i n}{h}
+```
+
+(up to the sign convention for the Fourier/Laplace exponential), hence lie strictly in the left half-plane. The transfer is bounded in the right half-plane and has finite limit at infinity. By the maximum principle,
+
+```math
+\boxed{\|\Phi_h\|_{H^\infty(\Re s>0)}\le1.}
+```
+
+Therefore each local Euler cell is passively subordinate to the Gamma ground cell.
+
+A canonical Julia colligation is supplied by the commuting contraction
+
+```math
+C_h=\Phi_h(D)
+```
+
+(or equivalently the positive multiplier `sqrt(a_h/a_0)` if only the energy factor is needed):
+
+```math
+\mathcal U_h=
+\begin{pmatrix}
+C_h & (I-C_h^*C_h)^{1/2}\\
+(I-C_hC_h^*)^{1/2} & -C_h^*
+\end{pmatrix}.
+```
+
+No Weil positivity or zero information is used in this local construction.
+
+---
+
+# 6. Why the local PASS does not globalize orthogonally
+
+Let
+
+```math
+r_h(0):=\lim_{z\to0}\frac{a_h(z)}{a_0(z)}.
+```
+
+From the formulas above,
+
+```math
+\boxed{
+r_h(0)
+=\frac{x^3\cosh x}{\sinh^3x},
+\qquad x=h/4.
+}
+```
+
+At a prime scale `h=log p`,
+
+```math
+r_{\log p}(0)
+\sim\frac{(\log p)^3}{16\sqrt p}.
+```
+
+Hence
+
+```math
+\sum_p r_{\log p}(0)=\infty.
+```
+
+Indeed for large primes this term dominates a positive constant multiple of `1/p`, and Euler's prime harmonic series diverges.
+
+Consequently the naive orthogonal row
+
+```math
+f\mapsto(C_{\log p}f)_p
+```
+
+from a single Gamma-ground port to all prime outputs is unbounded. Equivalently, local passive subordination does not furnish a global orthogonal direct-sum storage.
+
+This sharpens the remaining gate:
+
+```text
+local relative storage: solved exactly;
+global relative scale storage: still open and necessarily non-diagonal/renormalized.
+```
+
+The Riemann-R continuum or an overlap-cone scale coupling is therefore not optional if this branch is to produce an Object-X mechanism.
diff --git a/audits/P11_EULER_GAMMA_SCATTERING_PHASE_2026-09-13.md b/audits/P11_EULER_GAMMA_SCATTERING_PHASE_2026-09-13.md
new file mode 100644
index 00000000..db92e3e2
--- /dev/null
+++ b/audits/P11_EULER_GAMMA_SCATTERING_PHASE_2026-09-13.md
@@ -0,0 +1,481 @@
+# P11 Audit — Euler/Gamma scattering-phase realization
+
+**Datum:** 13. September 2026  
+**Basis:** Prime AR(1)-Poisson / Gamma harmonic bridge.  
+**Rolle:** theorem-level unitary phase representation des zentrierten Fensteroperators.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein Positivitaets-/RH-Beweis; allgemeine Zeta-Streuinterpretationen sind aus anderen Kontexten bekannt, daher keine Publikationsneuheit behauptet.
+
+---
+
+## 0. Kurzurteil
+
+Der zentrierte nichtpolare Weil-Multiplikator auf einem kompakten Fenster ist exakt die Phasenableitung eines **expliziten unitären endlichen Euler/Gamma-Streufaktors**.
+
+Fuer jede Primzahl definiere
+
+```math
+\boxed{
+S_p(z)
+:=
+\frac{1-p^{-1/2}e^{iz\log p}}
+     {1-p^{-1/2}e^{-iz\log p}}.
+}
+```
+
+Fuer reelles `z` gilt `|S_p(z)|=1` und
+
+```math
+\boxed{
+-i\overline{S_p(z)}S_p'(z)
+=
+\frac{d}{dz}\arg S_p(z)
+=
+(\log p)\left[1-P_{p^{-1/2}}^{\mathbb D}(z\log p)\right].
+}
+```
+
+Archimedisch setze
+
+```math
+\boxed{
+S_\infty(z)
+:=
+\pi^{-iz}
+\frac{\Gamma(1/4+iz/2)}
+     {\Gamma(1/4-iz/2)}.
+}
+```
+
+Auch `|S_infty(z)|=1` fuer reelles `z`, und exakt
+
+```math
+\boxed{
+-i\overline{S_\infty(z)}S_\infty'(z)
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi.
+}
+```
+
+Fuer ein Fenster `[-a,a]` definiere den endlichen Streufaktor
+
+```math
+\boxed{
+S_a(z)
+:=
+S_\infty(z)
+\prod_{p\le e^{2a}}S_p(z).
+}
+```
+
+Dann ist fuer alle reellen `z`
+
+```math
+\boxed{
+-i\overline{S_a(z)}S_a'(z)
+=
+B_{\infty}(z)
++
+\sum_{p\le e^{2a}}B_p(z),
+}
+```
+
+wobei `B_infty` der zentrierte archimedische und `B_p` der **vollstaendige** zentrierte `p`-Power-Multiplikator ist.
+
+Auf Funktionen mit Traeger in `[-a,a]` ist dies in der quadratischen Form exakt der zentrierte COMMON-JUMP-/Weil-Operator; die ueber den geometrischen Cutoff hinaus ergaenzten Prime powers tragen exakt null.
+
+Auf NULLPOL gilt daher
+
+```math
+\boxed{
+Q_W(v)
+=
+\int_{\mathbb R}
+|\widehat v(z)|^2
+\left[-i\overline{S_a(z)}S_a'(z)\right]dz.
+}
+```
+
+Der Weiloperator ist damit ein skalarer **Wigner--Smith / phase-delay operator** des expliziten Fenster-Streufaktors.
+
+Status:
+
+```text
+local Euler scattering factors S_p                     ✓[M]
+archimedean Gamma scattering factor S_infty             ✓[M]
+window product S_a unitary on real axis                 ✓[M]
+phase derivative = centered COMMON-JUMP multiplier      ✓[M]
+NULLPOL Weil form = Wigner-Smith quadratic form         ✓[M]
+positive scattering dilation / monotone time delay      ?[O]
+full positive Object X                                  ?[O]
+RH                                                      ?[O]
+publication novelty                                     ?[O]
+```
+
+---
+
+# 1. Lokaler Prime-Streufaktor
+
+Fixiere `p`, setze
+
+```math
+q=p^{-1/2},
+\qquad
+\theta=z\log p.
+```
+
+Dann
+
+```math
+S_p(z)
+=\frac{1-qe^{i\theta}}{1-qe^{-i\theta}}.
+```
+
+Fuer reelles `theta` sind Zaehler und Nenner komplex konjugiert, also
+
+```math
+|S_p(z)|=1.
+```
+
+Logarithmisch differenziert nach `theta`:
+
+```math
+\frac{d}{d\theta}\log S_p
+=
+-i\left(
+\frac{qe^{i\theta}}{1-qe^{i\theta}}
++
+\frac{qe^{-i\theta}}{1-qe^{-i\theta}}
+\right).
+```
+
+Somit
+
+```math
+\frac{d}{d\theta}\arg S_p
+=-2\operatorname{Re}
+\frac{qe^{i\theta}}{1-qe^{i\theta}}.
+```
+
+Aus
+
+```math
+P_q^{\mathbb D}(\theta)
+=1+2\operatorname{Re}
+\frac{qe^{i\theta}}{1-qe^{i\theta}}
+```
+
+folgt
+
+```math
+\boxed{
+\frac{d}{d\theta}\arg S_p
+=1-P_q^{\mathbb D}(\theta).
+}
+```
+
+Mit `d theta/dz=log p`:
+
+```math
+\boxed{
+-i\overline{S_p(z)}S_p'(z)
+=(\log p)(1-P_q^{\mathbb D}(z\log p)).
+}
+```
+
+Nach dem Poisson-Audit ist dies exakt der zentrierte gesamte Weil-Prime-Power-Turm von `p`.
+
+---
+
+# 2. Beziehung zum lokalen Eulerfaktor
+
+Schreibe
+
+```math
+\zeta_p(s)=(1-p^{-s})^{-1}.
+```
+
+Dann gilt direkt
+
+```math
+\boxed{
+S_p(z)
+=
+\frac{\zeta_p(1/2+iz)}{\zeta_p(1/2-iz)}.
+}
+```
+
+Die P11-AR(1)-Poisson-Dichte ist somit die Phasengeschwindigkeit des Quotienten der beiden kritischen Randwerte des lokalen Eulerfaktors.
+
+Firewall: Der endliche Fensterfaktor verwendet nur endlich viele Primzahlen. Ein unkritisches Produkt ueber **alle** Primzahlen auf `Re(s)=1/2` wird nicht gebildet; dort konvergiert das Eulerprodukt nicht klassisch.
+
+---
+
+# 3. Archimedischer Streufaktor
+
+Setze
+
+```math
+S_\infty(z)
+=\pi^{-iz}
+\frac{\Gamma(1/4+iz/2)}
+     {\Gamma(1/4-iz/2)}.
+```
+
+Fuer reelles `z` sind die Gammawerte komplex konjugiert; daher
+
+```math
+|S_\infty(z)|=1.
+```
+
+Logarithmisch differenziert:
+
+```math
+\begin{aligned}
+\frac{d}{dz}\log S_\infty(z)
+&=-i\log\pi
++\frac{i}{2}\psi\left(\frac14+\frac{iz}{2}\right)
++\frac{i}{2}\psi\left(\frac14-\frac{iz}{2}\right)\\
+&=i\left[
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)
+-\log\pi
+\right].
+\end{aligned}
+```
+
+Somit
+
+```math
+\boxed{
+-i\overline{S_\infty(z)}S_\infty'(z)
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi.
+}
+```
+
+Dies ist exakt der zentrierte archimedische Multiplikator aus COMMON-JUMP.
+
+Ferner ist `S_infty` genau das Verhaeltnis des archimedischen completed-zeta-Faktors
+
+```math
+\pi^{-s/2}\Gamma(s/2)
+```
+
+an `s=1/2+iz` und `s=1/2-iz`.
+
+---
+
+# 4. Fenster-Streumatrix
+
+Fuer `a>0` setze
+
+```math
+\boxed{
+S_a(z)
+=S_\infty(z)
+\prod_{p\le e^{2a}}S_p(z).
+}
+```
+
+Das Produkt ist endlich und fuer reelles `z` unitär:
+
+```math
+|S_a(z)|=1.
+```
+
+Daher ist
+
+```math
+\boxed{
+\tau_a(z):=-i\overline{S_a(z)}S_a'(z)
+}
+```
+
+reell. Durch logarithmische Ableitung
+
+```math
+\boxed{
+\tau_a(z)
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi
++
+\sum_{p\le e^{2a}}
+(\log p)
+\left[1-P_{p^{-1/2}}^{\mathbb D}(z\log p)\right].
+}
+```
+
+---
+
+# 5. Warum ganze Prime-Tuerme trotz Fenster exakt sind
+
+COMMON-JUMP verwendet geometrisch nur Prime powers mit
+
+```math
+k\log p\le2a.
+```
+
+Im Faktor `S_p` steckt dagegen formal der ganze geometrische Turm `k>=1`. Dies ist auf dem Testfunktionsraum kein Fehler:
+
+Fuer `v` mit Traeger in `[-a,a]` und `k log p>2a` gilt
+
+```math
+\|K_{k\log p}v\|^2-2\|v\|^2=0.
+```
+
+Unter Fouriertransformation bedeutet dies
+
+```math
+\int|\widehat v(z)|^2\cos(kz\log p)dz=0.
+```
+
+Daher duerfen die fehlenden hohen Potenzen **innerhalb der quadratischen Form** exakt ergaenzt werden.
+
+Somit gilt fuer alle solchen `v`
+
+```math
+\boxed{
+\langle v,(X_a^*X_a-\Gamma_aI)v\rangle
+=
+\int|\widehat v(z)|^2\tau_a(z)dz.
+}
+```
+
+Auf NULLPOL ist die linke Seite exakt `Q_W(v)`.
+
+---
+
+# 6. Wigner--Smith-Lesart
+
+Fuer eine skalare unitäre Streumatrix `S(z)` ist der Wigner--Smith-/phase-delay-Ausdruck
+
+```math
+Q_S(z)=-iS(z)^*\frac{dS}{dz}(z).
+```
+
+Da `S_a` skalar ist,
+
+```math
+Q_{S_a}(z)=\tau_a(z).
+```
+
+Daher:
+
+```math
+\boxed{
+Q_W(v)
+=
+\langle\widehat v,
+Q_{S_a}\widehat v\rangle_{L^2(\mathbb R)}
+\qquad
+(v\in\mathscr D_{NP,a}).
+}
+```
+
+Dies ist eine exakte Operatoridentitaet, keine physikalische Analogie.
+
+### Firewall
+
+Wigner-Zeitverzoegerungen koennen lokal negativ sein. Aus Unitaritaet von `S_a` folgt **keine** Positivitaet von `Q_{S_a}`. NP-GAP bleibt daher unangetastet offen.
+
+---
+
+# 7. Relation zur Poisson-/Blaschke-Geometrie
+
+Fuer reelles `0<q<1` sei der Disk-Blaschkefaktor
+
+```math
+b_q(w)=\frac{w-q}{1-qw}.
+```
+
+Auf `w=e^{i\theta}` gilt
+
+```math
+\frac{d}{d\theta}\arg b_q(e^{i\theta})
+=P_q^{\mathbb D}(\theta).
+```
+
+Auf dem Rand kann `S_p` daher als der Quotient
+
+```math
+S_p\sim\frac{w}{b_q(w)}
+```
+
+gelesen werden; seine Phasenableitung ist `1-P_q`.
+
+Archimedisch ist
+
+```math
+\frac{2\mu}{z^2+\mu^2}
+```
+
+die Phasenableitung des Half-plane-Blaschke-Faktors
+
+```math
+B_\mu(z)=\frac{z-i\mu}{z+i\mu}.
+```
+
+Die Gamma-Resolventenleiter ist damit zugleich eine Blaschke-/harmonic-measure-Leiter.
+
+Dies erklaert die enge Naehe der neuen Architektur zu Hardy-/de-Branges-/scattering-Methoden, ohne deren Positivitaetssaetze rueckwaerts zu importieren.
+
+---
+
+# 8. Vergleich mit allgemeiner Literatur
+
+Streumatrizen mit Zeta-Quotienten und Wigner-Zeitverzoegerungen sind in anderen spektralen/geometrischen Kontexten bekannt, insbesondere in der Streutheorie hyperbolischer Flaechen. Daher wird die allgemeine Idee `zeta ratio -> scattering phase` **nicht** als neu beansprucht.
+
+Der projektinterne neue Punkt ist die exakt hergeleitete Kette
+
+```text
+P11 AR(1) covariance
+ -> disk Poisson symbol
+ -> exact centered local Euler tower
+ -> finite-window Euler scattering phase
+```
+
+parallel zu
+
+```text
+critical-half Gamma resolvent ladder
+ -> half-plane Poisson comb
+ -> exact archimedean scattering phase,
+```
+
+und ihre Identifikation mit dem bereits bewiesenen COMMON-JUMP-Fensteroperator.
+
+Publikationsneuheit dieser spezifischen Kombination bleibt `?[O]`.
+
+---
+
+# 9. Neuer Object-X-Gate
+
+Die jetzt kanonische Frage lautet:
+
+> Gibt es ein explizites positives scattering system / conservative dilation, dessen Wigner--Smith-Operator nach einem **vorwaerts konstruierten** Compression-/Schur-Schritt genau `Q_{S_a}` auf `Ran L_{1/2}` liefert?
+
+Die Daten dafuer sind jetzt vollstaendig explizit:
+
+```text
+boundary/source:     L_{1/2}, E_±,
+prime channels:      disk Blaschke points q_p=p^{-1/2},
+arch channels:       half-plane points i mu_m,
+window scattering:   S_a,
+Weil operator:       -i S_a^* S_a'.
+```
+
+Ein positiver konservativer Dilationmechanismus waere ein echter Object-X-Kandidat. Ein theorem-level No-Go fuer diese Klasse waere ebenfalls wertvoll.
+
+---
+
+# 10. Firewalls
+
+Nicht behaupten:
+
+- `tau_a(z)>=0` punktweise;
+- Unitaritaet impliziere Wigner--Smith-Positivitaet;
+- das endliche Eulerprodukt konvergiere zum vollen Eulerprodukt auf der kritischen Linie;
+- die funktionale Gleichung liefere die Fensterform ohne Renormierungs-/Cutoffkontrolle;
+- ein bekanntes hyperbolisches Streumodell sei bereits unser Object X;
+- NP-GAP, Object X oder RH seien geloest;
+- Publikationsneuheit sei geklaert.
diff --git a/audits/P11_GAMMA_AS_RIEMANN_R_CONTINUUM_OF_CONDITIONED_CELLS_2026-09-15.md b/audits/P11_GAMMA_AS_RIEMANN_R_CONTINUUM_OF_CONDITIONED_CELLS_2026-09-15.md
new file mode 100644
index 00000000..b00ae3c1
--- /dev/null
+++ b/audits/P11_GAMMA_AS_RIEMANN_R_CONTINUUM_OF_CONDITIONED_CELLS_2026-09-15.md
@@ -0,0 +1,171 @@
+# P11 Audit — Gamma ladder as a Riemann-R continuum of conditioned P11 cells
+
+**Datum:** 15. September 2026
+
+## 0. Result
+
+Let `h>0`, `k>=1`, `t=kh`, and define the P11-type primitive squared amplitude
+
+```math
+\alpha_{h,k}^2:=h e^{-3t/2}.
+```
+
+For the Gamma ladder
+
+```math
+\mu_m=2m+1/2,
+\qquad m>=0,
+```
+
+define
+
+```math
+\boxed{
+\beta_{h,k,m}^2
+:=\frac{h e^{-\mu_m t}}{e^t-1}
+=\frac{\alpha_{h,k}^2}{1-e^{-t}}e^{-2mt}.
+}
+```
+
+With the positive Riemann-R density
+
+```math
+\rho(h)=\frac{d}{dh}R(e^h),
+```
+
+one has for every suitable source `v`
+
+```math
+\boxed{
+\int_0^\infty\rho(h)
+\sum_{k>=1}\beta_{h,k,m}^2\|K_{kh}v\|^2dh
+=
+\int_0^\infty e^{-\mu_m t}\|K_tv\|^2dt.
+}
+```
+
+Thus every positive Gamma resolvent mode is a positive continuum average of the same root-conditioned logarithmic lattice cell family.
+
+**Status:** exact positive identity `✓[M]`; no RH/Object-X conclusion.
+
+---
+
+## 1. Proof
+
+The continuum-cell audit gives the unweighted pushforward identity
+
+```math
+\boxed{
+\int_0^\infty\rho(h)
+\sum_{k>=1}h\,\delta_{kh}(dt)dh
+=(e^t-1)dt.
+}
+```
+
+Multiply the test integrand at `t` by
+
+```math
+\frac{e^{-\mu_m t}}{e^t-1}\|K_tv\|^2.
+```
+
+The factor `e^t-1` cancels and yields exactly
+
+```math
+\int_0^\infty e^{-\mu_m t}\|K_tv\|^2dt.
+```
+
+The second expression for `beta` follows from
+
+```math
+e^t-1=e^t(1-e^{-t})
+```
+
+and `mu_m=2m+1/2`.
+
+---
+
+## 2. Ground mode and P11 root-conditioning
+
+For `m=0`,
+
+```math
+\boxed{
+\beta_{h,k,0}^2
+=\frac{\alpha_{h,k}^2}{1-e^{-kh}}.
+}
+```
+
+For a genuine prime spacing `h=log p`, put `q=p^-1/2`. The normalized stationary OU node at depth `k` has root correlation `q^k`; after conditioning out the common root, its residual variance is
+
+```math
+\boxed{1-q^{2k}=1-p^{-k}.}
+```
+
+Hence `beta_(p,k,0)^2` is precisely the primitive P11 squared amplitude divided by the root-conditioned residual variance.
+
+---
+
+## 3. Martingale multiplicity produces the Weil weight
+
+The P11 rest multiplicity at channel depth `k` is exactly
+
+```math
+m_{p,k}=p^k-1.
+```
+
+Therefore
+
+```math
+\begin{aligned}
+(p^k-1)\beta_{p,k,0}^2
+&=(p^k-1)
+\frac{(\log p)p^{-3k/2}}{1-p^{-k}}\\
+&=(\log p)p^{-k/2}\\
+&=\frac{\Lambda(p^k)}{\sqrt{p^k}}.
+\end{aligned}
+```
+
+Thus
+
+```math
+\boxed{
+\text{P11 rest multiplicity}
+\times
+\text{root-conditioned primitive precision cell}
+=
+\text{Weil p-power weight}.
+}
+```
+
+This is an exact local identity, not a fitted exponent or asymptotic.
+
+---
+
+## 4. Higher Gamma modes
+
+For fixed `(h,k)`,
+
+```math
+\beta_{h,k,m}^2
+=\beta_{h,k,0}^2 e^{-2mkh}.
+```
+
+So the entire Gamma ladder is generated from the ground conditioned cell by a universal geometric round-trip damping in the separation `t=kh`.
+
+Summing `m>=0` gives
+
+```math
+\sum_{m>=0}\beta_{h,k,m}^2
+=\frac{h e^{-3kh/2}}
+{(1-e^{-kh})(1-e^{-2kh})}.
+```
+
+Consequently the complete archimedean positive layer is a positive Riemann-R continuum of this two-stage conditioned/round-trip cell family.
+
+---
+
+## 5. Firewall
+
+The identity explains local weights and multiplicities but does not compare the discrete prime sampling of the cells with their continuous Riemann-R average. That discrete-versus-continuum storage/sampling comparison remains the global RH-hard gate.
+
+In particular, no claim is made that multiplying or averaging these positive cells makes the centered Weil form positive. The earlier no-gos for diagonal and ordinary cascade coupling remain in force.
diff --git a/audits/P11_GAMMA_LADDER_AS_STRONG_AR1_CELL_LIMITS_2026-09-15.md b/audits/P11_GAMMA_LADDER_AS_STRONG_AR1_CELL_LIMITS_2026-09-15.md
new file mode 100644
index 00000000..201acd42
--- /dev/null
+++ b/audits/P11_GAMMA_LADDER_AS_STRONG_AR1_CELL_LIMITS_2026-09-15.md
@@ -0,0 +1,112 @@
+# P11 Audit — Gamma ladder as strong AR(1) cell limits
+
+**Datum:** 15. September 2026
+
+## Result
+
+For `mu>0` and `h>0` define
+
+```math
+A_{h,\mu}
+:=\sum_{k\ge1}h e^{-\mu kh}K_{kh}^*K_{kh}.
+```
+
+This is the positive logarithmic-grid jump cell with AR(1) transition
+
+```math
+r=e^{-\mu h}.
+```
+
+Then
+
+```math
+\boxed{
+A_{h,\mu}\xrightarrow[h\downarrow0]{\rm strong}
+A_{0,\mu}
+:=\frac2\mu D^2(D^2+\mu^2)^{-1}.
+}
+```
+
+Moreover
+
+```math
+\boxed{\|A_{h,\mu}-A_{0,\mu}\|=\frac2\mu}
+```
+
+for every `h>0`. Thus every mode has the same strong-but-never-norm-continuous sampling boundary behavior.
+
+For the Gamma ladder
+
+```math
+\mu_m=2m+1/2,
+```
+
+and the Critical-half P11 parameter `q=e^{-h/2}`,
+
+```math
+\boxed{e^{-\mu_m h}=q^{4m+1}.}
+```
+
+Hence the complete Gamma resolvent ladder is the family of strong `h=0` limits of logarithmic AR(1) cells with transition powers
+
+```text
+q, q^5, q^9, q^13, ...
+```
+
+The ground mode is exactly the P11 transition itself.
+
+**Status:** `✓[M]`; no RH/Object-X conclusion.
+
+---
+
+## Multiplier proof
+
+Let `r=e^{-mu h}`. The multiplier of `A_(h,mu)` is
+
+```math
+a_{h,\mu}(z)
+=h\left[
+\frac{1+r}{1-r}-P_r(zh)
+\right].
+```
+
+For fixed `z`, as `h->0`,
+
+```math
+h\frac{1+r}{1-r}\to\frac2\mu,
+```
+
+and
+
+```math
+hP_r(zh)\to\frac{2\mu}{z^2+\mu^2}.
+```
+
+Therefore
+
+```math
+\boxed{
+a_{h,\mu}(z)\to
+\frac{2z^2}{\mu(z^2+\mu^2)}.}
+```
+
+This is exactly the Gamma-mode multiplier.
+
+The Poisson bounds give
+
+```math
+0\le a_{h,\mu}(z)
+\le\frac{4hr}{1-r^2}
+=\frac{2h}{\sinh(\mu h)}
+<\frac2\mu.
+```
+
+Thus dominated convergence proves strong convergence.
+
+At the alias frequencies `z_n=2 pi n/h`, one has `a_(h,mu)(z_n)=0`, while the limiting multiplier tends to `2/mu` as `n->infinity`. Since both multipliers lie in `[0,2/mu]`, the operator-norm difference is exactly `2/mu`.
+
+---
+
+## Interpretation firewall
+
+This is a structural discretization theorem, not an RH mechanism. The higher Gamma masses are already known from the digamma expansion; expressing them as AR(1) cell limits does not create a new positivity theorem. Its use is architectural: finite-prime P11 cells and archimedean Gamma cells belong to one semigroup family, while the unavoidable aliasing explains why the gluing must be strong/source-local rather than norm-uniform.
diff --git a/audits/P11_GAMMA_TAIL_FRAME_NOGO_2026-09-15.md b/audits/P11_GAMMA_TAIL_FRAME_NOGO_2026-09-15.md
new file mode 100644
index 00000000..fba26dc0
--- /dev/null
+++ b/audits/P11_GAMMA_TAIL_FRAME_NOGO_2026-09-15.md
@@ -0,0 +1,253 @@
+# P11 Audit — Higher-Gamma tail-frame no-go
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** enger Klassen-No-Go fuer die Idee, dass die hoeheren Gamma-Moden den vom P11-Boundary-Shorting verlorenen Tail coerciv ergaenzen.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein No-Go gegen dynamische Gamma-Kopplung, Filterbank-Storage oder nichtlokale Transfermechanismen.
+
+---
+
+## 0. Kurzurteil
+
+Die Gamma-Exponentialleiter
+
+```math
+ g_m(t)=e^{-(2m+1/2)t},
+ \qquad m>=0,
+```
+
+ist im Critical-half-Tailraum `L^2(0,infty)` vollstaendig. Der Ground-Modus ist
+
+```math
+g_0(t)=e^{-t/2},
+```
+
+also exakt der OU-Boundary-Modus des P11-Shortings.
+
+Nach orthogonaler Projektion der hoeheren Moden auf `g_0^perp` spannen diese zwar dicht das verlorene Tail-Komplement, aber **nicht mit einer radiusuniformen unteren Frame-Schranke**.
+
+Damit ist die Architekturklasse
+
+```text
+P11 keeps g_0
++ higher Gamma exponentials statically frame the missing tail
++ uniform coercivity in the window radius
+```
+
+rigoros ausgeschlossen.
+
+Status:
+
+```text
+Gamma exponentials complete in L2 tail                ✓[M]
+projected higher modes dense in g0^perp                ✓[M]
+uniform lower frame bound on g0^perp                   ×[M]
+dynamic/filter-bank coupling                           ?[O]
+```
+
+---
+
+# 1. Compactification to a moment system
+
+Let
+
+```math
+\mu_m=2m+\frac12,
+\qquad
+g_m(t)=e^{-\mu_m t}.
+```
+
+Use
+
+```math
+x=e^{-2t},
+\qquad
+dt=-\frac{dx}{2x}.
+```
+
+Then
+
+```math
+g_m(t)=x^{m+1/4}.
+```
+
+For a linear combination,
+
+```math
+\int_0^\infty
+\left|\sum_m a_mg_m(t)\right|^2dt
+=
+\frac12\int_0^1
+\left|\sum_m a_mx^m\right|^2x^{-1/2}dx.
+```
+
+Thus the Gamma ladder is unitarily equivalent, up to the fixed factor `x^(1/4)`, to the monomials in
+
+```math
+L^2((0,1),\tfrac12x^{-1/2}dx).
+```
+
+Polynomials are dense in this weighted `L^2`, so
+
+```math
+\boxed{\overline{span}\{g_m:m>=0\}=L^2(0,\infty).}
+```
+
+---
+
+# 2. Projected higher modes span the ground complement
+
+The ground mode has norm
+
+```math
+\|g_0\|^2=\int_0^\infty e^{-t}dt=1.
+```
+
+For `m>=1`,
+
+```math
+\langle g_m,g_0\rangle
+=
+\int_0^\infty e^{-(2m+1)t}dt
+=
+\frac1{2m+1}.
+```
+
+Define
+
+```math
+\boxed{
+\widetilde g_m
+=g_m-\frac1{2m+1}g_0.
+}
+```
+
+Then `tilde g_m` lies in `g_0^perp`. Because the full Gamma ladder is complete, orthogonal projection onto `g_0^perp` gives
+
+```math
+\boxed{
+\overline{span}\{\widetilde g_m:m>=1\}=g_0^\perp.
+}
+```
+
+So the higher Gamma modes can detect every missing P11-tail direction in the weak completeness sense.
+
+---
+
+# 3. No lower frame bound
+
+Completeness is not coercivity. Let `f_R` be any unit vector supported in `[R,R+1]`. By Cauchy--Schwarz,
+
+```math
+|\langle f_R,g_m\rangle|^2
+\le
+\int_R^{R+1}e^{-2\mu_m t}dt
+\le
+\frac{e^{-2\mu_mR}}{2\mu_m}.
+```
+
+Therefore
+
+```math
+\sum_{m>=1}|\langle f_R,g_m\rangle|^2
+\le
+\sum_{m>=1}
+\frac{e^{-(4m+1)R}}{4m+1}
+\longrightarrow0.
+```
+
+On the other hand
+
+```math
+\|f_R\|=1.
+```
+
+Hence there is no `A>0` such that
+
+```math
+A\|f\|^2
+\le
+\sum_{m>=1}|\langle f,g_m\rangle|^2
+```
+
+for all tail functions `f`.
+
+---
+
+# 4. The same failure on g0^perp
+
+The vector `f_R` has ground overlap bounded by
+
+```math
+|\langle f_R,g_0\rangle|^2
+\le\int_R^{R+1}e^{-t}dt
+\le e^{-R}.
+```
+
+Set
+
+```math
+h_R
+=f_R-\langle f_R,g_0\rangle g_0.
+```
+
+Then `h_R in g_0^perp` and
+
+```math
+\|h_R\|\to1.
+```
+
+Moreover
+
+```math
+\langle h_R,\widetilde g_m\rangle
+=\langle f_R,\widetilde g_m\rangle.
+```
+
+The `g_m` part is controlled as above. The subtracted ground part contributes at most
+
+```math
+\frac{|\langle f_R,g_0\rangle|}{2m+1},
+```
+
+whose square-sum is `O(e^{-R})` because
+
+```math
+\sum_{m>=1}(2m+1)^{-2}<\infty.
+```
+
+Thus
+
+```math
+\boxed{
+\sum_{m>=1}
+|\langle h_R,\widetilde g_m\rangle|^2
+\to0,
+\qquad
+\|h_R\|\to1.
+}
+```
+
+No lower frame bound exists even on the precise P11-lost subspace `g_0^perp`.
+
+---
+
+# 5. Strategic meaning
+
+The stopped P11 precision retains the normalized ground boundary mode and loses its orthogonal tail complement. The Gamma ladder is algebraically rich enough to span that complement, but its raw exponential modes become exponentially blind to energy translated far out in the tail coordinate.
+
+Therefore no all-radius proof can be based merely on
+
+```text
+completeness of the higher Gamma ladder
++ static coefficient estimates.
+```
+
+The surviving possibilities remain dynamical:
+
+```text
+- the causal Gamma filter bank with internal states;
+- a storage inequality coupled to the P11 stopped boundary state;
+- a relative/scattering realization of the arithmetic Volterra transfer.
+```
diff --git a/audits/P11_HAMILTONIAN_DISCREPANCY_FLOW_2026-09-15.md b/audits/P11_HAMILTONIAN_DISCREPANCY_FLOW_2026-09-15.md
new file mode 100644
index 00000000..fbc574f5
--- /dev/null
+++ b/audits/P11_HAMILTONIAN_DISCREPANCY_FLOW_2026-09-15.md
@@ -0,0 +1,361 @@
+# P11 Audit — Prime discrepancy as a stopped-OU Hamiltonian defect
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level dynamische Form der im NULLPOL-Diskrepanz-Audit isolierten arithmetischen Maßdifferenz.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein Vorzeichen-, Positivitaets-, Object-X- oder RH-Beweis.
+
+---
+
+## 0. Kurzurteil
+
+Die signierte Prime-Diskrepanz
+
+```math
+\Delta
+=
+\nu_P-e^{t/2}dt,
+\qquad
+\nu_P
+=
+\sum_{n=p^k}\frac{\Lambda(n)}{\sqrt n}\delta_{\log n},
+```
+
+ist nicht nur am OU-Root sichtbar. Sie wird **vollstaendig** durch die Differenz der positiven Hamiltonian-Dichten zweier gestoppter Critical-half-OU-Flows kodiert.
+
+Der universelle gestoppte Kernel
+
+```math
+G_L(s,t)
+=e^{-(s+t)/2+\min(s,t,L)}
+```
+
+hat den Rang-1-Hamiltonianvektor
+
+```math
+\boxed{
+g_L(t)
+=e^{-(t-L)/2}\mathbf1_{t\ge L}.
+}
+```
+
+Definiere
+
+```math
+\boxed{
+H_P(L)
+:=\|g_L\|_{L^2(\nu_P)}^2
+=
+\int_{[L,\infty)}e^{-(t-L)}d\nu_P(t),
+}
+```
+
+und fuer den kanonischen Pol-Compensator
+
+```math
+\boxed{
+H_0(L)
+:=\|g_L\|_{L^2(e^{t/2}dt)}^2
+=
+2e^{L/2}.
+}
+```
+
+Dann
+
+```math
+\boxed{
+H_\Delta(L)
+:=H_P(L)-H_0(L)
+=
+\int_{[L,\infty)}e^{-(t-L)}d\Delta(t).
+}
+```
+
+und im Stieltjes-/Distributionssinn
+
+```math
+\boxed{
+d\Delta(L)
+=H_\Delta(L)dL-dH_\Delta(L).
+}
+```
+
+Somit bestimmt der Hamiltonian-Defekt `H_Delta` die gesamte Prime-Diskrepanz exakt.
+
+Bei `L=0` ergibt sich der P11-Roottrace-Wert
+
+```math
+H_\Delta(0)
+=-\zeta'/\zeta(3/2)-2.
+```
+
+Status:
+
+```text
+prime Hamiltonian H_P >= 0                              ✓[M]
+continuum compensator Hamiltonian H_0 > 0               ✓[M]
+H_Delta = exponential tail transform of Delta           ✓[M]
+Delta = H_Delta dL - dH_Delta                            ✓[M]
+L=0 recovers P11/OU root-trace discrepancy              ✓[M]
+fixed sign / domination H_P <= H_0 or converse           ×[M] (diagnostically false; no theorem needed for route)
+positive coupling of the two Hamiltonians                ?[O]
+```
+
+---
+
+# 1. Hamiltonian vector of the stopped kernel
+
+From
+
+```math
+G_L(s,t)
+=e^{-(s+t)/2}
++\int_0^L
+\mathbf1_{u\le s,t}
+ e^{-(s-u)/2}e^{-(t-u)/2}du,
+```
+
+one has, away from the null boundary surfaces,
+
+```math
+\partial_LG_L(s,t)
+=
+\mathbf1_{L<s,t}
+ e^{-(s-L)/2}e^{-(t-L)/2}.
+```
+
+Thus
+
+```math
+\boxed{
+\partial_LG_L=g_Lg_L^*,
+\qquad
+g_L(t)=e^{-(t-L)/2}\mathbf1_{t\ge L}.
+}
+```
+
+This is the universal positive rank-one depth increment.
+
+---
+
+# 2. Prime and continuum Hamiltonian densities
+
+For any positive locally finite measure `mu` for which the tail integral converges, the scalar Hamiltonian density generated by `g_L` is
+
+```math
+H_\mu(L)
+=\int_{[L,\infty)}e^{-(t-L)}d\mu(t).
+```
+
+For the prime-power measure,
+
+```math
+\boxed{
+H_P(L)
+=e^L
+\sum_{\log n\ge L}
+\frac{\Lambda(n)}{n^{3/2}}.
+}
+```
+
+This converges absolutely for every `L>=0`.
+
+For the continuous compensator,
+
+```math
+\begin{aligned}
+H_0(L)
+&=
+\int_L^\infty
+ e^{-(t-L)}e^{t/2}dt\\
+&=
+e^L\int_L^\infty e^{-t/2}dt\\
+&=2e^{L/2}.
+\end{aligned}
+```
+
+Both Hamiltonians are pointwise positive.
+
+Their difference is
+
+```math
+\boxed{
+H_\Delta(L)
+=H_P(L)-H_0(L)
+=
+\int_{[L,\infty)}e^{-(t-L)}d\Delta(t).
+}
+```
+
+---
+
+# 3. Inversion: H_Delta determines Delta exactly
+
+Let generally
+
+```math
+H(L)=\int_{[L,\infty)}e^{-(t-L)}d\mu(t).
+```
+
+Between atoms, if `dmu=rho(L)dL`, differentiation gives
+
+```math
+H'(L)=H(L)-\rho(L).
+```
+
+At an atom of mass `a` at `L_0`, increasing the lower endpoint through `L_0` removes exactly that atom, so
+
+```math
+H(L_0+)-H(L_0-)=-a.
+```
+
+Both facts are summarized by the Stieltjes identity
+
+```math
+\boxed{
+d\mu(L)=H(L)dL-dH(L).
+}
+```
+
+Applying this to the signed measure `Delta` gives
+
+```math
+\boxed{
+d\Delta(L)=H_\Delta(L)dL-dH_\Delta(L).
+}
+```
+
+Thus no arithmetic information is lost by replacing `Delta` with its stopped-OU Hamiltonian defect.
+
+---
+
+# 4. Root value and the P11 trace bridge
+
+At `L=0`,
+
+```math
+\begin{aligned}
+H_P(0)
+&=\sum_n\frac{\Lambda(n)}{n^{3/2}}
+=-\frac{\zeta'}{\zeta}(3/2),\\
+H_0(0)&=2.
+\end{aligned}
+```
+
+Hence
+
+```math
+\boxed{
+H_\Delta(0)
+=-\frac{\zeta'}{\zeta}(3/2)-2.
+}
+```
+
+This is exactly the `sigma=1/2` root-trace discrepancy from `P11_NULLPOL_PRIME_DISCREPANCY_CANCELLATION_2026-09-15.md`.
+
+The previous scalar root identity is therefore merely the `L=0` endpoint of a complete Hamiltonian flow.
+
+---
+
+# 5. Spatial identification with the P11 boundary depth
+
+The P11 masks use the physical depth
+
+```math
+L_R(x)=2(R-|x|).
+```
+
+Thus the same variable `L` simultaneously parametrizes
+
+```text
+1. the positive stopped-OU innovation flow;
+2. the P11 visible branch depth;
+3. the two-sided COMMON-JUMP overlap cone;
+4. the exponentially smoothed prime-vs-pole discrepancy H_Delta(L).
+```
+
+This is an exact common coordinate, not a fitted scaling.
+
+---
+
+# 6. No easy sign theorem
+
+It would be tempting to hope for a pointwise comparison
+
+```math
+H_P(L)\le H_0(L)
+```
+
+or the reverse. That would turn the Hamiltonian defect into a signed positive/negative flow.
+
+Direct numerical evaluation of the absolutely convergent expression for `H_P` shows sign changes of
+
+```math
+H_\Delta(L)=H_P(L)-H_0(L)
+```
+
+already at modest `L`; therefore no such global fixed-sign route should be pursued.
+
+This numerical observation is used only to kill the proposed easy strategy; no theorem-level arithmetic statement is inferred from it.
+
+---
+
+# 7. RH firewall
+
+The cumulative discrepancy
+
+```math
+D(T)=\Delta([0,T])
+```
+
+satisfies, by integrating
+
+```math
+d\Delta=H_\Delta dL-dH_\Delta,
+```
+
+the relation
+
+```math
+D(T)
+=
+H_\Delta(0)-H_\Delta(T)
++\int_0^T H_\Delta(L)dL
+```
+
+(up to the harmless endpoint convention at atoms).
+
+Consequently any polynomial bound
+
+```math
+H_\Delta(L)=O(L^B)
+```
+
+would imply a polynomial bound for `D(T)` and hence the RH-scale weighted prime-number-theorem error identified in the preceding discrepancy audit.
+
+Thus proving simple polynomial control of the Hamiltonian defect is already essentially as hard as RH; the dynamical reformulation isolates the difficulty but does not trivialize it.
+
+---
+
+# 8. Sharpened Object-X gate
+
+The arithmetic obstruction can now be stated dynamically:
+
+```math
+\boxed{
+\text{two positive OU Hamiltonians }H_P,H_0
+\text{ have a sign-changing difference }H_\Delta
+\text{ that encodes all of }\Delta.
+}
+```
+
+The remaining useful question is not whether one Hamiltonian dominates the other pointwise. It is whether a **larger conservative/positive boundary system** can couple them so that their signed Hamiltonian defect appears as a Schur boundary term while the total parent energy stays positive.
+
+Any such construction must:
+
+1. be defined from the prime lattice, continuous compensator and critical-half OU kernel before invoking Weil positivity;
+2. reproduce the full `L`-flow, not only the root value `L=0`;
+3. include the higher-rank boundary-tail structure ruled out by the scalar-Hub and primewise-contraction no-gos;
+4. survive the exact Prime-2 mixed-pair calibration.
diff --git a/audits/P11_HARD_AUDIT_ERRATA_AND_PROLATE_BARRIER_2026-09-15.md b/audits/P11_HARD_AUDIT_ERRATA_AND_PROLATE_BARRIER_2026-09-15.md
new file mode 100644
index 00000000..f5122da2
--- /dev/null
+++ b/audits/P11_HARD_AUDIT_ERRATA_AND_PROLATE_BARRIER_2026-09-15.md
@@ -0,0 +1,370 @@
+# P11 Hard Audit — Errata, parity-normalization check, and prolate barrier
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** destruktiver Gegencheck der Critical-half/Prolate-Front.  
+**Registry:** unverändert.  
+**Nonclaim:** kein neuer RH-/Object-X-/Publikationsclaim.
+
+---
+
+## 0. Kurzurteil
+
+Der harte Gegencheck zerstört die Kernreduktionen nicht, korrigiert aber zwei wichtige Punkte:
+
+1. Die im Prolate-Audit als exakt bezeichnete Orbitkettenlaenge
+
+```math
+\lfloor L/t\rfloor+1
+```
+
+ist an den diskreten Stellen `L/t in N` um eins zu gross. Die exakte wesentliche Maximalkettenlaenge ist
+
+```math
+\boxed{N(t)=\lceil L/t\rceil.}
+```
+
+Die daraus im alten Audit gezogene Lower-Frame-Schranke bleibt konservativ gueltig, war an diesen Stellen aber nicht scharf und darf dort nicht als exakt bezeichnet werden.
+
+2. Der zunaechst vermutete Faktor-2-Fehler im even/odd-Arb-Checker ist **kein Zertifikatsfehler**. Der Checker benutzt auf der positiven Frequenzhalbachse absichtlich den verdoppelten Paritaetskernel. Halbgeradenfaktor und Kernskalierung kompensieren sich exakt sowohl in der Spur als auch im generalisierten Ritz-Paar `(A,G)`.
+
+Der wichtigere strategische Befund ist negativ:
+
+> Die aktuelle gewichtete-Prolate-Zertifizierung ist als feste-Fenster-Methode sauber, erbt aber fuer grosse Fenster die doppelt-exponentielle Frequenzbarriere des Prime-Combs. Sie ist daher ein guter Regressionstest der Architektur, aber fuer sich allein kein plausibler globaler RH-Mechanismus.
+
+Status:
+
+```text
+Critical-half Range/Green theorem                         survives ✓[M]
+P11 sampled OU/tree Gram identity                         survives ✓[M]
+finite Gamma/Poisson/scattering identities                survives ✓[M]
+rank-2 NULLPOL Paley-Wiener shorting                      survives ✓[M]
+trace-minus-Ritz lemma                                    survives ✓[M]
+old orbit-chain exactness claim                           corrected
+suspected parity factor-2 checker bug                     withdrawn
+fixed-window weighted-prolate reduction                   survives ✓[M]_part
+weighted-prolate as scalable all-window RH mechanism      strongly disfavored
+all-window NP-GAP / Object X / RH                         ?[O]
+```
+
+---
+
+# 1. Exact correction of the translation-orbit chain formula
+
+Let `I` be an open interval of length `L>0` and
+
+```math
+A_t=P_I(T_t+T_{-t})P_I,
+\qquad t>0.
+```
+
+Decompose `I` modulo the translation lattice `t Z`. For almost every residue class the fibre is a finite path. Its essential maximal number of vertices is
+
+```math
+\boxed{
+N(t)=\left\lceil\frac{L}{t}\right\rceil.
+}
+```
+
+Indeed, if
+
+```math
+L/t=m+r,
+\qquad m\in\mathbb N_0,
+\qquad 0<r<1,
+```
+
+there is a positive-measure family of fibres with `m+1=ceil(L/t)` points. If `L/t=m` is an integer, the endpoint fibres that could contain `m+1` points occur only on a null set (and the interval is open); almost every maximal fibre has exactly `m` points.
+
+Example:
+
+```math
+I=(0,1),\qquad t=1/2.
+```
+
+Almost every fibre has two vertices, not three. Thus the old formula
+
+```math
+floor(L/t)+1
+```
+
+fails exactly when `L/t` is integral.
+
+Since the path adjacency matrix `P_N` has top eigenvalue
+
+```math
+2\cos\frac{\pi}{N+1},
+```
+
+we obtain the exact operator norm
+
+```math
+\boxed{
+\|A_t\|
+=2\cos\frac{\pi}{\lceil L/t\rceil+1}.
+}
+```
+
+This also covers `t>=L`: then `ceil(L/t)=1` and the right side is zero.
+
+Since
+
+```math
+K_t^*K_t=2I-A_t,
+```
+
+the exact support-only lower bound is
+
+```math
+\boxed{
+\|K_t v\|_2^2
+\ge
+\left(
+2-2\cos\frac{\pi}{\lceil L/t\rceil+1}
+\right)\|v\|_2^2.
+}
+```
+
+The old floor-based formula used a weakly larger `N` at the exceptional ratios and therefore produced a weakly smaller lower bound. Hence the old inequality remains safe; only the exactness/sharpness claim is corrected.
+
+---
+
+# 2. Parity normalization: the suspected factor-2 bug is not real
+
+For the full Paley--Wiener kernel
+
+```math
+K_a(z,w)=\frac{\sin a(z-\bar w)}{\pi(z-\bar w)},
+```
+
+the correctly normalized parity kernels are
+
+```math
+\boxed{
+K_a^{\pm}(z,w)
+=\frac12\bigl(K_a(z,w)\pm K_a(z,-w)\bigr).
+}
+```
+
+The Arb checker defines instead
+
+```math
+\widetilde K_a^{\pm}=K_a(z,w)\pm K_a(z,-w)=2K_a^{\pm}
+```
+
+and integrates only over `z>0`.
+
+For an even weight `w(z)` the full trace in the parity sector is
+
+```math
+\operatorname{tr}D_\pm
+=
+\int_{\mathbb R}w(z)K_{NP}^{\pm}(z,z)\,dz
+=
+2\int_0^\infty w(z)K_{NP}^{\pm}(z,z)\,dz
+=
+\int_0^\infty w(z)\widetilde K_{NP}^{\pm}(z,z)\,dz.
+```
+
+So the checker trace is exactly normalized.
+
+For a finite kernel-section Ritz space, let `(A,G)` be the true full-line form and Gram matrices for the normalized parity kernel. Using the doubled kernel and the positive half-line gives
+
+```math
+\widetilde G=2G,
+\qquad
+\widetilde A=2A.
+```
+
+Hence
+
+```math
+\widetilde G^{-1}\widetilde A=G^{-1}A,
+```
+
+and the generalized Ritz eigenvalues, the Ritz trace, and the positivity test
+
+```math
+UG-A\succ0
+```
+
+are unchanged up to the common positive factor `2`.
+
+Therefore the earlier suspicion "checker missing 1/2" is withdrawn. The implementation should nevertheless be documented explicitly as a **positive-half-line doubled-kernel convention**, because otherwise the normalization looks incorrect on inspection.
+
+---
+
+# 3. Independent normalized diagnostic rerun at `a=1/2`
+
+The `a=1/2` multiplier is
+
+```math
+\tau_{1/2}(z)
+=
+\Re\psi\left(\frac14+\frac{iz}{2}\right)
+-\log\pi
+-\sqrt2\log2\cos(z\log2).
+```
+
+Recomputing with the *normalized* parity kernels and explicit factor `2` for the positive half-line reproduces the checker geometry.
+
+Using the current rationalized kernel-section node sets gives the following floating diagnostics:
+
+```text
+even: c=1.7, U=1.693
+  tr(D)                  ~ 4.69894069457
+  top Ritz eigenvalue    ~ 1.69060811940
+  Ritz trace             ~ 4.69523756358
+  T - trR + U            ~ 1.69670313099
+  diagnostic margin      ~ 0.00329686901
+
+odd: c=1.4, U=1.27
+  tr(D)                  ~ 2.72702973494
+  top Ritz eigenvalue    ~ 1.25079005295
+  Ritz trace             ~ 2.71472869855
+  T - trR + U            ~ 1.28230103640
+  diagnostic margin      ~ 0.11769896360
+```
+
+These numbers are **diagnostic only**. In particular, the even margin is small enough that no theorem-level PASS is claimed until an Arb run encloses all root partitions, trace integrals, Ritz traces and Cholesky pivots.
+
+---
+
+# 4. Why the weighted-prolate route does not scale naively
+
+For fixed `a`, write the finite prime comb
+
+```math
+P_a(z)
+=
+2\sum_{\log n\le2a}
+\frac{\Lambda(n)}{\sqrt n}\cos(z\log n)
+```
+
+and its total mass
+
+```math
+A_a
+=
+2\sum_{\log n\le2a}
+\frac{\Lambda(n)}{\sqrt n}.
+```
+
+The centered multiplier is
+
+```math
+\tau_a(z)
+=
+\Re\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi-P_a(z).
+```
+
+At high frequency
+
+```math
+\Re\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi
+=
+\log\frac{|z|}{2\pi}+O(z^{-2}).
+```
+
+The finite prime comb satisfies
+
+```math
+|P_a(z)|\le A_a.
+```
+
+Moreover the phases can recur arbitrarily close to simultaneous alignment for the finitely many independent prime logarithms, so a uniform pointwise treatment cannot replace `A_a` by a substantially smaller universal constant.
+
+Therefore a crude but structurally unavoidable pointwise tail firewall for
+
+```math
+(c-\tau_a)_+
+```
+
+has scale
+
+```math
+\boxed{
+T_a\gtrsim 2\pi\exp(A_a+c).
+}
+```
+
+By the prime number theorem, with `X=e^{2a}`,
+
+```math
+\sum_{n\le X}\frac{\Lambda(n)}{\sqrt n}
+\sim2\sqrt X,
+```
+
+so
+
+```math
+\boxed{A_a\sim4e^a.}
+```
+
+Consequently
+
+```math
+\boxed{
+T_a
+\approx
+\exp\bigl((4+o(1))e^a\bigr),
+}
+```
+
+doubly exponential in the window radius.
+
+This matches the barrier identified in the current compact-window literature for pointwise-envelope certificates. Thus the weighted-prolate defect is excellent for a fixed small window but, without an additional structural identity, inherits the same large-window time-band-limiting obstruction.
+
+---
+
+# 5. Literature collision / novelty firewall
+
+The hard audit also strengthens the novelty firewall:
+
+- Connes--Consani already use prolate spheroidal functions, compressed scaling operators and Toeplitz methods in Weil positivity at the archimedean place.
+- The current 2026 compact-window work proves unconditional positivity beyond the classical `log 2` range by a certified finite reduction and explicitly identifies the Landau--Widom/time-band-limiting barrier.
+- The two moment conditions at exponents `+-1/2` occur explicitly in the operator-theoretic Weil-positivity literature.
+
+Therefore the following are **not** publication novelty claims of this project:
+
+```text
+prolate/time-band limiting + Weil positivity
+Paley-Wiener finite-window formulation
+moment conditions at +-1/2
+Toeplitz/compression methods for local Weil positivity
+finite certified matrix reductions for fixed windows
+```
+
+The project-specific pieces that remain worth auditing are narrower:
+
+```text
+1. NULLPOL as the exact compact-support range of L_{1/2};
+2. the exact P11 prime-power ledger as a sampled critical-half OU/tree Gram;
+3. the way that P11/tree, Green boundary charges and COMMON-JUMP incidence data
+   might descend to one forward positive dilation without using Weil positivity
+   as an input.
+```
+
+No novelty is asserted for these either until a dedicated literature audit is complete.
+
+---
+
+# 6. Strategic consequence
+
+`NP-PROLATE-1` at `a=1/2` should be retained as a **regression / architecture consistency gate**, not promoted to the main global front.
+
+A successful Arb certificate would show that the Critical-half/NULLPOL reduction is numerically and analytically coherent on one nontrivial arithmetic window. But current literature already proves stronger fixed-window positivity, so such a certificate is not by itself mathematical progress toward RH.
+
+The main research question should therefore be sharpened to:
+
+```math
+\boxed{
+\text{Can the P11 OU/tree + Green-boundary geometry produce a radius-consistent}
+\atop
+\text{positive dilation / Schur complement that avoids pointwise resolution of }\tau_a?
+}
+```
+
+A candidate is only interesting if its positivity is **structural in `a`** and does not require resolving the doubly-exponential set of high-frequency prime-phase recurrences window by window.
+
+That is the next genuine Object-X gate.
diff --git a/audits/P11_HARD_AUDIT_SURVIVOR_LEDGER_2026-09-15.md b/audits/P11_HARD_AUDIT_SURVIVOR_LEDGER_2026-09-15.md
new file mode 100644
index 00000000..3fde5873
--- /dev/null
+++ b/audits/P11_HARD_AUDIT_SURVIVOR_LEDGER_2026-09-15.md
@@ -0,0 +1,382 @@
+# P11 / Objekt X — Hard-Audit Survivor Ledger
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** knappe Uebergabe nach dem destruktiven Audit dieses Forschungsastes.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein RH-Beweis, kein Object-X-Abschluss, kein Publikationsneuheitsclaim.
+
+---
+
+## 0. Urteil
+
+Nach absichtlich destruktiver Gegenpruefung bleibt ein kleinerer, aber mathematisch klarerer Kern uebrig.
+
+Der Ast hat **kein neues RH-Kriterium** gefunden. Die endgueltige Frequenzbereichsform setzt sich exakt zum klassischen Lagarias-Objekt
+
+```math
+G(s)=\frac{\xi'}{\xi}\left(\frac12+s\right)
+```
+
+zusammen. Bekannt ist
+
+```math
+RH
+\iff
+\operatorname{Re}G(s)>0
+\qquad(\operatorname{Re}s>0).
+```
+
+Der moegliche projektspezifische Wert liegt daher nur in einer konstruktiven Zeit-/Zustandsraumgeometrie, die diese bekannte positive-real-Funktion **vorwaerts** aus den bereits vorhandenen P11-/OU-/Gamma-Daten realisiert.
+
+---
+
+# 1. Ueberlebende exakte Saetze
+
+### S1. NULLPOL ist Critical-half Green-Range
+
+```math
+C_c^\infty(-a,a)\cap\ker M(0)\cap\ker M(1)
+=
+(-\partial_x^2+1/4)C_c^\infty(-a,a).
+```
+
+Die zwei Mellinbedingungen sind exakt das Verschwinden der beiden aeusseren Green-Ladungen fuer den Kern `exp(-|x-y|/2)`.
+
+**Status:** `✓[M]`.
+
+### S2. P11 Prime-Power-Ledger ist sampled OU/Green
+
+Nach Weil-Diagonalnormalisierung:
+
+```math
+R_q(j,k)=q^{|j-k|},
+\qquad q=p^{-1/2}.
+```
+
+Hub + Innovationen ergeben exakt den stationaeren AR(1)-Kern.
+
+**Status:** `✓[M]` fuer den Kanalindex-Ledger; keine Gleichsetzung mit dem vollen raeumlichen P11-Operator.
+
+### S3. P11-Masken sind stopped OU
+
+Mit
+
+```math
+L_R(x)=2(R-|x|),
+\qquad
+m_p(x)=\lfloor L_R(x)/\log p\rfloor,
+```
+
+ist die lokale normalisierte Kovarianz
+
+```math
+R_q^{(m)}(j,k)=q^{j+k-2\min(j,k,m)}.
+```
+
+Die Tiefeninkremente sind positiv Rang eins.
+
+**Status:** `✓[M]`.
+
+### S4. Universeller stopped Critical-half-Kern
+
+```math
+G_L(s,t)
+=\exp\left(-\frac{s+t}{2}+\min(s,t,L)\right).
+```
+
+P11 ist seine exakte logarithmische Gitterabtastung. Seine inverse Kovarianz ist in der Kanaltiefe erneut `-partial_t^2+1/4`.
+
+**Status:** `✓[M]`.
+
+### S5. Exaktes boundary-tail shorting
+
+Diskret:
+
+```math
+Y_m=\sqrt{1-q^2}\sum_{k\ge m}q^{k-m}X_k
+```
+
+und
+
+```math
+\sum_{r=1}^{m-1}
+\frac{\|Y_r-qY_{r+1}\|^2}{1-q^2}
++\|Y_m\|^2
+=
+\sum_{r<m}\|X_r\|^2+\|Y_m\|^2.
+```
+
+Der aeussere Tail wird kontraktiv auf genau einen normierten exponentiellen Boundary-Modus gekuerzt.
+
+Kontinuierlich gilt dieselbe Identitaet fuer
+
+```math
+Y(u)=\int_u^\infty e^{-(t-u)/2}X(t)dt.
+```
+
+**Status:** `✓[M]`.
+
+### S6. Ein universelles Jump-Feld fuer alle Primzahlen
+
+```math
+X_v(t)=e^{-t/4}K_tv
+```
+
+liefert
+
+```math
+\sqrt{w_{p,k}}K_{k\log p}v
+=\sqrt{\log p}\,X_v(k\log p).
+```
+
+Jeder Primast tastet dasselbe kontinuierliche Feld auf seinem logarithmischen Gitter ab.
+
+**Status:** `✓[M]`.
+
+### S7. Prime-Pole-Diskrepanznormalform
+
+```math
+\Delta
+=\sum_{n=p^k}\frac{\Lambda(n)}{\sqrt n}\delta_{\log n}
+-e^{t/2}dt.
+```
+
+Auf NULLPOL hebt sich der glatte Prime-Hauptterm exakt gegen den Gamma-Grundmodus auf. Der grosse fensterabhaengige Diagonalskalar verschwindet aus der window-free Darstellung.
+
+**Status:** `✓[M]`.
+
+### S8. Critical-half Finite-Part-Transfer
+
+Der kanonische Abel-Finite-Part-Tail ist
+
+```math
+J_\Delta(L)
+=e^{L/2}\left(
+L-\gamma-
+\sum_{\log n<L}\frac{\Lambda(n)}n
+\right),
+```
+
+mit
+
+```math
+J_\Delta(0)=-\gamma,
+\qquad
+d\Delta=\frac12J_\Delta dL-dJ_\Delta.
+```
+
+Damit koppelt die Diskrepanz exakt an den first-order Critical-half-Port
+
+```math
+D_+=\partial_x+\frac12.
+```
+
+**Status:** `✓[M]`.
+
+### S9. Kausale Volterra-Faktorisierung
+
+Mit
+
+```math
+(V_Jv)(x)=\int_{-\infty}^xJ_\Delta(x-y)v(y)dy
+```
+
+und `D_-=D_+^*` gilt
+
+```math
+D_-V_J+V_J^*D_+
+=2\gamma I+\mathcal C_\Delta.
+```
+
+**Status:** `✓[M]`.
+
+### S10. Hoehere Gamma-Schicht ist eine stabile kausale Filterbank
+
+Fuer
+
+```math
+\mu_m=2m+\frac12,
+\qquad m\ge1,
+\qquad y_m=(\partial_x+\mu_m)^{-1}v,
+```
+
+ist
+
+```math
+\int_0^\infty h_1(t)\|K_tv\|^2dt
+=
+\sum_{m\ge1}\frac2{\mu_m}\|y_m'\|^2.
+```
+
+**Status:** `✓[M]`.
+
+### S11. Causal NULLPOL-Normalform
+
+```math
+Q_W(v)
+=
+\sum_{m\ge1}\frac2{\mu_m}\|y_m'\|^2
+-c_*\|v\|^2
+-2\operatorname{Re}\langle D_+v,V_Jv\rangle,
+```
+
+mit
+
+```math
+c_*=\log(8\pi)+\frac\pi2-4-\gamma>0.
+```
+
+**Status:** `✓[M]` als Identitaet; Positivitaet offen.
+
+### S12. Eulerfaktor ist der adjungierte P11-AR(1)-Resolvent
+
+```math
+T_q^*=\sqrt{1-q^2}(I-qS)^{-1}.
+```
+
+Am Root-Impuls ist die Transferfunktion
+
+```math
+\frac1{1-qz}.
+```
+
+Mit `q=p^{-1/2}` und `z=e^{-s\log p}` ist dies exakt
+
+```math
+\frac1{1-p^{-(s+1/2)}}.
+```
+
+**Status:** `✓[M]`; der Eulerfaktor an sich ist klassisch, die Identifikation mit dem vorhandenen P11-Choleskyfaktor ist der projektinterne Brueckenpunkt.
+
+### S13. Strong-Terminal-Geometrie
+
+Normalisiert gilt
+
+```math
+\|R_q-R_q^{(m)}\|=\frac{1+q}{1-q},
+```
+
+also strong, aber keine Normkonvergenz.
+
+Mit Weilgewichten gilt im festen Innenraum fuer Tiefe `L>=2`
+
+```math
+\sup_p\|C_p-C_p^{(m_p(L))}\|
+\le C_0Le^{-L/2}.
+```
+
+Die bewegte Randzone `L_R(x) approx 0` erklaert, warum global Strong-Terminal natuerlich ist, waehrend feste innere Quellen exponentiell gut konvergieren.
+
+**Status:** `✓[M]` fuer den Kanal-Gram-Mechanismus; kein neuer voller raeumlicher C6-Satz.
+
+---
+
+# 2. Explizit geschlossene / verworfene Klassen
+
+```text
+pointwise scattering multiplier positivity                    ×[M]
+primewise contractive P11 innovation -> bare jump shorting     ×[M]
+reverse primewise contraction                                  ×[M]
+scalar Hub repair of stopped-tail defect                       ×[M]
+local positive critical-rate finite-part Hamiltonian           ×[M]
+static higher-Gamma lower-frame repair of lost OU tail         ×[M]
+fixed-window prolate as a scalable all-window RH strategy      demoted
+```
+
+Der Hard Audit hat ausserdem die alte Orbitketten-Exaktheitsformel korrigiert:
+
+```math
+N(t)=\lceil L/t\rceil
+```
+
+(statt `floor(L/t)+1` an den ganzzahligen Verhaeltnissen).
+
+---
+
+# 3. Literaturkollision / was NICHT neu ist
+
+Nicht als Neuheit beanspruchen:
+
+```text
+- RH als positive-real/Herglotz-Kriterium fuer xi'/xi;
+- Lagarias-Positivitaet;
+- Zeta-bezogene Kontroll-/Transfer-Systeme allgemein;
+- Prolate/Paley-Wiener/Toeplitz-Methoden fuer Weil-Positivitaet;
+- Eulerprodukt als Fredholm-/Diagonal-Determinantenschreibweise;
+- AR(1), OU, Green-Kerne oder passive Realisierungstheorie fuer sich.
+```
+
+Die Frequenzbereichsverdichtung ist exakt
+
+```math
+\boxed{
+\frac{\xi'}{\xi}\left(\frac12+s\right)
+=
+(s-\tfrac12)\widehat J_\Delta(s)
++
+\sum_{m\ge1}\frac{s}{\mu_m(s+\mu_m)}
+-\frac{c_*}{2}.
+}
+```
+
+Das zugehoerige positive-real RH-Kriterium ist klassisch. Der Forschungswert dieses Astes kann deshalb nur in der **konstruktiven Herleitung der Komponenten aus P11/stopped-OU/overlap-cone** liegen.
+
+---
+
+# 4. Der eine verbleibende Gate
+
+Die einzige Route, die nach dem Hard Audit noch als substantiell genug fuer Objekt X erscheint, lautet:
+
+```math
+\boxed{
+\text{Konstruiere direkt aus der gestoppten P11-OU-Filtration,}
+\atop
+\text{dem kontinuierlichen Pol-Compensator und der Gamma-Filterbank}
+\text{einen positiven Storage-/konservativen Zustandsraum, dessen}
+\text{Transfer automatisch }\xi'/\xi(1/2+s)\text{ ist.}
+}
+```
+
+Der Zustandsraum muss **vor** jeder Verwendung von Weil-Positivitaet, Lagarias-Positivitaet oder RH definiert sein.
+
+### PASS
+
+Ein kanonischer, coefficient-free positiver Block / Storage-Satz wird aus den vorhandenen OU-/Boundary-Daten abgeleitet und liefert die kausale Normalform. Dann waere die rechte-Halbebenen-Passivitaet strukturell und wuerde ueber das bekannte Lagarias-Kriterium RH implizieren.
+
+### FAIL
+
+Wenn bereits die natuerliche Storage-Klasse, die nur aus
+
+```text
+- stopped OU innovation states,
+- normalized boundary-tail states,
+- continuous compensator,
+- higher Gamma filter states,
+- canonical overlap-cone incidence
+```
+
+besteht, den exakten `R=1` Prime-2 mixed witness nicht reproduzieren kann, wird diese gesamte Architekturklasse als No-Go geschlossen.
+
+Keine freien Kalibrierkoeffizienten duerfen nach dem Witness angepasst werden.
+
+---
+
+# 5. Empfehlung fuer externen Hauptagenten
+
+Nicht die 30+ Audits einzeln lesen. Zuerst genau diese Datei, danach nur die Primarproofs fuer S1, S3/S4, S5/S6, S8/S9, S12/S13.
+
+Destruktiver Reviewauftrag:
+
+```text
+A. Pruefe jede Survivor-Identitaet unabhaengig.
+B. Suche zuerst nach Normalisierungs-/Randtermfehlern in S5, S8 und S9.
+C. Pruefe, ob die P11-Euler-Resolventidentitaet S12 ueber das skalare
+   Impulsmodell hinaus irgendwo ueberinterpretiert wird.
+D. Pruefe die Strong-vs-Norm-Aussage S13 mit den echten Weilgewichten.
+E. Akzeptiere das Lagarias-Kriterium als bekannte Zielidentitaet; bewerte nur,
+   ob die konkrete P11/OU-Geometrie einen nichtzirkulaeren Storage-Kandidaten
+   liefert.
+```
+
+Das ist der aktuelle ehrliche Forschungsstand.
\ No newline at end of file
diff --git a/audits/P11_LAGARIAS_TRANSFER_DECOMPOSITION_2026-09-15.md b/audits/P11_LAGARIAS_TRANSFER_DECOMPOSITION_2026-09-15.md
new file mode 100644
index 00000000..e862cd78
--- /dev/null
+++ b/audits/P11_LAGARIAS_TRANSFER_DECOMPOSITION_2026-09-15.md
@@ -0,0 +1,403 @@
+# P11 Audit — Lagarias transfer decomposition of the causal normal form
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Identifikation der neuen kausalen Normalform mit dem klassischen positive-real RH-Objekt.  
+**Registry:** unveraendert.  
+**Nonclaim:** das positive-real-Kriterium ist klassisch (Lagarias); kein Neuheits- oder RH-Beweis.
+
+---
+
+## 0. Kurzurteil
+
+Die neue Volterra-/Gamma-Normalform setzt sich im Laplace-Bereich exakt zur verschobenen logarithmischen Ableitung der Riemann-Xi-Funktion zusammen.
+
+Definiere
+
+```math
+\boxed{
+G(s)
+:=\frac{\xi'}{\xi}\left(\frac12+s\right).
+}
+```
+
+Mit
+
+```math
+\mu_m=2m+\frac12,
+\qquad
+c_*=\log(8\pi)+\frac\pi2-4-\gamma,
+```
+
+und der Laplacetransformierten des arithmetischen Volterra-Kerns gilt fuer den zunaechst absolut konvergenten Bereich und danach meromorph fortgesetzt:
+
+```math
+\boxed{
+G(s)
+=
+(s-\tfrac12)\widehat J_\Delta(s)
++
+\sum_{m=1}^\infty
+\frac{s}{\mu_m(s+\mu_m)}
+-
+\frac{c_*}{2}.
+}
+```
+
+Der Term
+
+```math
+\frac{s}{\mu_m(s+\mu_m)}
+```
+
+ist die Laplace-/positive-real-Version genau desselben stabilen Gamma-Filters, dessen Zeitbereichsdissipation in der Weilform als
+
+```math
+\frac2{\mu_m}\|y_m'\|^2
+```
+
+auftritt.
+
+Das klassische Kriterium von Lagarias lautet, in dieser verschobenen Variable,
+
+```math
+\boxed{
+RH
+\iff
+\operatorname{Re}G(s)>0
+\quad\text{fuer alle }\operatorname{Re}s>0.
+}
+```
+
+Damit ist die aktuelle Object-X-Frage **kein neues RH-Kriterium**. Sie ist die wesentlich engere konstruktive Frage, ob die bereits unabhaengig gefundene P11/stopped-OU/overlap-cone-Geometrie eine vorwaerts definierte passive Realisierung genau dieser bekannten positive-real-Funktion liefert.
+
+Status:
+
+```text
+exact transfer decomposition into J_Delta + Gamma bank  ✓[M]
+mu_0 + pole collapse to the constant 2                   ✓[M]
+constant equals -c_*/2 after assembly                    ✓[M]
+Lagarias positive-real RH criterion                       known/classical
+P11/OU constructive passive realization                  ?[O]
+```
+
+---
+
+# 1. Completed logarithmic derivative
+
+For
+
+```math
+z=\frac12+s,
+```
+
+the completed function is
+
+```math
+\xi(z)
+=\frac12z(z-1)\pi^{-z/2}\Gamma(z/2)\zeta(z).
+```
+
+Therefore
+
+```math
+\begin{aligned}
+G(s)
+&=\frac1{s+1/2}
+ +\frac1{s-1/2}
+ -\frac12\log\pi\\
+&\quad+
+\frac12\psi\left(\frac14+\frac s2\right)
++\frac{\zeta'}{\zeta}\left(\frac12+s\right).
+\end{aligned}
+```
+
+---
+
+# 2. Arithmetic relative-determinant term
+
+From the previous causal-transfer audit,
+
+```math
+\widehat J_\Delta(s)
+=
+\frac{
+-\gamma
++\zeta'/\zeta(s+1/2)
++1/(s-1/2)
+}{s-1/2}.
+```
+
+Hence
+
+```math
+\boxed{
+\frac1{s-1/2}
++\frac{\zeta'}{\zeta}(s+1/2)
+=
+\gamma+(s-1/2)\widehat J_\Delta(s).
+}
+```
+
+This is the prime/pole part of the completed logarithmic derivative.
+
+---
+
+# 3. Gamma ladder in positive-real form
+
+The digamma partial fraction expansion gives
+
+```math
+\frac12
+\left[
+\psi\left(\frac14+\frac s2\right)-\psi(1/4)
+\right]
+=
+\sum_{m=0}^\infty
+\left(
+\frac1{\mu_m}-\frac1{s+\mu_m}
+\right),
+```
+
+where
+
+```math
+\mu_m=2m+\frac12.
+```
+
+Equivalently,
+
+```math
+\boxed{
+\frac12
+\left[
+\psi\left(\frac14+\frac s2\right)-\psi(1/4)
+\right]
+=
+\sum_{m=0}^\infty
+\frac{s}{\mu_m(s+\mu_m)}.
+}
+```
+
+Since
+
+```math
+\kappa_*=\log\pi-\psi(1/4),
+```
+
+we get
+
+```math
+\boxed{
+-\frac12\log\pi
++\frac12\psi\left(\frac14+\frac s2\right)
+=
+-\frac{\kappa_*}{2}
++
+\sum_{m=0}^\infty
+\frac{s}{\mu_m(s+\mu_m)}.
+}
+```
+
+For `Re s>0`, each Gamma summand has positive real part:
+
+```math
+\operatorname{Re}
+\frac{s}{\mu(s+\mu)}
+=
+\frac{
+\operatorname{Re}(s)(\operatorname{Re}(s)+\mu)
++(\operatorname{Im}s)^2
+}{
+\mu|s+\mu|^2
+}
+>0.
+```
+
+---
+
+# 4. Ground-mode / pole collapse
+
+The ground Gamma mass is
+
+```math
+\mu_0=\frac12.
+```
+
+Its term is
+
+```math
+\frac{s}{(1/2)(s+1/2)}
+=\frac{2s}{s+1/2}.
+```
+
+Together with the remaining completed-zeta pole factor,
+
+```math
+\frac1{s+1/2},
+```
+
+one gets the exact constant
+
+```math
+\boxed{
+\frac1{s+1/2}
++
+\frac{2s}{s+1/2}
+=2.
+}
+```
+
+Thus no `s`-dependence from this pair remains.
+
+---
+
+# 5. Final assembly
+
+Combining Sections 1--4 gives
+
+```math
+G(s)
+=
+(s-1/2)\widehat J_\Delta(s)
++
+\sum_{m=1}^\infty\frac{s}{\mu_m(s+\mu_m)}
++
+2+\gamma-\frac{\kappa_*}{2}.
+```
+
+But
+
+```math
+c_*=\kappa_*-4-2\gamma,
+```
+
+so
+
+```math
+2+\gamma-\frac{\kappa_*}{2}
+=-\frac{c_*}{2}.
+```
+
+Therefore
+
+```math
+\boxed{
+G(s)
+=
+(s-\tfrac12)\widehat J_\Delta(s)
++
+\sum_{m=1}^\infty\frac{s}{\mu_m(s+\mu_m)}
+-
+\frac{c_*}{2}.
+}
+```
+
+This is exactly the frequency/Laplace counterpart of the causal dissipation-feedback decomposition of the Weil quadratic form.
+
+---
+
+# 6. Collision with the classical Lagarias criterion
+
+Jeffrey Lagarias proved/studied the classical positivity characterization
+
+```math
+RH
+\iff
+\operatorname{Re}\frac{\xi'}{\xi}(z)>0
+\quad\text{for }\operatorname{Re}z>1/2.
+```
+
+After `z=1/2+s`, this is
+
+```math
+\boxed{
+RH
+\iff
+\operatorname{Re}G(s)>0
+\quad(\operatorname{Re}s>0).
+}
+```
+
+Thus the abstract statement
+
+```text
+"find a passive realization of G"
+```
+
+is not itself new; it is another expression of a known positive-real RH criterion.
+
+The project-specific content of the present chain is the exact decomposition of `G` into components that arose independently from the finite-window/P11 analysis:
+
+```text
+arithmetic channel:   (s-1/2) Jhat_Delta(s)
+higher Gamma bank:    sum_{m>=1} s/[mu_m(s+mu_m)]
+fixed leakage:        -c_*/2
+```
+
+and the fact that the same arithmetic channel was obtained geometrically from the stopped OU boundary-depth system.
+
+---
+
+# 7. Boundary Herglotz interpretation under RH
+
+If RH holds, the nontrivial zeros are
+
+```math
+\rho=\frac12+i\gamma.
+```
+
+The Hadamard/logarithmic-derivative representation then has the schematic positive-real form
+
+```math
+G(s)
+=\sum_\gamma \frac{m_\gamma}{s-i\gamma}
+```
+
+with the standard symmetric regularization. For `Re s>0`, every summand has positive real part.
+
+Its boundary real part as `Re s downarrow 0` is, distributionally, a positive measure supported at the ordinates `gamma`. This is the same zero-side positivity that underlies Weil's quadratic criterion.
+
+This observation is classical/Herglotz in nature; it is recorded only to show why a successful passive state-space realization would be the correct kind of Object X.
+
+---
+
+# 8. Updated novelty firewall
+
+Known / not claimed as project novelty:
+
+```text
+- RH as positivity of Re(xi'/xi) in the right half-strip;
+- positive-real/Herglotz interpretation;
+- zeta-related transfer-system viewpoints in general;
+- causal-system reformulations by themselves.
+```
+
+Potentially project-specific and still requiring independent literature audit:
+
+```text
+- derivation of the arithmetic impulse kernel J_Delta from the P11 stopped-OU
+  boundary geometry;
+- exact universal boundary-tail shorting of the P11 masked filtration;
+- the time-domain decomposition matching the higher Gamma filter bank and
+  fixed leakage constant with no fitted coefficients.
+```
+
+No novelty is asserted yet.
+
+---
+
+# 9. Sharpened research gate
+
+The only version of the passivity route that would be substantive is:
+
+```math
+\boxed{
+\text{derive a positive storage realization directly from the concrete}
+\atop
+\text{P11 stopped-OU / overlap-cone geometry, and then verify that its}
+\text{transfer is }G(s),\text{ without assuming Lagarias positivity.}
+}
+```
+
+Invoking a generic Herglotz/passive realization theorem would be circular, because the required positive-real property is already equivalent to RH.
diff --git a/audits/P11_NP_GAP_PROLATE_COMPRESSION_2026-09-13.md b/audits/P11_NP_GAP_PROLATE_COMPRESSION_2026-09-13.md
new file mode 100644
index 00000000..0e9e402d
--- /dev/null
+++ b/audits/P11_NP_GAP_PROLATE_COMPRESSION_2026-09-13.md
@@ -0,0 +1,578 @@
+# P11 Audit — NP-GAP as a rank-2-shorted prolate Birman--Schwinger problem
+
+**Datum:** 13. September 2026  
+**Basisbranch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Strukturreduktion plus klar getrennte numerische Diagnostik.  
+**Registry:** unverändert.  
+**Nonclaim:** kein globaler NP-GAP-/Object-X-/RH-Beweis.
+
+---
+
+## 0. Kurzurteil
+
+Die aktuelle Critical-half-/Scattering-Front kann auf jedem festen Fenster als **kompaktes Prolate-/Birman--Schwinger-Problem** formuliert werden.
+
+Die zwei NULLPOL-Bedingungen werden dabei nicht nur als Nebenbedingungen behandelt, sondern exakt als Rang-2-Shorting des Paley--Wiener-Reproduktionskerns.
+
+Fuer das Laborfenster `a=1/2` zeigt die numerische Diagnostik, dass diese Rang-2-Kompression die Spur des gesamten negativen Multiplikatorblocks sehr stark reduziert (ungefaehr von `2.775` auf `0.450`). Das ist **keine Zertifizierung**, aber ein starker Hinweis, dass die richtige positive Architektur eine komprimierte Prolate-/Hardy-Geometrie und keine punktweise Faktorisierung ist.
+
+Status:
+
+```text
+centered-gap monotonicity in a                         ✓[M]
+exact translation-orbit chain bound                    ✓[M]
+NULLPOL reproducing-kernel rank-2 shorting              ✓[M]
+negative multiplier block trace class                   ✓[M]
+explicit trace formula                                  ✓[M]
+prolate/Birman--Schwinger reduction                     ✓[M]_part
+finite-mode / trace diagnostics at a=1/2                diagnostic only
+sharp NP-GAP all a                                      ?[O]
+Object X / RH                                           ?[O]
+```
+
+---
+
+## 1. Centered gap is monotone under window exhaustion
+
+Let
+
+```math
+\mathscr D_{NP,a}
+= C_c^\infty(-a,a)\cap\ker M(0)\cap\ker M(1).
+```
+
+Define the centered Rayleigh infimum
+
+```math
+\boxed{
+g(a)
+:=
+\inf_{0\ne v\in\mathscr D_{NP,a}}
+\frac{Q_W(v)}{\|v\|_2^2}.
+}
+```
+
+If `0<a<b`, then
+
+```math
+\mathscr D_{NP,a}\subset\mathscr D_{NP,b}.
+```
+
+Moreover the centered COMMON-JUMP form is cutoff-gauge invariant on old vectors:
+
+```math
+Q_{W,b}(v)=Q_{W,a}(v)
+\qquad(v\in\mathscr D_{NP,a}).
+```
+
+Therefore
+
+```math
+\boxed{
+g(b)\le g(a).}
+```
+
+Thus new prime thresholds do **not** raise the centered gap. They only rewrite the same global Weil form on a larger support class.
+
+Equivalently, if
+
+```math
+\lambda_{NP}(a)
+=\inf\frac{\|X_av\|^2}{\|v\|^2},
+```
+
+then
+
+```math
+\boxed{
+g(a)=\lambda_{NP}(a)-\Gamma_a}
+```
+
+is the invariant monotone quantity.
+
+Strategic consequence: the all-window problem is an exhaustion/direct-limit problem, not an induction in which each newly entering prime repairs the previous radius.
+
+---
+
+## 2. Exact orbit-chain lower bound for one translation jump
+
+Let `I=(-a,a)` have length `L=2a`, let `P_I` be multiplication by `1_I`, and let
+
+```math
+A_t:=P_I(T_t+T_{-t})P_I.
+```
+
+For fixed `t>0`, decompose `I` into orbits modulo `t`. Every orbit meets `I` in a finite chain. The maximal chain length is
+
+```math
+N(t)=\left\lfloor\frac{L}{t}\right\rfloor+1
+\qquad(0<t\le L),
+```
+
+while for `t>L` every chain has length one.
+
+On a chain of length `N`, `A_t` is the adjacency matrix of the path graph `P_N`, whose largest eigenvalue is
+
+```math
+2\cos\frac{\pi}{N+1}.
+```
+
+Hence
+
+```math
+\boxed{
+\|A_t\|
+=2\cos\frac{\pi}{N(t)+1}
+\qquad(0<t\le L),
+}
+```
+
+and `A_t=0` for `t>L`.
+
+Since
+
+```math
+K_t^*K_t=2I-A_t,
+```
+
+we obtain the exact support-only bound
+
+```math
+\boxed{
+\|K_tv\|_2^2
+\ge
+c_L(t)\|v\|_2^2,
+}
+```
+
+with
+
+```math
+c_L(t)=
+2-2\cos\frac{\pi}{N(t)+1}
+\quad(0<t\le L),
+```
+
+and `c_L(t)=2` for `t>L`.
+
+Checks:
+
+- if `t>L/2`, then `N=2` and `c_L(t)=1`;
+- if `t>L`, then `c_L(t)=2`;
+- as `t\downarrow0`, `c_L(t)\sim \pi^2t^2/L^2`.
+
+This gives an exact discrete-orbit form of the finite-window nonlocal Poincare mechanism. At `a=1/2`, integrating this bound against `h(t)dt` is rigorous in principle but numerically weaker than the previously derived Schur/resolvent bound; it is therefore retained as a structural lemma, not as the main positivity route.
+
+---
+
+## 3. Paley--Wiener formulation and the finite-window multiplier
+
+Use the unitary Fourier transform. Let
+
+```math
+PW_a=\mathcal F L^2(-a,a).
+```
+
+On NULLPOL define
+
+```math
+\mathcal N_a
+:=
+\{F\in PW_a:F(i/2)=F(-i/2)=0\}.
+```
+
+By the Critical-half range theorem,
+
+```math
+\boxed{
+\mathcal N_a=(z^2+1/4)PW_a
+}
+```
+
+on the smooth test class.
+
+For the canonical finite-window COMMON-JUMP representation the centered multiplier is
+
+```math
+\boxed{
+\tau_a(z)
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi
+-2\sum_{\log n\le2a}
+\frac{\Lambda(n)}{\sqrt n}\cos(z\log n).
+}
+```
+
+Then
+
+```math
+\boxed{
+Q_W(v)=\int_{\mathbb R}|F(z)|^2\tau_a(z)\,dz,
+\qquad F=\widehat v\in\mathcal N_a.
+}
+```
+
+The multiplier satisfies
+
+```math
+\tau_a(z)=\log\frac{|z|}{2\pi}+O_a(1)
+\qquad(|z|\to\infty),
+```
+
+so its negative set is compact.
+
+---
+
+## 4. Exact NULLPOL reproducing-kernel shorting
+
+The Paley--Wiener reproducing kernel is
+
+```math
+\boxed{
+K_a(z,w)
+=
+\frac{\sin(a(z-\overline w))}
+{\pi(z-\overline w)}.
+}
+```
+
+Let
+
+```math
+\lambda_+=i/2,
+\qquad
+\lambda_-=-i/2.
+```
+
+Define the two kernel vectors
+
+```math
+k_\pm(z)=K_a(z,\lambda_\pm)
+```
+
+and the `2x2` Gram matrix
+
+```math
+G_{\sigma\tau}=K_a(\lambda_\sigma,\lambda_\tau).
+```
+
+Explicitly,
+
+```math
+\boxed{
+G
+=\frac1\pi
+\begin{pmatrix}
+\sinh a & a\\
+a & \sinh a
+\end{pmatrix}.
+}
+```
+
+Since `sinh a>a` for `a>0`, `G` is strictly positive.
+
+The orthogonal projection from `PW_a` onto the codimension-2 NULLPOL space therefore has reproducing kernel
+
+```math
+\boxed{
+K_{NP,a}(z,w)
+=
+K_a(z,w)
+-k(z)^*G^{-1}k(w),
+}
+```
+
+where
+
+```math
+k(z)=(k_+(z),k_-(z))^T.
+```
+
+Thus NULLPOL is an **exact rank-2 shorting** of Paley--Wiener geometry.
+
+This is the reproducing-kernel counterpart of
+
+```math
+\mathscr D_{NP,a}=L_{1/2}C_c^\infty(-a,a).
+```
+
+---
+
+## 5. The entire negative danger is a compact trace-class block
+
+Split
+
+```math
+\tau_a=\tau_{a,+}-\tau_{a,-},
+\qquad
+\tau_{a,\pm}\ge0.
+```
+
+On `\mathcal N_a` set
+
+```math
+A_{a,-}
+:=P_{\mathcal N_a}M_{\tau_{a,-}}P_{\mathcal N_a}.
+```
+
+Because `tau_{a,-}` has compact support and `K_{NP,a}(z,z)` is locally bounded, `A_{a,-}` is positive trace class. Its trace is exactly
+
+```math
+\boxed{
+\operatorname{tr}A_{a,-}
+=
+\int_{\mathbb R}
+\tau_{a,-}(z)K_{NP,a}(z,z)\,dz.
+}
+```
+
+In particular
+
+```math
+\|A_{a,-}\|
+\le
+\operatorname{tr}A_{a,-}.
+```
+
+More generally, for every bounded Borel set `E`, the NULLPOL concentration operator
+
+```math
+C_{a,E}
+=P_{\mathcal N_a}M_{1_E}P_{\mathcal N_a}
+```
+
+is trace class with
+
+```math
+\boxed{
+\operatorname{tr}C_{a,E}
+=\int_EK_{NP,a}(z,z)\,dz.
+}
+```
+
+This is precisely a rank-2-shorted prolate/time-band concentration operator.
+
+---
+
+## 6. Birman--Schwinger / prolate reduction
+
+Let
+
+```math
+A_{a,+}
+=P_{\mathcal N_a}M_{\tau_{a,+}}P_{\mathcal N_a}.
+```
+
+Then
+
+```math
+Q_W(F)=\langle F,(A_{a,+}-A_{a,-})F\rangle.
+```
+
+Because `tau_a(z)\to+infty`, the positive part controls the complement of a bounded frequency set. The only possible negative spectrum of the compressed Weil form is therefore generated by the compact block `A_{a,-}`.
+
+Whenever a positive reference lower bound for `A_{a,+}` is fixed, the remaining test is a compact Birman--Schwinger norm condition of the form
+
+```math
+\boxed{
+\left\|
+A_{a,+}^{-1/2}A_{a,-}A_{a,+}^{-1/2}
+\right\|\le1.
+}
+```
+
+The exact implementation may use a slightly shifted/coercive positive reference operator if convenient; the key theorem-level point is that the negative side is trace class and belongs to an explicit prolate shorting.
+
+Therefore a rigorous proof can be organized by:
+
+1. certify the finitely many leading eigenvalues of the shorted concentration/Birman--Schwinger operator;
+2. control the remaining tail by its trace or Hilbert--Schmidt norm;
+3. keep the high-frequency positive part analytically outside the finite numerical block.
+
+This is a non-circular route: no Weil positivity is used as input.
+
+---
+
+## 7. Special laboratory `a=1/2`
+
+At
+
+```math
+a=1/2
+```
+
+only the prime power `n=2` appears in the canonical finite-window multiplier:
+
+```math
+\boxed{
+\tau_{1/2}(z)
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi
+-\sqrt2\log2\,\cos(z\log2).
+}
+```
+
+The exact threshold is
+
+```math
+\Gamma_{1/2}
+=\kappa_*+\sqrt2\log2.
+```
+
+### 7.1 Prime shift has an exact two-edge block
+
+Let `t=log2`. Since
+
+```math
+1/2<t<1=2a,
+```
+
+every `t`-orbit in the support interval has length at most two. If
+
+```math
+C_t=P_IT_tP_I,
+```
+
+then
+
+```math
+C_t^2=0,
+```
+
+and `C_t+C_t^*` is fiberwise the `2x2` path adjacency matrix on the two edge intervals, with spectrum contained in
+
+```text
+{-1,0,+1}.
+```
+
+Hence the centered prime contribution
+
+```math
+-w_2(C_t+C_t^*)
+```
+
+is an explicit matched-edge defect of operator norm exactly `w_2=(log2)/sqrt2`.
+
+This isolates the only arithmetic negative block at `a=1/2`.
+
+### 7.2 Finite gamma-mode diagnostic
+
+Using the positive resolvent expansion
+
+```math
+\Phi_\infty(D)
+=\sum_{m\ge0}
+\frac2{\mu_m}D^2(D^2+\mu_m^2)^{-1},
+\qquad
+\mu_m=2m+1/2,
+```
+
+an independent Ritz diagnostic on smooth Critical-half lifted trial spaces indicates that the first `40` positive gamma modes together with the `p=2` jump already cross the exact threshold `Gamma_{1/2}`; the remaining gamma tail is positive.
+
+This is **diagnostic only**: a finite Ritz lower eigenvalue is not a lower bound for the full operator. The significance is strategic: a proof at `a=1/2` can plausibly be reduced to a finite rational-resolvent certification plus a positive tail, rather than requiring the whole infinite gamma ladder at once.
+
+### 7.3 Rank-2 NULLPOL trace reduction diagnostic
+
+For `a=1/2`, numerical quadrature of the exact trace formula gives approximately
+
+```text
+unshorted PW negative trace     2.775
+NULLPOL-shorted negative trace  0.450
+```
+
+for the canonical finite multiplier above.
+
+Thus the two exact NULLPOL evaluations remove roughly `84%` of the dangerous negative trace in this laboratory.
+
+Again: these numbers are **not theorem certificates** until enclosed with interval arithmetic. The exact theorem is the trace formula itself.
+
+A pure one-line trace domination using only
+
+```math
+\|A_-\|\le tr(A_-)
+```
+
+and
+
+```math
+\|C_E\|\le tr(C_E)
+```
+
+is still too coarse; at least the leading shorted prolate eigenvalue(s) must be controlled rather than replacing the whole concentration norm by its trace.
+
+---
+
+## 8. Narrow No-Go for a naive Blaschke cancellation
+
+The prime scattering factor can be written
+
+```math
+S_p(z)
+=
+\frac{w}{b_q(w)},
+\qquad
+w=e^{iz\log p},
+\qquad
+q=p^{-1/2},
+```
+
+where
+
+```math
+b_q(w)=\frac{w-q}{1-qw}
+```
+
+is a degree-one Blaschke factor.
+
+Its central zero corresponds to `z=i/2`, suggesting a rank-one model-space relation. However the exponential covering produces the full periodic family
+
+```math
+z=i/2+\frac{2\pi k}{\log p},
+\qquad k\in\mathbb Z.
+```
+
+The single NULLPOL condition `F(i/2)=0` does not annihilate the entire periodic prime defect. Equivalently, in physical space one can choose `u` as two small matched bumps separated by `log p`; then `v=L_{1/2}u` is automatically NULLPOL while the prime shift correlation remains nonzero.
+
+Therefore:
+
+```text
+"central NULLPOL zero alone kills the full prime Blaschke defect"  ×[M]
+```
+
+The correct object is the full shorted Paley--Wiener/prolate compression above.
+
+---
+
+## 9. New preferred gate
+
+Replace the previous vague `CRIT-HALF-COMPRESS-1` by the precise gate
+
+```text
+NP-PROLATE-1 / a=1/2
+```
+
+Goal:
+
+1. work with the exact rank-2-shorted kernel `K_{NP,1/2}`;
+2. isolate the compact negative set of `tau_{1/2}`;
+3. certify the leading eigenvalues of the corresponding weighted prolate/Birman--Schwinger operator with Arb or another rigorous interval method;
+4. bound the residual spectrum by trace/Hilbert--Schmidt tails;
+5. use analytic high-frequency positivity outside the certified block.
+
+PASS would give a genuinely forward proof of the sharp NULLPOL lower-frame bound at `a=1/2` in the new Critical-half architecture.
+
+FAIL would provide an explicit negative eigenfunction and therefore a decisive local counterexample to the current route.
+
+---
+
+## 10. Firewalls
+
+Do not claim from this audit alone:
+
+- NP-GAP at `a=1/2` is proved;
+- the diagnostic traces are certified intervals;
+- a finite Ritz PASS proves the infinite-dimensional operator inequality;
+- the negative block is finite rank (it is trace class, generally infinite rank);
+- the central Blaschke zero by itself removes the prime defect;
+- all-window NP-GAP, Object X, or RH is solved;
+- publication novelty is established.
diff --git a/audits/P11_NP_PROLATE_TRACE_RITZ_CERTIFICATE_2026-09-13.md b/audits/P11_NP_PROLATE_TRACE_RITZ_CERTIFICATE_2026-09-13.md
new file mode 100644
index 00000000..2410386b
--- /dev/null
+++ b/audits/P11_NP_PROLATE_TRACE_RITZ_CERTIFICATE_2026-09-13.md
@@ -0,0 +1,387 @@
+# P11 Audit — Trace-minus-Ritz certificate for the shorted NP prolate defect
+
+**Datum:** 13. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level certification lemma plus unc ertified `a=1/2` diagnostics.  
+**Registry:** unveraendert.  
+**Nonclaim:** the numerical margins below are not yet Arb-certified and therefore are not yet a theorem-level NP-GAP PASS.
+
+---
+
+## 0. Purpose
+
+The previous prolate audit reduces the finite-window NULLPOL Weil form to a compact weighted prolate defect.
+
+For a fixed `a>0`, let
+
+```math
+\mathcal N_a
+=\{F\in PW_a:F(i/2)=F(-i/2)=0\},
+```
+
+and let `tau_a` be the canonical centered finite-window multiplier.
+
+For any `c>0` define
+
+```math
+\boxed{
+D_{a,c}
+:=
+P_{\mathcal N_a}
+M_{(c-\tau_a)_+}
+P_{\mathcal N_a}.
+}
+```
+
+Since `(c-tau_a)_+` has compact support, `D_{a,c}` is positive trace class.
+
+Pointwise
+
+```math
+\tau_a(z)
+\ge
+c-(c-\tau_a(z))_+,
+```
+
+hence for `F in N_a`
+
+```math
+\boxed{
+Q_W(F)
+\ge
+c\|F\|^2
+-\langle F,D_{a,c}F\rangle.
+}
+```
+
+Therefore
+
+```math
+\boxed{
+\|D_{a,c}\|<c
+\quad\Longrightarrow\quad
+Q_W(F)>0
+\ \text{for all }0\ne F\in\mathcal N_a.
+}
+```
+
+The point of this audit is to make `||D||<c` rigorously certifiable without an infinite-dimensional Galerkin error analysis.
+
+---
+
+# 1. Exact trace formula
+
+The rank-2-shorted NULLPOL reproducing kernel is
+
+```math
+K_{NP,a}(z,w)
+=
+K_a(z,w)-k(z)^*G^{-1}k(w),
+```
+
+with
+
+```math
+K_a(z,w)
+=\frac{\sin(a(z-\overline w))}
+{\pi(z-\overline w)}.
+```
+
+Hence
+
+```math
+\boxed{
+\operatorname{tr}D_{a,c}
+=
+\int_{\mathbb R}
+(c-\tau_a(z))_+
+K_{NP,a}(z,z)\,dz.
+}
+```
+
+This is a one-dimensional compact-support integral.
+
+---
+
+# 2. Trace-minus-Ritz upper certificate for the top eigenvalue
+
+Let the eigenvalues of `D=D_{a,c}` be
+
+```math
+\lambda_1\ge\lambda_2\ge\cdots\ge0.
+```
+
+Choose any `m`-dimensional subspace `V_m` of the Hilbert space and let
+
+```math
+\theta_1\ge\theta_2\ge\cdots\ge\theta_m\ge0
+```
+
+be the Ritz eigenvalues of the compression `P_{V_m}DP_{V_m}`.
+
+By the min-max principle,
+
+```math
+\boxed{
+\theta_j\le\lambda_j
+\qquad(1\le j\le m).
+}
+```
+
+Because `D` is positive trace class,
+
+```math
+\operatorname{tr}D
+=\sum_{j\ge1}\lambda_j.
+```
+
+Therefore
+
+```math
+\begin{aligned}
+\lambda_1
+&=\operatorname{tr}D-\sum_{j\ge2}\lambda_j\\
+&\le
+\operatorname{tr}D-\sum_{j=2}^{m}\lambda_j\\
+&\le
+\operatorname{tr}D-\sum_{j=2}^{m}\theta_j.
+\end{aligned}
+```
+
+Thus the exact certification lemma is
+
+```math
+\boxed{
+\operatorname{tr}D_{a,c}
+-
+\sum_{j=2}^{m}\theta_j
+<c
+\quad\Longrightarrow\quad
+\|D_{a,c}\|<c.
+}
+```
+
+Combining with §0:
+
+```math
+\boxed{
+\operatorname{tr}D_{a,c}
+-
+\sum_{j=2}^{m}\theta_j
+<c
+\quad\Longrightarrow\quad
+Q_W>0\text{ on }\mathcal N_a\setminus\{0\}.
+}
+```
+
+This is useful because a rigorous certificate needs only:
+
+1. an **upper interval bound** for the scalar trace integral;
+2. rigorous **lower interval bounds** for finitely many Ritz eigenvalues `theta_2,...,theta_m`.
+
+No upper eigenvalue bound, no full-space Galerkin convergence theorem, and no estimate of the unresolved infinite Ritz tail are required.
+
+---
+
+# 3. Parity splitting reduces NULLPOL to one shorting condition per sector
+
+The multiplier `tau_a` is even. Hence `D_{a,c}` commutes with parity and decomposes as
+
+```math
+D_{a,c}=D_{a,c}^{even}\oplus D_{a,c}^{odd}.
+```
+
+For an even Paley--Wiener function,
+
+```math
+F(-i/2)=F(i/2),
+```
+
+so the two NULLPOL conditions reduce to the single condition
+
+```math
+F(i/2)=0.
+```
+
+For an odd function,
+
+```math
+F(-i/2)=-F(i/2),
+```
+
+and again NULLPOL is one scalar condition.
+
+Thus the global rank-2 shorting is exactly
+
+```text
+one rank-1 shorting in the even sector
++
+one rank-1 shorting in the odd sector.
+```
+
+The trace-minus-Ritz certificate can and should therefore be applied separately in the two parity sectors.
+
+---
+
+# 4. `a=1/2` diagnostics
+
+For
+
+```math
+a=1/2
+```
+
+the canonical finite-window multiplier is
+
+```math
+\tau_{1/2}(z)
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)
+-\log\pi
+-\sqrt2\log2\cos(z\log2).
+```
+
+Numerical Nyström diagnostics using the **exact shorted reproducing kernel** give for the pure negative block `D_{1/2,0}` approximately
+
+```text
+even leading eigenvalue  0.2258375288...
+odd  leading eigenvalue  0.2059078829...
+```
+
+and rapid spectral falloff after the first mode in each parity.
+
+A more useful shifted defect is `D_{a,c}`.
+
+### Even sector
+
+Near
+
+```text
+c = 1.675
+```
+
+the diagnostic top eigenvalue is
+
+```text
+||D_even|| ≈ 1.665877817
+```
+
+so
+
+```text
+||D_even||/c ≈ 0.994553921.
+```
+
+The direct margin is only about `9.1e-3`, so a rigorous upper norm enclosure must be reasonably sharp.
+
+However the trace-minus-Ritz certificate is favorable: diagnostic Ritz data indicate that using approximately
+
+```text
+m = 12
+```
+
+already gives
+
+```text
+tr(D_even) - sum_{j=2}^{12} theta_j
+≈ 1.66903
+< 1.675,
+```
+
+with margin about
+
+```text
+5.9e-3.
+```
+
+### Odd sector
+
+Near
+
+```text
+c = 1.25
+```
+
+the diagnostic top eigenvalue is
+
+```text
+||D_odd|| ≈ 1.115100777,
+```
+
+so the margin is large.
+
+Already roughly
+
+```text
+m = 5
+```
+
+Ritz dimensions are diagnostically enough for the trace-minus-Ritz upper certificate.
+
+### Firewall
+
+All numbers in this section are floating/Nyström diagnostics only. They motivate the finite certificate dimensions; they are not accepted as proof inputs.
+
+---
+
+# 5. Why this is stronger than a standard Ritz PASS
+
+A conventional finite Ritz calculation can only show that some finite-dimensional restriction is positive; it does not control the missing infinite-dimensional directions.
+
+The present certificate reverses the useful inequality:
+
+- Ritz eigenvalues of a positive compact operator are **lower bounds** for the corresponding exact eigenvalues;
+- subtracting those lower bounds from an exact trace gives an **upper bound** for the dangerous top eigenvalue.
+
+Thus the uncomputed infinite tail helps rather than hurts: it is positive and is already included in the trace.
+
+This is the crucial certification advantage.
+
+---
+
+# 6. Recommended rigorous implementation
+
+For each parity separately:
+
+1. choose a rational/interval value of `c` near the diagnostic optimum;
+2. enclose all roots of `tau_{1/2}(z)=c` that delimit the compact support of `(c-tau)_+`;
+3. certify the trace integral with Arb;
+4. freeze an explicit finite Ritz family (e.g. shorted kernel sections or low-order orthogonalized polynomial/prolate trial functions);
+5. interval-certify its Gram and weighted-operator matrices;
+6. obtain rigorous lower bounds for `theta_2,...,theta_m` by interval Cholesky/Sturm/generalized-eigenvalue bracketing;
+7. verify
+
+```math
+tr(D)-\sum_{j=2}^{m}\theta_j<c.
+```
+
+Suggested initial dimensions from diagnostics:
+
+```text
+odd:  m=5 or modestly larger for safety
+even: m=12, preferably 14--16 for interval margin
+```
+
+No statement should be promoted until the interval margins are strictly separated from zero.
+
+---
+
+# 7. Strategic status
+
+If the certificate closes, `a=1/2` becomes the first sharp NULLPOL NP-GAP theorem proved directly from the new Critical-half/prolate architecture.
+
+That would not be a new RH result by itself, and fixed-`a` positivity is not claimed to be RH-equivalent. Its importance would be architectural: it would demonstrate that the same rank-2-shortened Paley--Wiener geometry that arose from `L_{1/2}` is quantitatively strong enough to dominate the sign-changing Euler/Gamma scattering multiplier.
+
+If the certificate fails after rigorous optimization, the failure will identify the exact parity/eigenmode that obstructs the current prolate route.
+
+---
+
+# 8. Nonclaims
+
+Do not claim yet:
+
+- `a=1/2` has been interval-certified by this audit;
+- the floating margins are theorem data;
+- the all-window NP-GAP follows from one fixed radius;
+- the trace-minus-Ritz lemma alone proves RH;
+- publication novelty is established.
diff --git a/audits/P11_NULLPOL_PRIME_DISCREPANCY_CANCELLATION_2026-09-15.md b/audits/P11_NULLPOL_PRIME_DISCREPANCY_CANCELLATION_2026-09-15.md
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--- /dev/null
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@@ -0,0 +1,609 @@
+# P11 Audit — NULLPOL prime-discrepancy cancellation and OU root-trace bridge
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Umformung der NULLPOL-Weilform nach dem Hard Audit.  
+**Registry:** unverändert.  
+**Nonclaim:** kein Positivitäts-, Object-X-, RH- oder Publikationsneuheitsclaim.
+
+---
+
+## 0. Kurzurteil
+
+Auf NULLPOL kann der exponentiell wachsende glatte Hauptterm der Prime-Power-Maßes **exakt** gegen den Grundmodus `mu_0=1/2` der Gamma-/Green-Schicht eliminiert werden.
+
+Definiere die lokal endliche Prime-Power-Maßstruktur
+
+```math
+\nu_P
+:=
+\sum_{n=p^k\ge2}
+\frac{\Lambda(n)}{\sqrt n}\,\delta_{\log n}
+```
+
+und ihre polbereinigte Diskrepanz
+
+```math
+\boxed{
+\Delta
+:=
+\nu_P-e^{t/2}\,dt.
+}
+```
+
+Weiter setze
+
+```math
+h_1(t)
+:=
+h(t)-e^{-t/2}
+=
+\frac{e^{-5t/2}}{1-e^{-2t}}.
+```
+
+Dann gilt fuer jede kompakt getragene NULLPOL-Testfunktion `v` exakt
+
+```math
+\boxed{
+Q_W(v)
+=
+\int_0^\infty h_1(t)\|K_tv\|_2^2dt
+-(\kappa_*-4)\|v\|_2^2
+-
+\int_0^\infty
+\bigl(f_v(t)+f_v(-t)\bigr)\,d\Delta(t),
+}
+```
+
+wobei
+
+```math
+f_v=v*\widetilde v,
+\qquad
+K_t=T_{t/2}-T_{-t/2}.
+```
+
+Die Darstellung ist **fensterfrei**: fuer kompakt getragenes `v` ist der atomare Anteil automatisch lokal endlich. Der wachsende Fenster-Skalar
+
+```math
+2\sum_{\log n\le2a}\Lambda(n)/\sqrt n
+```
+
+tritt nicht mehr auf. Die gesamte radiusabhaengige arithmetische Schwierigkeit sitzt stattdessen in der signierten Diskrepanz `Delta`.
+
+Zusaetzlich besitzt `Delta` eine exakte OU-/P11-Rootspurinterpretation. In der normierten OU-Familie mit Rootkorrelation `e^{-sigma t}` gilt fuer `sigma>1/4`
+
+```math
+\boxed{
+\operatorname{tr}H_{P}^{(\sigma)}
+-
+\operatorname{tr}H_{0}^{(\sigma)}
+=
+-\frac{\zeta'}{\zeta}\left(\frac12+2\sigma\right)
+-
+\frac1{2\sigma-1/2}
+=
+\mathcal L\Delta(2\sigma).
+}
+```
+
+Bei der P11-/Critical-half-Skala `sigma=1/2` ist dies
+
+```math
+-\zeta'/\zeta(3/2)-2.
+```
+
+Das ist eine exakte Bruecke zwischen dem P11-Root und der neuen Diskrepanznormalform. Sie ist **noch keine positive Differenz- oder Schur-Komplement-Realisierung**.
+
+Status:
+
+```text
+NULLPOL growing/decaying correlation identity            ✓[M]
+prime-main / Gamma-ground cancellation                    ✓[M]
+window-free discrepancy normal form                       ✓[M]
+Laplace transform of Delta                                ✓[M]
+OU/P11 root-trace identity for Delta                      ✓[M]
+polynomial cumulative Delta bound <=> RH-scale PNT error  ✓[M] (standard equivalence route)
+positive Schur/dilation realization of Delta              ?[O]
+full Object X / RH                                        ?[O]
+publication novelty                                       ?[O]
+```
+
+---
+
+# 1. Starting point: the additive explicit formula
+
+Use the same normalization as `P11_NP_COMMON_JUMP_GRAM_2026-09-13.md`.
+For
+
+```math
+f_v=v*\widetilde v,
+\qquad
+\widetilde v(x)=\overline{v(-x)},
+```
+
+write
+
+```math
+F_v(t):=f_v(t)+f_v(-t).
+```
+
+The explicit formula reads
+
+```math
+\begin{aligned}
+Q_W(v)
+={}&
+\int_{\mathbb R}f_v(x)(e^{x/2}+e^{-x/2})dx\\
+&-
+\int_0^\infty F_v(t)\,d\nu_P(t)\\
+&-(\log4\pi+\gamma)f_v(0)\\
+&-
+\int_0^\infty
+\{F_v(t)-2e^{-t/2}f_v(0)\}h(t)dt,
+\end{aligned}
+```
+
+where
+
+```math
+h(t)=\frac{e^{-t/2}}{1-e^{-2t}}.
+```
+
+For compactly supported `v`, the `nu_P` integral is a finite sum.
+
+---
+
+# 2. NULLPOL turns the growing exponential correlation into the decaying one
+
+Define
+
+```math
+E_+(v)=\int e^{x/2}v(x)dx,
+\qquad
+E_-(v)=\int e^{-x/2}v(x)dx.
+```
+
+On NULLPOL,
+
+```math
+E_+(v)=E_-(v)=0.
+```
+
+By Fubini,
+
+```math
+\int_{\mathbb R}f_v(t)e^{t/2}dt
+=E_+(v)\overline{E_-(v)}=0,
+```
+
+and similarly
+
+```math
+\int_{\mathbb R}f_v(t)e^{-t/2}dt=0.
+```
+
+Splitting both identities at `t=0` and adding them gives
+
+```math
+\int_0^\infty
+(e^{t/2}+e^{-t/2})F_v(t)dt=0.
+```
+
+Hence
+
+```math
+\boxed{
+\int_0^\infty e^{t/2}F_v(t)dt
+=
+-\int_0^\infty e^{-t/2}F_v(t)dt.
+}
+```
+
+This is exact and uses only NULLPOL. No PNT, RH or asymptotic prime information is used.
+
+Equivalently, in the correlation notation `C_v(t)=<v,T_t v>` one has
+
+```math
+\boxed{
+\Re\int_0^\infty e^{t/2}C_v(t)dt
+=
+-\Re\int_0^\infty e^{-t/2}C_v(t)dt.
+}
+```
+
+The growing `e^{t/2}` mode is therefore conjugate, on NULLPOL, to the decaying Critical-half Green mode.
+
+---
+
+# 3. Exact cancellation with the Gamma ground mode
+
+Split
+
+```math
+h(t)=e^{-t/2}+h_1(t),
+\qquad
+h_1(t)=\frac{e^{-5t/2}}{1-e^{-2t}}.
+```
+
+Also write
+
+```math
+\nu_P=\Delta+e^{t/2}dt.
+```
+
+The prime term becomes
+
+```math
+-\int F_v\,d\nu_P
+=
+-\int F_v\,d\Delta
+-
+\int_0^\infty e^{t/2}F_v(t)dt.
+```
+
+By the NULLPOL identity in Section 2,
+
+```math
+-\int_0^\infty e^{t/2}F_v(t)dt
+=
++\int_0^\infty e^{-t/2}F_v(t)dt.
+```
+
+The `e^{-t/2}` piece of the archimedean integral is
+
+```math
+-\int_0^\infty
+\{F_v(t)-2e^{-t/2}f_v(0)\}e^{-t/2}dt
+=
+-\int_0^\infty e^{-t/2}F_v(t)dt
++2f_v(0),
+```
+
+because `int_0^infty e^{-t}dt=1`.
+
+The two nonlocal correlation terms cancel exactly.
+
+Thus
+
+```math
+Q_W(v)
+=
+-\int F_v\,d\Delta
++(2-\log4\pi-\gamma)f_v(0)
+-
+\int_0^\infty
+\{F_v(t)-2e^{-t/2}f_v(0)\}h_1(t)dt.
+```
+
+Using
+
+```math
+F_v(t)
+=2f_v(0)-\|K_tv\|_2^2
+```
+
+gives
+
+```math
+-\int
+\{F_v-2e^{-t/2}f_v(0)\}h_1dt
+=
+\int h_1(t)\|K_tv\|_2^2dt
+-2f_v(0)\int h_1(t)(1-e^{-t/2})dt.
+```
+
+From the already proved identity
+
+```math
+\kappa_*
+=
+\log4\pi+\gamma
++2\int_0^\infty h(t)(1-e^{-t/2})dt
+```
+
+and
+
+```math
+2\int_0^\infty e^{-t/2}(1-e^{-t/2})dt=2
+```
+
+one gets
+
+```math
+2\int_0^\infty h_1(t)(1-e^{-t/2})dt
+=
+\kappa_*-(\log4\pi+\gamma)-2.
+```
+
+Therefore the scalar coefficient is
+
+```math
+4-\kappa_*.
+```
+
+This proves
+
+```math
+\boxed{
+Q_W(v)
+=
+\int_0^\infty h_1(t)\|K_tv\|_2^2dt
+-(\kappa_*-4)\|v\|_2^2
+-
+\int_0^\infty F_v(t)\,d\Delta(t).
+}
+```
+
+Numerically only as orientation,
+
+```math
+\kappa_*-4
+=1.3721834192256655\ldots
+```
+
+No numerical value is needed in the proof.
+
+---
+
+# 4. The arithmetic discrepancy and its Laplace transform
+
+For `Re s>1/2`, absolute convergence gives
+
+```math
+\begin{aligned}
+\mathcal L\Delta(s)
+&:=\int_0^\infty e^{-st}\,d\Delta(t)\\
+&=
+\sum_{n\ge2}\frac{\Lambda(n)}{\sqrt n}n^{-s}
+-
+\int_0^\infty e^{-(s-1/2)t}dt\\
+&=
+-\frac{\zeta'}{\zeta}\left(s+\frac12\right)
+-
+\frac1{s-1/2}.
+\end{aligned}
+```
+
+Hence
+
+```math
+\boxed{
+\mathcal L\Delta(s)
+=
+-\frac{\zeta'}{\zeta}\left(s+\frac12\right)
+-
+\frac1{s-1/2}.
+}
+```
+
+The continuous compensator removes exactly the pole at `s+1/2=1` from the Euler logarithmic derivative.
+
+The cumulative discrepancy is
+
+```math
+\boxed{
+D(T)
+:=\Delta([0,T])
+=
+\sum_{n\le e^T}\frac{\Lambda(n)}{\sqrt n}
+-2(e^{T/2}-1).
+}
+```
+
+This is the weighted Chebyshev discrepancy in logarithmic coordinates.
+
+---
+
+# 5. Firewall: polynomial control of D is already RH-scale information
+
+Let
+
+```math
+S(X)=\sum_{n\le X}\frac{\Lambda(n)}{\sqrt n}.
+```
+
+Then `D(log X)=S(X)-2(sqrt(X)-1)`.
+
+If for some fixed `B`
+
+```math
+D(T)=O(T^B),
+```
+
+then
+
+```math
+S(X)=2(\sqrt X-1)+O((\log X)^B).
+```
+
+Stieltjes partial summation in the reverse direction gives
+
+```math
+\psi(X)
+=\sqrt X\,S(X)
+-\frac12\int_1^X S(t)t^{-1/2}dt,
+```
+
+hence
+
+```math
+\boxed{
+\psi(X)=X+O(\sqrt X(\log X)^B).
+}
+```
+
+Such a bound excludes nontrivial zeros with real part `>1/2`; by the functional equation it is RH-scale information.
+
+Conversely the standard RH bound
+
+```math
+\psi(X)=X+O(\sqrt X\log^2X)
+```
+
+and partial summation imply
+
+```math
+D(T)=O(T^3)
+```
+
+(up to harmless lower-end constants).
+
+Thus the new normal form has not made the hard arithmetic disappear: it has isolated it cleanly in `Delta`. Any easy claim that `D(T)` has polynomial growth would essentially be an RH proof and must be treated accordingly.
+
+---
+
+# 6. Exact OU/P11 root-trace bridge
+
+Use the normalized OU family from the Critical-half rigidity audit:
+
+```math
+\langle\Phi^{(\sigma)}_x,\Phi^{(\sigma)}_y\rangle
+=e^{-\sigma d(x,y)},
+\qquad \sigma>0.
+```
+
+At a prime-power node `t_n=log n`, the weighted feature has root coordinate
+
+```math
+\sqrt{w_n}\,e^{-\sigma t_n},
+\qquad
+w_n=\frac{\Lambda(n)}{\sqrt n}.
+```
+
+Therefore the trace of the root projection over the prime-power channel basis is, for `sigma>1/4`,
+
+```math
+\begin{aligned}
+\operatorname{tr}H_P^{(\sigma)}
+&=
+\sum_n w_n e^{-2\sigma t_n}\\
+&=
+\sum_n\Lambda(n)n^{-(1/2+2\sigma)}\\
+&=
+-\frac{\zeta'}{\zeta}\left(\frac12+2\sigma\right).
+\end{aligned}
+```
+
+Now put on a continuous reference ray the compensator measure
+
+```math
+d\lambda_0(t)=e^{t/2}dt.
+```
+
+The same OU root coordinate has squared continuous mass
+
+```math
+\begin{aligned}
+\operatorname{tr}H_0^{(\sigma)}
+&=
+\int_0^\infty e^{t/2}e^{-2\sigma t}dt\\
+&=
+\frac1{2\sigma-1/2}.
+\end{aligned}
+```
+
+Consequently
+
+```math
+\boxed{
+\operatorname{tr}H_P^{(\sigma)}
+-
+\operatorname{tr}H_0^{(\sigma)}
+=
+\mathcal L\Delta(2\sigma).
+}
+```
+
+At the Critical-half/P11 value `sigma=1/2`,
+
+```math
+\boxed{
+\operatorname{tr}H_P^{(1/2)}
+-
+\operatorname{tr}H_0^{(1/2)}
+=
+-\zeta'/\zeta(3/2)-2.
+}
+```
+
+The P11 exponent `3/4` is visible here at the feature level:
+
+```math
+\sqrt{w_n}\,e^{-t_n/2}
+=\sqrt{\Lambda(n)}\,n^{-3/4}.
+```
+
+For `n=p^k` this is precisely
+
+```math
+\sqrt{\log p}\,p^{-3k/4},
+```
+
+the P11 hub/root amplitude.
+
+Thus the P11 common root is an exact discrete OU sampling of the same pole-subtracted prime-vs-continuum trace family that appears in the discrepancy normal form.
+
+---
+
+# 7. What this does and does not prove
+
+What is now exact:
+
+```text
+Prime measure             nu_P
+minus pole compensator    e^{t/2}dt
+= arithmetic discrepancy Delta.
+
+NULLPOL converts the compensator correlation into the decaying e^{-t/2}
+Green correlation.
+
+That Green correlation cancels exactly the mu_0=1/2 Gamma ground mode.
+
+The P11 root trace, compared against the same continuous compensator root,
+recovers the Laplace transform of Delta.
+```
+
+This is a genuine structural bridge between three previously separate project objects:
+
+```text
+NULLPOL Green exterior cancellation
+Gamma ground mode
+P11 OU/tree root.
+```
+
+However the decisive positivity step is still absent. In particular:
+
+```math
+\operatorname{tr}H_P^{(\sigma)}
+-
+\operatorname{tr}H_0^{(\sigma)}
+```
+
+is a **difference** of positive traces. No positive block matrix, contraction, Schur complement or shorting has yet been constructed whose residual quadratic form is the full `Delta` term.
+
+A trace identity must not be promoted to an operator identity.
+
+---
+
+# 8. Next gate
+
+The sharpened Object-X question is now:
+
+```math
+\boxed{
+\text{Can the signed discrepancy pairing }
+\int F_v\,d\Delta
+\text{ be obtained as a forward Schur complement / compression}
+\atop
+\text{of a joint discrete-prime + continuous critical-half OU geometry?}
+}
+```
+
+Required firewall:
+
+1. the joint positive parent must be defined before using Weil positivity;
+2. the continuous component must be the canonical compensator `e^{t/2}dt`, not fitted from `Delta`;
+3. the construction must reproduce more than the scalar root trace — ideally the full autocorrelation pairing;
+4. failure is acceptable and should become a class no-go;
+5. no claim that this route is new in the literature until a dedicated audit is complete.
diff --git a/audits/P11_PASSIVE_EULER_CASCADE_COLLAPSE_NOGO_2026-09-15.md b/audits/P11_PASSIVE_EULER_CASCADE_COLLAPSE_NOGO_2026-09-15.md
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--- /dev/null
+++ b/audits/P11_PASSIVE_EULER_CASCADE_COLLAPSE_NOGO_2026-09-15.md
@@ -0,0 +1,65 @@
+# P11 Audit — Passive Euler cascade collapse no-go
+
+**Datum:** 15. September 2026
+
+## Result
+
+For a prime `p` put
+
+```math
+F_p(s)=\frac{1-p^{-1/2}}{1-p^{-(s+1/2)}}.
+```
+
+For `Re s>0`, `F_p` is Schur; on the positive real axis `0<F_p(s)<1` and `F_p(0)=1`.
+
+The inverse continuous Euler background is the Critical-half allpass
+
+```math
+b(s)=\frac{s-1/2}{s+1/2}.
+```
+
+Hence for every finite prime cutoff `P`,
+
+```math
+\Theta_P(s)=b(s)\prod_{p\le P}F_p(s)
+```
+
+is Schur. Its standard de-Branges/KYP cascade storage is positive and nonorthogonal across the cells.
+
+However, for every fixed real `s>0`,
+
+```math
+\boxed{\Theta_P(s)\to0\qquad(P\to\infty).}
+```
+
+Indeed
+
+```math
+-\log F_p(s)
+=\log\frac{1-p^{-(s+1/2)}}{1-p^{-1/2}}
+=p^{-1/2}(1-p^{-s})+O(p^{-1}),
+```
+
+and the prime sum of the leading positive term diverges because `sum_p p^-1/2 = infinity`, while `p^-s -> 0`. Therefore the product tends to zero.
+
+Consequently the Cayley positive-real transfer
+
+```math
+(1+Theta_P)/(1-Theta_P)
+```
+
+tends to the trivial constant `1` on the positive real axis (and any locally uniform analytic limit of the Schur products is the zero function).
+
+## Meaning
+
+The most direct nonorthogonal passive cross-prime architecture — ordinary cascade of the canonical local Schur cells, together with the Critical-half allpass — loses all arithmetic information in the infinite-prime limit.
+
+A nontrivial global transfer therefore requires a **relative renormalization against the positive Riemann-R continuum background**. Such a renormalization is not automatically passivity preserving and cannot be inserted as a free scalar normalization.
+
+```text
+orthogonal local-cell sum                 insufficient
+ordinary passive local-cell cascade       ×[M] in the direct-limit sense
+relative discrete/continuum storage        ?[O]
+```
+
+No RH/Object-X conclusion is claimed.
diff --git a/audits/P11_POSITIVE_CONTINUUM_EULER_CELL_DECOMPOSITION_2026-09-15.md b/audits/P11_POSITIVE_CONTINUUM_EULER_CELL_DECOMPOSITION_2026-09-15.md
new file mode 100644
index 00000000..1542e4c0
--- /dev/null
+++ b/audits/P11_POSITIVE_CONTINUUM_EULER_CELL_DECOMPOSITION_2026-09-15.md
@@ -0,0 +1,83 @@
+# P11 Audit — Positive continuum Euler-cell decomposition
+
+**Datum:** 15. September 2026
+
+## Ergebnis
+
+Fuer `h>0` setze
+
+```math
+\nu_h(dt)=\sum_{k\ge1} h e^{-kh/2}\,\delta_{kh}(dt),
+```
+
+und
+
+```math
+\rho(h)=\sum_{m\ge1}\frac{h^{m-1}}{m!\,\zeta(m+1)}.
+```
+
+Dann `rho(h)>0` und exakt
+
+```math
+\boxed{\int_0^\infty \rho(h)\nu_h(dt)dh=(e^{t/2}-e^{-t/2})dt.}
+```
+
+Daher
+
+```math
+\boxed{e^{t/2}dt=e^{-t/2}dt+\int_0^\infty\rho(h)\nu_h(dt)dh.}
+```
+
+Der erste Summand ist die Critical-half Gamma-Grunddichte. Die restliche glatte Pol-Hauptdichte ist ein positives Kontinuum derselben logarithmischen Lattice-Zellen, die fuer echte Primzahlen bei `h=log p` auftreten.
+
+## Beweis
+
+Fuer `t>0` gilt wegen positiver Terme und Tonelli
+
+```math
+\begin{aligned}
+t\sum_{k\ge1}\frac{\rho(t/k)}{k^2}
+&=\sum_{m\ge1}\frac{t^m}{m!\zeta(m+1)}
+  \sum_{k\ge1}k^{-(m+1)}\\
+&=\sum_{m\ge1}\frac{t^m}{m!}=e^t-1.
+\end{aligned}
+```
+
+Nach `t=kh` in der Zellmischung folgt die Massdichte
+
+```math
+e^{-t/2}(e^t-1)=e^{t/2}-e^{-t/2}.
+```
+
+## Prime-Diskrepanz
+
+Mit
+
+```math
+\nu_P=\sum_p\nu_{\log p},
+\qquad
+\Delta=\nu_P-e^{t/2}dt,
+```
+
+folgt
+
+```math
+\boxed{\Delta=\sum_p\nu_{\log p}-\int_0^\infty\rho(h)\nu_hdh-e^{-t/2}dt.}
+```
+
+Also ist die arithmetische Diskrepanz: diskrete Abtastung derselben Euler-/AR(1)-Zellfamilie bei `h=log p` minus positives Continuum derselben Zellen minus genau ein Gamma-Grundmodus.
+
+Fuer jede Zelle existiert der bereits bewiesene eine-Zustand-Mediator
+
+```math
+B_h=\sqrt{\frac{h q(1+q)}{1-q}}(T_h-I)(I-qT_h)^{-1},
+\qquad q=e^{-h/2}.
+```
+
+Damit benutzen echte Primseite und glatter Compensator erstmals **dieselbe lokale passive Zustandsklasse**. Dies entgeht dem frueheren Rank-1-No-Go eines einzelnen Continuum-Roots.
+
+## Firewall
+
+Die Identitaet liefert keine positive Ordnung zwischen diskreter Prime-Summe und Continuum-Mischung. Die unzentrierten positiven Energien divergieren im grossen `h`; nur relative/zentrierte Paarungen sind lokal endlich. Ein Object-X-Schluss braucht daher weiterhin eine kanonische relative Storage-/Shorting-Kopplung.
+
+**Status:** Zellmischung `✓[M]`; positive globale Kopplung `?[O]`; kein RH-/Object-X-Claim.
diff --git a/audits/P11_PRIMEWISE_INNOVATION_SHORTING_NOGO_2026-09-15.md b/audits/P11_PRIMEWISE_INNOVATION_SHORTING_NOGO_2026-09-15.md
new file mode 100644
index 00000000..7c5aae9a
--- /dev/null
+++ b/audits/P11_PRIMEWISE_INNOVATION_SHORTING_NOGO_2026-09-15.md
@@ -0,0 +1,375 @@
+# P11 Audit — Primewise innovation-shorting no-go on the physical jump range
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** enger Klassen-No-Go fuer den direkten P11 -> COMMON-JUMP Abstieg.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein No-Go gegen gekoppelte Root-/Gamma-/Fenster-/Maskenkonstruktionen.
+
+---
+
+## 0. Kurzurteil
+
+Der naheliegende Versuch, die diagonale COMMON-JUMP-Primegeometrie direkt als kontraktives Shorting der P11-AR(1)-Innovationsgeometrie zu erhalten, scheitert bereits auf **einem einzelnen Primast** und sogar nach Einschraenkung auf die tatsaechlich von Translationen erzeugten Kanalsequenzen.
+
+Fuer
+
+```math
+q=p^{-1/2}\in(0,1)
+```
+
+sei
+
+```math
+(T_qX)_m
+=\sqrt{1-q^2}\sum_{k\ge m}q^{k-m}X_k,
+```
+
+so dass
+
+```math
+T_q^*T_q=R_q-uu^*,
+\qquad
+R_q(j,k)=q^{|j-k|},
+\qquad
+u_k=q^k.
+```
+
+Die COMMON-JUMP-Kanaele eines einzelnen Fouriermodus mit Phase
+
+```math
+\theta=z\log p
+```
+
+haben, nach Herausziehen des gemeinsamen Faktors `sqrt(log p)`, die Form
+
+```math
+\boxed{
+X_k(\theta)
+=q^{k/2}\,2\sin\frac{k\theta}{2}.
+}
+```
+
+Definiere
+
+```math
+\mathcal R_q(\theta)
+:=
+\frac{\|T_qX(\theta)\|_{\ell^2}^2}
+{\|X(\theta)\|_{\ell^2}^2}
+```
+
+fuer `theta` mit `X(theta) != 0`.
+
+Dann gilt fuer jedes `0<q<1`
+
+```math
+\boxed{
+\lim_{\theta\to0}\mathcal R_q(\theta)>1,
+\qquad
+\mathcal R_q(\pi)<1.
+}
+```
+
+Damit expandiert `T_q` einige physikalische Jump-Richtungen und kontrahiert andere. Folglich existiert weder in der Richtung
+
+```text
+COMMON-JUMP bare channel -> P11 innovation
+```
+
+noch in der Rueckrichtung ein primeweiser kontraktiver Identifikator, der auf dem gesamten translationsgenerierten Kanalraum einfach `T_q` bzw. `T_q^{-1}` realisiert.
+
+Status:
+
+```text
+physical translation-channel parametrization      ✓[M]
+low-frequency expansion of T_q                    ✓[M]
+pi-frequency contraction of T_q                   ✓[M]
+primewise contraction/shorting route              ×[M]
+coupled Root+Gamma+mask route                     ?[O]
+```
+
+---
+
+# 1. Physical channel range
+
+For the prime `p`, put
+
+```math
+h=\log p,
+\qquad
+q=e^{-h/2}=p^{-1/2}.
+```
+
+The common-jump channel at prime power `p^k` is
+
+```math
+K_{kh}=T_{kh/2}-T_{-kh/2}.
+```
+
+On a Fourier mode of frequency `z`, its scalar multiplier is
+
+```math
+2i\sin\frac{kzh}{2}.
+```
+
+The Weil weight is
+
+```math
+w_{p,k}=(\log p)q^k.
+```
+
+After discarding the harmless common factors `sqrt(log p)` and `i`, the weighted channel sequence is therefore exactly
+
+```math
+\boxed{
+X_k(\theta)=2q^{k/2}\sin(k\theta/2),
+\qquad
+\theta=zh.
+}
+```
+
+Thus this no-go is not tested on arbitrary `ell^2`: it is tested on the actual one-parameter physical channel curve generated by the translation representation.
+
+---
+
+# 2. Low-frequency limit
+
+As `theta -> 0`,
+
+```math
+X_k(\theta)
+=k\theta q^{k/2}+O(\theta^3k^3q^{k/2}).
+```
+
+Hence the limiting ratio is the ratio for
+
+```math
+x_k=kq^{k/2}.
+```
+
+Write
+
+```math
+r=\sqrt q\in(0,1),
+\qquad
+a=qr=r^3.
+```
+
+For `x_k=kr^k`, the tail transform is
+
+```math
+\begin{aligned}
+(T_qx)_m
+&=\sqrt{1-q^2}\sum_{k\ge m}q^{k-m}kr^k\\
+&=\sqrt{1-q^2}\,r^m
+\left(
+\frac{m}{1-r^3}
++
+\frac{r^3}{(1-r^3)^2}
+\right).
+\end{aligned}
+```
+
+Using
+
+```math
+\sum_{m\ge1}q^m=\frac q{1-q},
+\quad
+\sum_{m\ge1}mq^m=\frac q{(1-q)^2},
+\quad
+\sum_{m\ge1}m^2q^m=\frac{q(1+q)}{(1-q)^3},
+```
+
+one obtains after simplification
+
+```math
+\boxed{
+\lim_{\theta\to0}\mathcal R_q(\theta)
+=
+\frac{(1+r)
+\left(r^8+2r^7+4r^6+6r^5+8r^4+6r^3+4r^2+2r+1\right)}
+{(1-r)(r^2+r+1)^4}.
+}
+```
+
+Subtracting `1` gives the manifestly positive expression
+
+```math
+\boxed{
+\lim_{\theta\to0}\mathcal R_q(\theta)-1
+=
+\frac{r^3
+\left(
+2r^6+6r^5+12r^4+16r^3+17r^2+11r+4
+\right)}
+{(1-r)(r^2+r+1)^4}
+>0.
+}
+```
+
+Thus the P11 innovation map strictly **expands** sufficiently low-frequency physical jump directions.
+
+---
+
+# 3. The phase `theta=pi`
+
+At `theta=pi`,
+
+```math
+X_{2m}=0,
+\qquad
+X_{2m+1}=2(-1)^m q^{m+1/2}.
+```
+
+The bare diagonal norm is
+
+```math
+\boxed{
+\|X(\pi)\|^2
+=\frac{4q}{1-q^2}.
+}
+```
+
+Since
+
+```math
+T_q^*T_q=R_q-uu^*,
+```
+
+the AR(1) part is
+
+```math
+\langle X,R_qX\rangle
+=
+\frac{4q(1-q^3)}{(1-q^2)(1+q^3)},
+```
+
+while the root component is
+
+```math
+|\langle u,X\rangle|^2
+=
+\frac{4q^3}{(1+q^3)^2}.
+```
+
+Therefore
+
+```math
+\boxed{
+\mathcal R_q(\pi)
+=
+\frac{(1-q)(1+q^4)}
+{(1+q)(1-q+q^2)^2}.
+}
+```
+
+Moreover
+
+```math
+\boxed{
+1-\mathcal R_q(\pi)
+=
+\frac{q^2(2q^3-2q^2+q+1)}
+{(1+q)(1-q+q^2)^2}
+>0.
+}
+```
+
+Indeed
+
+```math
+2q^3-2q^2+q+1
+=1+q-2q^2(1-q)>0
+```
+
+for `0<q<1` (and positivity is immediate also from the displayed polynomial on the interval).
+
+Thus `T_q` strictly **contracts** the physical channel at `theta=pi`.
+
+---
+
+# 4. Contractive feature-shorting consequences
+
+Suppose one tries to realize the bare COMMON-JUMP channel geometry from the P11 innovation target by a primewise contraction `C_p` satisfying
+
+```math
+C_p(T_qX(\theta))=X(\theta)
+```
+
+for every physical phase `theta`.
+
+Contractivity would require
+
+```math
+\|X(\theta)\|\le\|T_qX(\theta)\|
+```
+
+for every `theta`, i.e.
+
+```math
+\mathcal R_q(\theta)\ge1.
+```
+
+This fails at `theta=pi`.
+
+Conversely, a primewise contraction carrying bare channels to P11 innovations,
+
+```math
+C_pX(\theta)=T_qX(\theta),
+```
+
+would require
+
+```math
+\mathcal R_q(\theta)\le1
+```
+
+for every `theta`, which fails for sufficiently small nonzero `theta`.
+
+Therefore both naive primewise contractive directions are excluded.
+
+---
+
+# 5. Scope firewall
+
+This no-go excludes only the class in which:
+
+1. each prime branch is treated independently;
+2. the P11 root has already been separated/conditioned out;
+3. no archimedean/continuous compensator is coupled to the branch;
+4. no spatial P11 masks `P_{Omega_{p,k-1,R}}` are used;
+5. the intertwiner is required to be a contraction between the bare COMMON-JUMP channel norm and the P11 innovation norm on the full physical translation-channel range.
+
+It does **not** exclude:
+
+```text
+- a joint root + innovation construction;
+- cancellation against the continuous compensator e^{t/2}dt;
+- a spatially masked/radius-dependent Schur complement;
+- coupling between prime branches through the common root;
+- coupling to the higher Gamma channel h_1;
+- a larger conservative/unitary colligation whose compression is not primewise.
+```
+
+---
+
+# 6. Strategic consequence
+
+The new prime-discrepancy audit shows that after NULLPOL the natural arithmetic object is
+
+```math
+Delta=nu_P-e^{t/2}dt.
+```
+
+The present no-go shows simultaneously that the **innovation branch alone** cannot be identified contractively with the COMMON-JUMP diagonal geometry.
+
+Thus any surviving Object-X architecture based on P11 must use at least one of the pieces that the naive branchwise reduction discards:
+
+```math
+\boxed{
+\text{common root / continuous compensator / spatial masks / Gamma coupling}.
+}
+```
+
+This sharply narrows the next gate: test whether the common P11 root plus the canonical continuous compensator can be assembled into a forward positive parent whose Schur complement produces the full discrepancy pairing, not merely its scalar root trace.
diff --git a/audits/P11_PRIME_AR1_POISSON_GAMMA_HARMONIC_BRIDGE_2026-09-13.md b/audits/P11_PRIME_AR1_POISSON_GAMMA_HARMONIC_BRIDGE_2026-09-13.md
new file mode 100644
index 00000000..c94d27cd
--- /dev/null
+++ b/audits/P11_PRIME_AR1_POISSON_GAMMA_HARMONIC_BRIDGE_2026-09-13.md
@@ -0,0 +1,468 @@
+# P11 Audit — Prime AR(1) Poisson / Gamma harmonic bridge
+
+**Datum:** 13. September 2026  
+**Basis:** COMMON-JUMP, P11 AR(1), Critical-half Green/tree und Diagonal-finite-part Audits.  
+**Rolle:** theorem-level harmonische Reorganisation der zentrierten Weil-Seite.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein NP-GAP-/RH-Beweis; keine Publikationsprioritaet behauptet.
+
+---
+
+## 0. Kurzurteil
+
+Fuer jede feste Primzahl ist der **exakte zentrierte Weil-Beitrag des gesamten Prime-Power-Turms** die Differenz zwischen der flachen Dichte `1` und der Poisson-/AR(1)-Spektraldichte des bereits in P11 gefundenen Kac--Murdock--Szego-Kerns.
+
+Setze
+
+```math
+q_p=p^{-1/2},
+\qquad
+\theta=z\log p.
+```
+
+Der Disk-Poisson-Kern ist
+
+```math
+\boxed{
+P_q^{\mathbb D}(\theta)
+:=
+\frac{1-q^2}{1-2q\cos\theta+q^2}
+=
+1+2\sum_{k\ge1}q^k\cos(k\theta).
+}
+```
+
+Dann gilt exakt
+
+```math
+\boxed{
+-2\log p\sum_{k\ge1}p^{-k/2}\cos(kz\log p)
+=
+\log p\left[1-P_{p^{-1/2}}^{\mathbb D}(z\log p)\right].
+}
+```
+
+Dies ist nicht nur formal mit P11 verwandt: `P_q^D` ist genau das Toeplitz-Symbol der P11-AR(1)-Kovarianz
+
+```math
+R_q(j,k)=q^{|j-k|}.
+```
+
+Parallel ist jeder **zentrierte Gamma-Resolventenmodus** ein negativer Poisson-Kern des oberen Halbraums:
+
+```math
+\boxed{
+A_m(z)-\frac{2}{\mu_m}
+=-2\pi P_{\mu_m}^{\mathbb H}(z),
+}
+```
+
+mit
+
+```math
+P_y^{\mathbb H}(x)
+:=\frac1\pi\frac{y}{x^2+y^2},
+\qquad
+\mu_m=2m+\frac12.
+```
+
+Die gesamte zentrierte Weil-Seite ist damit eine **Disk-/Half-plane-Poisson-Geometrie**:
+
+- Prime towers: `flat - disk Poisson`;
+- Gamma ladder: finite-part-renormalisierte `- half-plane Poisson`-Summe;
+- beide sind Standardmodelle derselben harmonischen Mass-/Harmonic-measure-Geometrie, konform durch Cayley-Transformation verwandt.
+
+Status:
+
+```text
+full p-power tower = log p (1 - AR1 Poisson symbol)    ✓[M]
+P11 AR1 covariance is exact Weil prime spectral object  ✓[M]
+Gamma centered modes = half-plane Poisson kernels       ✓[M]
+arch finite-part Poisson comb                           ✓[M]
+compact-window infinite k completion                    ✓[M]
+unified harmonic-measure architecture                   ✓[M]_part
+sharp NP-GAP / positivity                               ?[O]
+full Object X / RH                                      ?[O]
+publication novelty                                     ?[O]
+```
+
+---
+
+# 1. Prime-Power-Turm auf einem kompakten Fenster
+
+Fixiere eine Primzahl `p` und schreibe
+
+```math
+\ell_p=\log p,
+\qquad
+q=q_p=p^{-1/2}.
+```
+
+Die Prime-Power-Gewichte sind
+
+```math
+w_{p,k}=\frac{\log p}{p^{k/2}}
+=(\log p)q^k.
+```
+
+Der zentrierte COMMON-JUMP-Beitrag ist
+
+```math
+\mathcal Q_p(v)
+:=
+\sum_{k\ge1}w_{p,k}
+\left(
+\|K_{k\ell_p}v\|_2^2-2\|v\|_2^2
+\right).
+```
+
+Fuer ein `v` mit Traeger in `[-a,a]` ist diese unendliche Schreibweise **exakt**, nicht asymptotisch: sobald
+
+```math
+k\ell_p>2a,
+```
+
+sind die beiden verschobenen Traeger disjunkt und
+
+```math
+\|K_{k\ell_p}v\|_2^2=2\|v\|_2^2.
+```
+
+Alle hinzugenommenen Terme sind daher identisch null. Ist bereits `log p>2a`, verschwindet der ganze Prime-Turm in der zentrierten Form.
+
+---
+
+# 2. Fourier-Symbol des Prime-Turms
+
+Aus
+
+```math
+K_t^*K_t=2I-T_t-T_{-t}
+```
+
+folgt unter der unitaeren Fouriertransformation
+
+```math
+\|K_tv\|^2-2\|v\|^2
+\longleftrightarrow
+-2\cos(zt).
+```
+
+Daher besitzt der `p`-Turm das Symbol
+
+```math
+\begin{aligned}
+B_p(z)
+&=-2\log p\sum_{k\ge1}q^k\cos(kz\ell_p).
+\end{aligned}
+```
+
+Die Reihe konvergiert fuer jedes `p` absolut, da `0<q<1`.
+
+---
+
+# 3. Exakte Poisson-Summation
+
+Der klassische Disk-Poisson-Kern besitzt fuer `0<q<1` die Fourierreihe
+
+```math
+\boxed{
+P_q^{\mathbb D}(\theta)
+=1+2\sum_{k\ge1}q^k\cos(k\theta)
+=\frac{1-q^2}{1-2q\cos\theta+q^2}.
+}
+```
+
+Mit `theta=z log p` ergibt sich unmittelbar
+
+```math
+\boxed{
+B_p(z)
+=(\log p)
+\left[
+1-P_{q_p}^{\mathbb D}(z\log p)
+\right].
+}
+```
+
+Explizit:
+
+```math
+\boxed{
+B_p(z)
+=(\log p)
+\left[
+1-
+\frac{1-p^{-1}}
+{1-2p^{-1/2}\cos(z\log p)+p^{-1}}
+\right].
+}
+```
+
+Aequivalent:
+
+```math
+B_p(z)
+=-2\log p\,
+\operatorname{Re}
+\frac{p^{-1/2-iz}}{1-p^{-1/2-iz}}.
+```
+
+Bis auf die uebliche Vorzeichenkonvention ist dies der Realteil der logarithmischen Ableitung des lokalen Eulerfaktors am kritischen Mittelpunkt.
+
+---
+
+# 4. P11-AR(1) ist exakt dasselbe Spektralobjekt
+
+P11 hat unabhaengig die normierte Prime-Power-Kovarianz
+
+```math
+\boxed{R_q(j,k)=q^{|j-k|}}
+```
+
+mit `q=p^{-1/2}` gefunden.
+
+Der zugehoerige Toeplitz-Operator auf der Halbgeraden besitzt das Symbol
+
+```math
+\boxed{
+\sum_{m\in\mathbb Z}q^{|m|}e^{im\theta}
+=P_q^{\mathbb D}(\theta).
+}
+```
+
+Somit ist der Zusammenhang nicht bloss
+
+```text
+Weil weights ~ AR(1)-looking coefficients,
+```
+
+sondern exakt
+
+```math
+\boxed{
+\text{centered Weil p-tower symbol}
+=
+\log p\,
+(\text{flat symbol}-\text{P11 AR(1) symbol}).
+}
+```
+
+Dies gibt dem in POS-DIL aufgetretenen Motiv `flat minus AR(1)` eine vollstaendige Spektralbedeutung auf dem gesamten Prime-Power-Turm.
+
+---
+
+# 5. Gamma-Moden sind Half-plane-Poisson-Kerne
+
+Aus dem Critical-half-Gamma-Audit:
+
+```math
+A_m(z)
+=
+\frac{2}{\mu_m}\frac{z^2}{z^2+\mu_m^2},
+\qquad
+\mu_m=2m+\frac12.
+```
+
+Zerlege die Diagonalmasse:
+
+```math
+A_m(z)
+=
+\frac{2}{\mu_m}
+-
+\frac{2\mu_m}{z^2+\mu_m^2}.
+```
+
+Der Poisson-Kern des oberen Halbraums mit Pol `iy` lautet
+
+```math
+P_y^{\mathbb H}(x)
+=\frac1\pi\frac{y}{x^2+y^2}.
+```
+
+Damit
+
+```math
+\boxed{
+A_m(z)-\frac{2}{\mu_m}
+=-2\pi P_{\mu_m}^{\mathbb H}(z).
+}
+```
+
+Jeder zentrierte Gamma-Modus ist also exakt ein negativer harmonischer-Mass-Kern auf der reellen Randachse.
+
+---
+
+# 6. Renormalisierter archimedischer Poisson-Kamm
+
+Der Diagonal-finite-part-Audit hat bewiesen
+
+```math
+\kappa_*
+=
+\lim_{N\to\infty}
+\left[
+\sum_{m=0}^N\frac{2}{\mu_m}
+-
+\log\frac{N+1}{\pi}
+\right].
+```
+
+Daher gilt fuer die zentrierte archimedische Symbolfunktion
+
+```math
+\boxed{
+\Phi_\infty(z)-\kappa_*
+=
+\lim_{N\to\infty}
+\left[
+\log\frac{N+1}{\pi}
+-2\pi\sum_{m=0}^N
+P_{\mu_m}^{\mathbb H}(z)
+\right].
+}
+```
+
+Der Grenzwert ist exakt
+
+```math
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi.
+```
+
+Somit ist der archimedische zentrierte Operator eine finite-part-renormalisierte Poisson-Kamm-Geometrie an den Hoehen
+
+```math
+\mu_m=2m+\frac12.
+```
+
+---
+
+# 7. Einheitliche harmonische Normalform auf NULLPOL
+
+Fuer `v` mit Traeger in `[-a,a]` kann die Prime-Seite wegen §1 primeweise auf ganze Potenztuerme erweitert werden. Es tragen nur endlich viele Primzahlen `p<=e^{2a}` nichttrivial bei.
+
+Auf NULLPOL besitzt die Weilform daher die harmonische Symbolbeschreibung
+
+```math
+\boxed{
+\begin{aligned}
+Q_W(v)
+&=
+\int_{\mathbb R}|\widehat v(z)|^2
+\Bigg[
+\underbrace{
+\lim_{N\to\infty}
+\left(
+\log\frac{N+1}{\pi}
+-2\pi\sum_{m=0}^N P_{\mu_m}^{\mathbb H}(z)
+\right)
+}_{\text{archimedischer finite-part Poisson-Kamm}}\\
+&\qquad\qquad
++
+\underbrace{
+\sum_{p\le e^{2a}}
+(\log p)
+\left(
+1-P_{p^{-1/2}}^{\mathbb D}(z\log p)
+\right)
+}_{\text{endliche Summe von AR(1)/Disk-Poisson-Defekten}}
+\Bigg]dz,
+\end{aligned}
+}
+```
+
+mit der jeweils im Projekt verwendeten unitaeren Fourier-Normalisierung.
+
+Wichtig: die punktweise Prime-Summe wird hier nur ueber die endlich vielen `p<=e^{2a}` geschrieben; die unendliche `k`-Ergaenzung ist formexakt wegen der Trägerlokalitaet.
+
+---
+
+# 8. Disk und oberer Halbraum sind dieselbe harmonische Geometrie
+
+`P_q^D` und `P_y^H` sind die beiden Standard-Poisson-Kerne fuer harmonische Fortsetzung
+
+```text
+Einheitsdisk  <---- Cayley ---->  oberer Halbraum.
+```
+
+Unter der Cayley-Transformation gehen harmonische Masse und Poisson-Kerne ineinander ueber, mit dem ueblichen Rand-Jacobian.
+
+Damit sind Prime- und Gamma-Seite jetzt nicht nur durch dasselbe `K_t`-Jump-Feature und denselben Critical-half-Green-Grundmodus verbunden, sondern auch durch dieselbe **harmonic-measure kernel class**.
+
+Die Parameter spielen unterschiedliche Rollen:
+
+```text
+Prime p:      disk radius q_p=p^{-1/2}, periodic boundary angle z log p;
+Gamma m:      half-plane height mu_m=2m+1/2, real boundary variable z.
+```
+
+Es wird keine kuenstliche direkte Identifikation `q_p <-> mu_m` behauptet.
+
+---
+
+# 9. Neue Object-X-Lesart
+
+Die bisherige Kette wird nun:
+
+```text
+critical-half L_{1/2}
+   |
+   +-- pole charges E_± = oriented Green boundary transfer
+   |
+   +-- P11 q^{|j-k|} = OU/tree covariance
+   |       |
+   |       +-- Toeplitz symbol = disk Poisson P_q
+   |       +-- exact Weil p-tower = log p (1-P_q)
+   |
+   +-- Gamma ladder mu_m
+           |
+           +-- centered modes = -2pi half-plane Poisson
+           +-- kappa_* = finite-part diagonal renormalization
+```
+
+Dies ist eine vorwaerts definierte gemeinsame harmonische Architektur. Der verbleibende Engpass ist nicht mehr die Suche nach irgendeinem gemeinsamen Prime-/Archimedes-Kern, sondern die Positivitaet der **renormalisierten harmonischen Defektform** auf
+
+```math
+\operatorname{Ran}L_{1/2}.
+```
+
+---
+
+# 10. Was dies noch nicht beweist
+
+Poisson-Kerne sind positiv als harmonische Dichten, aber die Weilform enthaelt
+
+```text
+flat - Poisson
+```
+
+beziehungsweise einen finite-part-renormalisierten negativen Poisson-Kamm. Daraus folgt keine automatische Positivitaet.
+
+Insbesondere wird nicht behauptet:
+
+- `1-P_q^D` sei punktweise nichtnegativ;
+- der archimedische finite-part Poisson-Kamm sei punktweise nichtnegativ;
+- NP-GAP sei geschlossen;
+- eine Cayley-Transformation allein beweise RH;
+- der P11 Tree ersetze die raeumlichen Fensteroperatoren;
+- Publikationsneuheit sei geklaert.
+
+---
+
+# 11. Naechster Gate
+
+Die neue bevorzugte Frage lautet:
+
+> Gibt es eine **gemeinsame positive harmonic-extension / Dirichlet-to-Neumann dilation**, deren Boundary-Schur-Komplement exakt die Disk-/Half-plane-Poisson-Defektform auf `Ran L_{1/2}` liefert?
+
+Dies ist nicht als rueckwaerts definierte Dilation zulaessig. Gesucht ist eine explizite Konstruktion aus
+
+```text
+OU/tree edges,
+critical-half Green propagation,
+Disk-/Half-plane Poisson extension,
+und den vorhandenen COMMON-JUMP incidence operators K_t.
+```
+
+Ein Erfolg waere ein echter Object-X-Kandidat; ein theorem-level No-Go fuer diese Dilationklasse waere ebenfalls substanzieller Fortschritt.
diff --git a/audits/P11_PRIME_ONLY_CELL_SAMPLING_NORMAL_FORM_2026-09-15.md b/audits/P11_PRIME_ONLY_CELL_SAMPLING_NORMAL_FORM_2026-09-15.md
new file mode 100644
index 00000000..97a30501
--- /dev/null
+++ b/audits/P11_PRIME_ONLY_CELL_SAMPLING_NORMAL_FORM_2026-09-15.md
@@ -0,0 +1,226 @@
+# P11 Audit — Prime-only AR(1) cell sampling normal form
+
+**Datum:** 15. September 2026
+
+## 0. Result
+
+Let `v` be smooth, compactly supported, and NULLPOL. For `h>0` put
+
+```math
+q=e^{-h/2},
+\qquad
+B_h=\sqrt{\frac{hq(1+q)}{1-q}}(T_h-I)(I-qT_h)^{-1},
+```
+
+and define the centered cell response
+
+```math
+\boxed{
+C_v(h)
+:=\|B_hv\|_2^2
+-2\frac{hq}{1-q}\|v\|_2^2.
+}
+```
+
+If `supp(v) subset [-a,a]`, then `C_v(h)=0` for `h>2a`.
+
+With
+
+```math
+\rho(h)=\sum_{m\ge1}\frac{h^{m-1}}{m!\zeta(m+1)},
+\qquad
+\mu_m=2m+1/2,
+```
+
+and
+
+```math
+E_{Gamma,>=1}(v)
+=\sum_{m\ge1}\frac2{\mu_m}
+\left\|\partial(\partial+\mu_m)^{-1}v\right\|_2^2,
+```
+
+the NULLPOL Weil form is exactly
+
+```math
+\boxed{
+Q_W(v)
+=
+\sum_p C_v(\log p)
+-\frac12\int_0^\infty\rho(h)C_v(h)dh
++E_{Gamma,>=1}(v)
+-(\kappa_*-4)\|v\|_2^2.
+}
+```
+
+The exterior sum is now over **primes only**. Every prime-power tower has been compressed into the one-state AR(1)/Euler cell `B_logp`.
+
+Status: identity `✓[M]`; positivity `?[O]`; no novelty/RH claim.
+
+---
+
+## 1. Centered cell as a Lambert transform
+
+Let
+
+```math
+F_v(t)=f_v(t)+f_v(-t),
+\qquad f_v=v*\widetilde v.
+```
+
+The prime-tower compression gives
+
+```math
+\|B_hv\|^2
+=\sum_{k\ge1}h e^{-kh/2}\|K_{kh}v\|^2.
+```
+
+Since
+
+```math
+\|K_tv\|^2=2\|v\|^2-F_v(t),
+```
+
+we obtain
+
+```math
+\boxed{
+C_v(h)
+=-\sum_{k\ge1}h e^{-kh/2}F_v(kh).
+}
+```
+
+For `h>2a`, all `kh` lie outside the autocorrelation support and `C_v(h)=0`.
+
+---
+
+## 2. Continuum cell average
+
+The positive continuum Euler-cell audit proved
+
+```math
+\int_0^\infty\rho(h)\nu_h(dt)dh
+=(e^{t/2}-e^{-t/2})dt,
+```
+
+where
+
+```math
+\nu_h=\sum_{k\ge1}h e^{-kh/2}\delta_{kh}.
+```
+
+Therefore
+
+```math
+I_\rho(v)
+:=\int_0^\infty\rho(h)C_v(h)dh
+=-\int_0^\infty F_v(t)(e^{t/2}-e^{-t/2})dt.
+```
+
+For NULLPOL the pole moments vanish, hence
+
+```math
+\int_0^\infty F_v(t)(e^{t/2}+e^{-t/2})dt=0.
+```
+
+Writing
+
+```math
+B_0(v):=\int_0^\infty e^{-t/2}F_v(t)dt,
+```
+
+we get
+
+```math
+\boxed{I_\rho(v)=2B_0(v).}
+```
+
+The Gamma ground-mode energy is
+
+```math
+E_{Gamma,0}(v)
+=\int_0^\infty e^{-t/2}\|K_tv\|^2dt
+=4\|v\|^2-B_0(v),
+```
+
+so
+
+```math
+\boxed{
+E_{Gamma,0}(v)
+=4\|v\|^2-\frac12 I_\rho(v).
+}
+```
+
+---
+
+## 3. Substitution in COMMON-JUMP
+
+On NULLPOL,
+
+```math
+Q_W(v)
+=
+\sum_p C_v(\log p)
++E_{Gamma,0}(v)
++E_{Gamma,>=1}(v)
+-\kappa_*\|v\|^2.
+```
+
+Insert the ground-mode identity from Section 2:
+
+```math
+Q_W(v)
+=
+\sum_p C_v(\log p)
+-\frac12\int\rho(h)C_v(h)dh
++E_{Gamma,>=1}(v)
+-(\kappa_*-4)\|v\|^2.
+```
+
+This proves the boxed formula in Section 0.
+
+---
+
+## 4. No information loss: Moebius inversion
+
+Define
+
+```math
+L_v(h):=-\frac{C_v(h)}h
+=\sum_{k\ge1}e^{-kh/2}F_v(kh).
+```
+
+Because `F_v` is compactly supported, every dilation sum below is finite at fixed `h`. Ordinary Moebius inversion gives
+
+```math
+\boxed{
+e^{-h/2}F_v(h)
+=\sum_{k\ge1}\mu(k)L_v(kh).
+}
+```
+
+Equivalently,
+
+```math
+F_v(h)
+=-e^{h/2}\sum_{k\ge1}\frac{\mu(k)}{kh}C_v(kh).
+```
+
+Thus the one-cell response `C_v(h)` is an invertible recoding of the original autocorrelation data; the prime-only normal form does not discard the prime-power information.
+
+---
+
+## 5. New gate
+
+The arithmetic part has become a sampling discrepancy for one universal cell family:
+
+```text
+actual arithmetic samples:      h = log p
+positive smooth reference:      rho(h) dh
+cell response:                  C_v(h)
+higher positive reservoir:      Gamma modes m>=1
+fixed leakage:                  kappa_* - 4
+```
+
+A successful Object-X route would need a forward positive storage/sampling theorem for this specific response class. The identity itself supplies no sign: `C_v(h)` is centered and may change sign.
diff --git a/audits/P11_PRIME_THRESHOLD_BAD_CHANNEL_2026-09-13.md b/audits/P11_PRIME_THRESHOLD_BAD_CHANNEL_2026-09-13.md
new file mode 100644
index 00000000..0d834fcd
--- /dev/null
+++ b/audits/P11_PRIME_THRESHOLD_BAD_CHANNEL_2026-09-13.md
@@ -0,0 +1,432 @@
+# P11 Audit — Prime-threshold bad channel and NULLPOL self-similarity
+
+**Datum:** 13. September 2026  
+**Basis:** `research/critical-half-green-tree-bridge-2026-09-13`.  
+**Rolle:** theorem-level Architekturfilter fuer `CRIT-HALF-COMPRESS / NP-GAP`.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein globaler NP-GAP-/Object-X-/RH-Beweis.
+
+---
+
+## 0. Kurzurteil
+
+Wenn ein einzelner Prime-Power-Shift mit Laenge `h` gerade in ein Fenster `I_a=(-a,a)` eingetreten ist,
+
+```math
+\frac h2<a<h,
+```
+
+ist seine komprimierte Translation ein nilpotenter partieller Shift. Der zentrierte Prime-Term besitzt dadurch eine exakte Kanalzerlegung
+
+```math
+-w(A+A^*)=-wP_+ + wP_-,
+```
+
+mit `w>0`, wobei `P_+` und `P_-` orthogonale Projektionen auf symmetrische bzw. antisymmetrische Randpaar-Kanaele sind.
+
+Der negative Kanal `Ran P_+` ist isometrisch zu einem **kleineren zentralen Fenster**
+
+```math
+I_b=(-b,b),
+\qquad
+b=a-\frac h2>0.
+```
+
+Noch staerker: Unter dieser Isometrie werden die beiden NULLPOL-Bedingungen exakt auf dieselben beiden NULLPOL-Bedingungen auf `I_b` transportiert. Daher bleibt der schlechte Prime-Kanal auf NULLPOL vollstaendig erhalten.
+
+Folgen:
+
+```text
+exact one-shift edge-channel diagonalization             ✓[M]
+negative prime channel = smaller-window symmetric copy   ✓[M]
+NULLPOL self-similarity on the negative channel           ✓[M]
+prime bad-channel bottom = -w also on NULLPOL             ✓[M]
+operator-norm-small threshold perturbation                ×[M]
+strong/form-local threshold mechanism still possible      ?[O]
+full NP-GAP / Object X / RH                               ?[O]
+```
+
+---
+
+# 1. Komprimierter Shift
+
+Sei
+
+```math
+I_a=(-a,a),
+\qquad
+H_a=L^2(I_a),
+```
+
+und `P_a` die orthogonale Projektion von `L^2(R)` auf `H_a`.
+
+Wir benutzen die unitäre Translation
+
+```math
+(T_hf)(x)=f(x-h)
+```
+
+und setzen
+
+```math
+A_{a,h}:=P_aT_hP_a|_{H_a}.
+```
+
+Angenommen
+
+```math
+\boxed{a<h<2a.}
+```
+
+Dann ist der Definitionsstreifen
+
+```math
+L_{a,h}=(-a,a-h)
+```
+
+und der Bildstreifen
+
+```math
+R_{a,h}=(-a+h,a).
+```
+
+Da `h>a`, sind diese beiden Streifen disjunkt.
+
+Der Operator `A_{a,h}` ist die unitäre Translation von `L_{a,h}` nach `R_{a,h}` und verschwindet auf dem orthogonalen Komplement des linken Streifens. Insbesondere
+
+```math
+A_{a,h}^*A_{a,h}=P_L,
+\qquad
+A_{a,h}A_{a,h}^*=P_R.
+```
+
+Da eine zweite Verschiebung um `h` insgesamt `2h>2a` betraegt, gilt exakt
+
+```math
+\boxed{A_{a,h}^2=0.}
+```
+
+---
+
+# 2. Exakte symmetrisch/antisymmetrische Zerlegung
+
+Setze
+
+```math
+J_{a,h}:=A_{a,h}+A_{a,h}^*.
+```
+
+Wegen `A^2=(A^*)^2=0` folgt
+
+```math
+J_{a,h}^2=P_L+P_R=:P_E.
+```
+
+Also hat `J_{a,h}` auf dem Randpaarraum `Ran P_E` nur die Eigenwerte `+1` und `-1` und verschwindet auf dem mittleren Rest.
+
+Definiere
+
+```math
+\boxed{
+P_+=\frac12(P_E+J_{a,h}),
+\qquad
+P_-=\frac12(P_E-J_{a,h}).
+}
+```
+
+Dann sind `P_+`, `P_-` orthogonale Projektionen und
+
+```math
+J_{a,h}=P_+-P_-.
+```
+
+Fuer einen Prime-Power-Kanal `n=p^k` mit
+
+```math
+h=\log n,
+\qquad
+w=\frac{\Lambda(n)}{\sqrt n},
+```
+
+ist sein zentrierter Weil-/COMMON-JUMP-Beitrag
+
+```math
+w\|K_hv\|^2-2w\|v\|^2
+=-w\langle v,(T_h+T_{-h})v\rangle.
+```
+
+Nach Fensterkompression daher exakt
+
+```math
+\boxed{
+C_{n,a}:=-wJ_{a,h}
+=-wP_+ + wP_-.
+}
+```
+
+Der einzige negative Prime-Kanal ist somit `Ran P_+`.
+
+---
+
+# 3. Der schlechte Kanal ist eine kleinere Fensterkopie
+
+Setze
+
+```math
+\boxed{b=a-\frac h2.}
+```
+
+Aus `h/2<a<h` folgt `0<b<h/2`.
+
+Fuer `g in L^2(-b,b)` definiere `S_+g in H_a` durch
+
+```math
+(S_+g)(y-h/2)=\frac{g(y)}{\sqrt2},
+\qquad
+(S_+g)(y+h/2)=\frac{g(y)}{\sqrt2},
+\qquad -b<y<b,
+```
+
+und `S_+g=0` auf dem mittleren Rest.
+
+Die beiden Kopien sind disjunkt, daher
+
+```math
+\|S_+g\|_{H_a}=\|g\|_{L^2(-b,b)}.
+```
+
+Aus der Definition folgt
+
+```math
+J_{a,h}S_+g=S_+g.
+```
+
+Somit ist `S_+` eine unitäre Abbildung
+
+```math
+\boxed{
+S_+:L^2(-b,b)\overset{\sim}{\longrightarrow}\operatorname{Ran}P_+.
+}
+```
+
+Auf diesem Kanal ist der Prime-Beitrag exakt
+
+```math
+\boxed{
+\langle S_+g,C_{n,a}S_+g\rangle
+=-w\|g\|^2.
+}
+```
+
+Analog liefert die antisymmetrische Kopie `S_-` den positiven Kanal `+w`.
+
+---
+
+# 4. NULLPOL ist auf dem schlechten Kanal selbstähnlich
+
+Erinnere
+
+```math
+E_+(f)=\int e^{x/2}f(x)dx,
+\qquad
+E_-(f)=\int e^{-x/2}f(x)dx.
+```
+
+Fuer `v=S_+g` ergibt direkte Substitution
+
+```math
+\begin{aligned}
+E_+(S_+g)
+&=\frac1{\sqrt2}\int_{-b}^b
+ g(y)\left(e^{(y-h/2)/2}+e^{(y+h/2)/2}\right)dy\\
+&=\sqrt2\cosh(h/4)E_+(g),
+\end{aligned}
+```
+
+und ebenso
+
+```math
+\boxed{
+E_-(S_+g)
+=\sqrt2\cosh(h/4)E_-(g).
+}
+```
+
+Da `cosh(h/4)>0`, gilt exakt
+
+```math
+\boxed{
+S_+g\in\mathscr D_{NP,a}
+\iff
+g\in\mathscr D_{NP,b}.
+}
+```
+
+Also
+
+```math
+\boxed{
+\operatorname{Ran}P_+\cap\mathscr D_{NP,a}
+=S_+\mathscr D_{NP,b}.
+}
+```
+
+Der negative Prime-Kanal wird durch NULLPOL **nicht** beseitigt; er enthaelt exakt eine kleinere Kopie derselben Nullpol-Geometrie.
+
+---
+
+# 5. Scharfe Unterkante des Prime-Terms auf NULLPOL
+
+Fuer jedes `b>0` ist `C_c^\infty(-b,b)\cap\mathscr D_{NP,b}` unendlichdimensional: zwei stetige lineare Momentbedingungen schneiden aus `C_c^\infty(-b,b)` nur Kodimension hoechstens zwei heraus.
+
+Waehle also ein nichttriviales `g` in dieser Klasse und normiere `\|g\|=1`. Dann ist `v=S_+g` Nullpol, `\|v\|=1`, und
+
+```math
+\langle v,C_{n,a}v\rangle=-w.
+```
+
+Andererseits gilt aus der Spektralzerlegung stets `C_{n,a}\succeq-wI`.
+
+Damit
+
+```math
+\boxed{
+\inf_{0\ne v\in\mathscr D_{NP,a}}
+\frac{\langle v,C_{n,a}v\rangle}{\|v\|^2}
+=-w
+\qquad(a>h/2,\ a<h).
+}
+```
+
+Die Nullpolbedingungen verbessern den isolierten Prime-Bottom also nicht.
+
+---
+
+# 6. Operatornorm-Sprung am Eintritt eines Prime-Power-Kanals
+
+Am exakten Cutoff
+
+```math
+a_0=h/2
+```
+
+haben `I_{a_0}` und `I_{a_0}+h` nur einen Randpunkt gemeinsam. In `L^2` ist daher
+
+```math
+A_{a_0,h}=0,
+\qquad
+C_{n,a_0}=0.
+```
+
+Fuer **jedes** `a>a_0` mit `a<h` besitzt dagegen `C_{n,a}` den Eigenwert `-w` und `+w` auf unendlichdimensionalen Randkanaelen. Somit
+
+```math
+\boxed{
+\|C_{n,a}\|=w
+\quad\text{fuer jedes }a>h/2,
+\qquad
+\|C_{n,h/2}\|=0.
+}
+```
+
+Also existiert **keine Operatornorm-kleine Fortsetzung ueber den Prime-Power-Cutoff**.
+
+Gleichzeitig schrumpfen die Randstreifen in Mass gegen null. Auf einem fest eingebetteten Ambientraum gilt daher fuer jedes feste `f in L^2`
+
+```math
+C_{n,a}f\to0
+\qquad(a\downarrow h/2),
+```
+
+also starke Konvergenz. Die richtige Schwellengeometrie ist folglich
+
+```text
+strong/form local,
+not operator-norm perturbative.
+```
+
+---
+
+# 7. Spezialfall CRIT-HALF-COMPRESS-1: a=1/2, p=2
+
+Hier
+
+```math
+h=\log2,
+\qquad
+w_2=\frac{\log2}{\sqrt2},
+\qquad
+b=\frac12-\frac{\log2}{2}
+=\frac{1-\log2}{2}.
+```
+
+Da
+
+```math
+\frac12<\log2<1,
+```
+
+liegen wir exakt im obigen nilpotenten Regime.
+
+Der Prime-2-Term besitzt also die scharfe Zerlegung
+
+```math
+\boxed{
+C_{2,1/2}
+=-\frac{\log2}{\sqrt2}P_+
++\frac{\log2}{\sqrt2}P_-.
+}
+```
+
+Der negative Kanal ist eine isometrische Kopie der Nullpolklasse auf dem kleineren Fenster
+
+```math
+\boxed{
+b=\frac{1-\log2}{2}\approx0.153426\ldots.}
+```
+
+Damit reduziert sich der echte positive Schur-Gate bei `a=1/2` auf die Frage, ob die archimedische Critical-half-Form (zusammen mit dem positiven `P_-`-Kanal und ihren Kreuzblöcken) diese exakt lokalisierte negative Kopie kontrolliert.
+
+---
+
+# 8. Strategische Konsequenz
+
+Ein naiver Plan
+
+```text
+prove NP-GAP below a prime threshold,
+then cross the threshold by a small operator-norm perturbation
+```
+
+ist ausgeschlossen.
+
+Der neue Prime-/Prime-Power-Kanal ist in Operatornorm sofort von Groesse `w_n`, auch auf NULLPOL.
+
+Zulaessig und strukturell passend bleiben dagegen:
+
+1. Schur-Komplement gegen den lokalisierten Randkanal `P_+`;
+2. strong-/Mosco-/form convergence am Schwellenwert;
+3. eine rekursive kleinere-Fensterkontrolle ueber die exakte Isometrie `S_+`;
+4. Kopplung des bad channel mit der Critical-half Green-/Gamma-Geometrie.
+
+Insbesondere legt die Selbstähnlichkeit
+
+```math
+\mathscr D_{NP,b}
+\overset{S_+}{\cong}
+\mathscr D_{NP,a}\cap\operatorname{Ran}P_+
+```
+
+eine **renormierungsartige Prime-threshold recursion** nahe. Ob die archimedische Form unter `S_+` eine passende positive Rekursion besitzt, ist der naechste nichtzirkulaere Gate.
+
+---
+
+# 9. Firewalls
+
+Nicht behauptet wird:
+
+- der gesamte Weiloperator habe am Cutoff einen Operatornormsprung; nur der einzelne neu aktivierte zentrierte Prime-Power-Kompressionsblock besitzt diese Eigenschaft in der gemeinsamen Ambientdarstellung;
+- starke/Form-Konvergenz scheitere;
+- die kleinere-Fenster-Selbstähnlichkeit allein beweise NP-GAP;
+- mehrere gleichzeitig aktive Prime-Kanaele seien bereits gemeinsam diagonalisiert;
+- Object X oder RH seien geloest.
diff --git a/audits/P11_ROOT_MASS_MONOTONE_TRANSPORT_2026-09-16.md b/audits/P11_ROOT_MASS_MONOTONE_TRANSPORT_2026-09-16.md
new file mode 100644
index 00000000..6c06687b
--- /dev/null
+++ b/audits/P11_ROOT_MASS_MONOTONE_TRANSPORT_2026-09-16.md
@@ -0,0 +1,334 @@
+# P11 Audit — Monotone root-mass transport to the smooth storage line
+
+**Datum:** 16. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Registry:** unveraendert.  
+**Status:** exact construction plus imported classical PNT estimate `✓[K/M]`; no RH/Object-X closure.
+
+---
+
+## 0. Result
+
+Let `n_1<n_2<...` run through all prime powers and put
+
+```math
+h_j=\log n_j,
+\qquad
+a_j=\frac{\Lambda(n_j)}{n_j}>0.
+```
+
+These are exactly the base-spacing weighted **squared P11 root amplitudes**:
+
+```math
+a_j=(\log p)\,|\langle\Phi_o,\Phi_{p,k}\rangle|^2
+\qquad(n_j=p^k).
+```
+
+Define cumulative mass coordinates
+
+```math
+s_0=\gamma,
+\qquad
+s_j=\gamma+\sum_{i\le j}a_i,
+```
+
+and intervals
+
+```math
+I_0=[0,\gamma),
+\qquad
+I_j=[s_{j-1},s_j)\quad(j\ge1).
+```
+
+Then
+
+```math
+|I_j|=a_j.
+```
+
+The classical prime number theorem with de la Vallee-Poussin error implies
+
+```math
+\boxed{
+s_j-h_j
+=O\!\left(\sqrt{h_j}\,e^{-c\sqrt{h_j}}\right),
+}
+```
+
+for some `c>0`; in particular
+
+```math
+\boxed{
+\sup_{L\in I_j}|L-h_j|\longrightarrow0.
+}
+```
+
+Thus the arithmetic root measure plus the Euler-constant boundary mass admits a canonical order-preserving equal-mass embedding into the smooth root-storage line. No zeros and no RH are used.
+
+Let
+
+```math
+\mathscr H_P^{root}
+=\mathbb C_{\gamma}\oplus\ell^2(a_j),
+```
+
+with norm
+
+```math
+\|(c,x)\|^2
+=\gamma|c|^2+\sum_ja_j|x_j|^2.
+```
+
+Define
+
+```math
+(U(c,x))(L)
+=c\,1_{I_0}(L)
++\sum_jx_j1_{I_j}(L).
+```
+
+Then
+
+```math
+\boxed{
+U:\mathscr H_P^{root}\to L^2(\mathbb R_+,dL)
+\text{ is an isometry.}
+}
+```
+
+Its range is the closed subspace of functions constant on each arithmetic mass cell. The orthogonal complement is a canonical positive continuum reservoir.
+
+Moreover, if the arithmetic delay semigroup is
+
+```math
+(D_P(s)x)_j=e^{-s h_j}x_j
+```
+
+and the continuum delay is
+
+```math
+(M_s f)(L)=e^{-sL}f(L),
+\qquad \Re s\ge0,
+```
+
+then for every fixed `s` with `Re s>=0`,
+
+```math
+\boxed{
+K_s:=M_sU-UD_P(s)
+}
+```
+
+is compact. Its nonzero singular values (up to the finite boundary coordinate) are exactly
+
+```math
+\sigma_j(s)^2
+=\frac1{a_j}\int_{I_j}
+|e^{-sL}-e^{-s h_j}|^2dL,
+```
+
+and `sigma_j(s)->0` by the localization above.
+
+This gives the first explicit global **relative** Hilbert coupling on the branch for which the discrete prime-power delay geometry and the smooth root line differ only by a compact operator. It does not imply positivity of the Weil form; the Critical-half rescaling and higher physical channel structure still have to be attached.
+
+---
+
+# 1. PNT input for the weighted cumulative root mass
+
+Let
+
+```math
+\psi(x)=\sum_{n\le x}\Lambda(n)=x+E(x).
+```
+
+Import the classical unconditional de la Vallee-Poussin estimate
+
+```math
+E(x)=O\left(xe^{-c\sqrt{\log x}}\right).
+```
+
+Partial summation gives
+
+```math
+A(x):=\sum_{n\le x}\frac{\Lambda(n)}n
+=\frac{\psi(x)}x+
+\int_1^x\frac{\psi(t)}{t^2}dt.
+```
+
+Hence
+
+```math
+A(x)
+=\log x+1+rac{E(x)}x
++\int_1^x\frac{E(t)}{t^2}dt.
+```
+
+The Laurent expansion of `-zeta'/zeta` at `1` gives the limiting constant
+
+```math
+A(x)-\log x\longrightarrow-\gamma.
+```
+
+Equivalently,
+
+```math
+\boxed{
+A(x)
+=\log x-\gamma
++\frac{E(x)}x
+-\int_x^\infty\frac{E(t)}{t^2}dt.
+}
+```
+
+The imported error estimate yields
+
+```math
+\frac{E(x)}x
+=O(e^{-c\sqrt{\log x}}),
+```
+
+and with `u=sqrt(log t)`,
+
+```math
+\int_x^\infty e^{-c\sqrt{\log t}}\frac{dt}{t}
+=2\int_{\sqrt{\log x}}^\infty ue^{-cu}du
+=O\left((1+\sqrt{\log x})e^{-c\sqrt{\log x}}\right).
+```
+
+Therefore
+
+```math
+\boxed{
+\gamma+A(x)
+=\log x
++O\left((1+\sqrt{\log x})e^{-c\sqrt{\log x}}\right).
+}
+```
+
+At `x=n_j`, this is precisely the claimed estimate for `s_j-h_j`.
+
+The estimate is `✓[K/M]`: the reduction is elementary, but it imports the classical unconditional PNT error theorem.
+
+---
+
+# 2. Entire arithmetic interval lies near its physical depth
+
+Since
+
+```math
+s_{j-1}=s_j-a_j,
+```
+
+and
+
+```math
+a_j=\frac{\Lambda(n_j)}{n_j}\to0,
+```
+
+one has
+
+```math
+\sup_{L\in I_j}|L-h_j|
+\le |s_j-h_j|+a_j\to0.
+```
+
+Thus the equal-mass transport does not merely match cumulative endpoints: each whole arithmetic cell collapses onto its original logarithmic depth asymptotically.
+
+---
+
+# 3. Isometric root-storage embedding
+
+By construction the intervals are pairwise disjoint and
+
+```math
+|I_0|=\gamma,
+\qquad |I_j|=a_j.
+```
+
+Therefore
+
+```math
+\begin{aligned}
+\|U(c,x)\|_{L^2(dL)}^2
+&=\gamma|c|^2+\sum_j a_j|x_j|^2\\
+&=\|(c,x)\|_{\mathscr H_P^{root}}^2.
+\end{aligned}
+```
+
+Since
+
+```math
+\sum_ja_j=\infty
+```
+
+by `A(x)~log x`, the intervals exhaust the whole half-line. The range of `U` is the piecewise-constant arithmetic-cell subspace, not all of `L^2(dL)`. Its orthogonal complement consists of zero-mean fluctuations inside the mass cells and supplies a canonical positive reservoir rather than being discarded.
+
+---
+
+# 4. Relative delay mismatch is compact
+
+For the normalized atomic basis vector supported on cell `j`, the mismatch has squared norm
+
+```math
+\sigma_j(s)^2
+=\frac1{a_j}\int_{I_j}
+|e^{-sL}-e^{-s h_j}|^2dL.
+```
+
+For `Re s>=0`, the map `L -> e^{-sL}` is Lipschitz on `R_+` with constant at most `|s|`. Hence
+
+```math
+\sigma_j(s)
+\le |s|\sup_{L\in I_j}|L-h_j|
+\longrightarrow0.
+```
+
+The mismatch images of distinct normalized atomic basis vectors have disjoint supports, hence are orthogonal. Therefore `K_s` is a diagonal-singular-value operator with singular values `sigma_j(s)` plus a finite-rank boundary contribution. Since `sigma_j(s)->0`,
+
+```math
+\boxed{K_s\text{ is compact}.}
+```
+
+The statement is pointwise in `s`; no `H^infty`-norm compactness is claimed.
+
+---
+
+# 5. Relation to the critical finite-part transfer
+
+The cumulative endpoint defect is exactly
+
+```math
+s(L)-L
+=\gamma+M_P(L)-L
+=-e^{-L/2}J_\Delta(L)
+```
+
+away from atom conventions, where
+
+```math
+M_P(L)=\sum_{\log n<L}\frac{\Lambda(n)}n.
+```
+
+Thus the monotone transport displacement is precisely the unscaled Critical-half finite-part defect already found in the Volterra audit.
+
+The new construction therefore gives that scalar discrepancy a concrete geometric role:
+
+```text
+J_Delta = Critical-half rescaling of the displacement between
+          arithmetic root-mass coordinates and smooth storage depth.
+```
+
+This is not a positivity theorem. It is a forward relative Hilbert embedding derived from the same stopped P11 root masses.
+
+---
+
+# 6. Remaining firewall
+
+Several facts remain open:
+
+1. Compact relative delay mismatch does not imply positive-realness of the assembled zeta transfer.
+2. The critical multiplication by `e^{L/2}` that converts root-square jumps `Lambda(n)/n` into Weil jumps `Lambda(n)/sqrt(n)` is unbounded on the plain root-storage line.
+3. The full physical P11/COMMON-JUMP channel has higher-rank tail directions, not only the scalar root coordinate.
+4. The construction must still reproduce the exact `R=1` Prime-2 mixed witness before it can count as an Object-X candidate.
+
+The next gate is therefore to determine whether the Critical-half source operator `D_+=partial+1/2` supplies exactly the graph norm needed to make the rescaled transport bounded, while retaining the compact relative delay geometry above.
diff --git a/audits/P11_ROOT_TRANSPORT_SAFE_TRANSFER_AND_CRITICAL_LIFT_FIREWALL_2026-09-16.md b/audits/P11_ROOT_TRANSPORT_SAFE_TRANSFER_AND_CRITICAL_LIFT_FIREWALL_2026-09-16.md
new file mode 100644
index 00000000..428f5980
--- /dev/null
+++ b/audits/P11_ROOT_TRANSPORT_SAFE_TRANSFER_AND_CRITICAL_LIFT_FIREWALL_2026-09-16.md
@@ -0,0 +1,293 @@
+# P11 Audit — Safe relative root transfer and the Critical-half lift firewall
+
+**Datum:** 16. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Registry:** unveraendert.  
+**Status:** relative root transfer `✓[K/M]`; bounded Critical-half lift `×[M]` for the plain root-storage class; no RH/Object-X closure.
+
+---
+
+## 0. Result
+
+Use the monotone root-mass cells from the preceding audit:
+
+```math
+h_j=\log n_j,
+\qquad
+a_j=\frac{\Lambda(n_j)}{n_j},
+```
+
+```math
+s_0=\gamma,
+\qquad
+s_j=\gamma+\sum_{i\le j}a_i,
+\qquad
+I_0=[0,\gamma),
+\qquad I_j=[s_{j-1},s_j).
+```
+
+Then for `Re s>0`,
+
+```math
+\boxed{
+\gamma+
+\sum_j a_j e^{-s h_j}
+-
+\int_0^\infty e^{-sL}dL
+=
+\gamma-\frac1s-\frac{\zeta'}{\zeta}(1+s).
+}
+```
+
+The cellwise transport representation is
+
+```math
+\boxed{
+\gamma-\int_0^\gamma e^{-sL}dL
++
+\sum_j
+\left[
+a_j e^{-s h_j}
+-
+\int_{I_j}e^{-sL}dL
+\right].
+}
+```
+
+The series of cell differences converges absolutely for every fixed `s` with `Re s>=0`, because
+
+```math
+\boxed{
+\sum_j a_j
+\sup_{L\in I_j}|L-h_j|<\infty.
+}
+```
+
+Thus the monotone transport gives a forward, unconditional relative storage/transfer realization of the pole-subtracted logarithmic derivative in the safe half-plane `Re(1+s)>=1`.
+
+However the Critical-half lift from root-square weights to Weil weights is
+
+```math
+a_j
+\mapsto
+e^{h_j/2}a_j
+=\frac{\Lambda(n_j)}{\sqrt{n_j}}.
+```
+
+On the plain root-storage line this is an unbounded exponential multiplier. No finite-order source Sobolev norm repairs this tail, since for every nonzero compactly supported source the large-shift jump norm satisfies
+
+```math
+\|K_t v\|^2=2\|v\|^2
+```
+
+for all sufficiently large `t`.
+
+Hence the plain monotone root-storage class cannot be Critical-half lifted by bounded diagonal rescaling. The remaining coupling must perform discrete/continuum cancellation **before** the critical amplification.
+
+---
+
+# 1. Safe relative transfer
+
+Since
+
+```math
+\sum_j a_j e^{-s h_j}
+=\sum_{n\ge2}\frac{\Lambda(n)}{n^{1+s}}
+=-\frac{\zeta'}{\zeta}(1+s),
+```
+
+and
+
+```math
+\int_0^\infty e^{-sL}dL=\frac1s,
+```
+
+the first boxed identity is immediate for `Re s>0`.
+
+Because the cells partition the half-line after `I_0`,
+
+```math
+\int_0^\infty e^{-sL}dL
+=\int_0^\gamma e^{-sL}dL
++\sum_j\int_{I_j}e^{-sL}dL,
+```
+
+which gives the cellwise difference form.
+
+---
+
+# 2. Absolute convergence on the boundary
+
+Let
+
+```math
+\delta_j=\sup_{L\in I_j}|L-h_j|.
+```
+
+The preceding PNT audit gives
+
+```math
+\delta_j
+\ll
+(1+\sqrt{h_j})e^{-c\sqrt{h_j}}+a_j.
+```
+
+Hence
+
+```math
+\sum_j a_j\delta_j
+\ll
+\sum_n\frac{\Lambda(n)}n
+(1+\sqrt{\log n})e^{-c\sqrt{\log n}}
++
+\sum_n\left(\frac{\Lambda(n)}n\right)^2.
+```
+
+The second sum converges absolutely. For the first, partial summation with
+
+```math
+\sum_{n\le e^H}\frac{\Lambda(n)}n=H+O(1)
+```
+
+reduces convergence to
+
+```math
+\int_0^\infty(1+\sqrt H)e^{-c\sqrt H}dH<\infty.
+```
+
+Thus
+
+```math
+\boxed{\sum_j a_j\delta_j<\infty.}
+```
+
+For fixed `s` with `Re s>=0`,
+
+```math
+|e^{-s h_j}-e^{-sL}|
+\le |s|\,|h_j-L|
+```
+
+for `L>=0`. Therefore
+
+```math
+\sum_j
+\left|
+a_j e^{-s h_j}-\int_{I_j}e^{-sL}dL
+\right|
+\le |s|\sum_j a_j\delta_j<\infty.
+```
+
+So the relative cell transfer has a direct absolutely convergent boundary representation, independent of analytic continuation of zeta.
+
+This is a useful distinction: the safe relative transfer is genuinely constructed from positive root masses and monotone transport; it is not defined backwards from a zeta continuation.
+
+---
+
+# 3. Critical-half lift
+
+The Weil atom at depth `h_j` is
+
+```math
+w_j
+=\frac{\Lambda(n_j)}{\sqrt{n_j}}
+=e^{h_j/2}a_j.
+```
+
+Thus passing from the root-square relative storage to the Weil layer requires the depth multiplier
+
+```math
+M_{crit}:f(L)\mapsto e^{L/4}f(L)
+```
+
+at the amplitude level, or `e^{L/2}` at the energy/measure level.
+
+This is unbounded on `L^2(dL)`.
+
+More importantly, it is unbounded on the actual universal source field. Recall
+
+```math
+X_v(t)=e^{-t/4}K_tv.
+```
+
+At large `t` relative to the support of a nonzero compact source,
+
+```math
+\|K_tv\|^2=2\|v\|^2.
+```
+
+Hence
+
+```math
+\|X_v(t)\|^2
+=2e^{-t/2}\|v\|^2,
+```
+
+which is integrable, but after critical energy rescaling
+
+```math
+e^{t/2}\|X_v(t)\|^2
+=2\|v\|^2,
+```
+
+whose integral over the tail diverges.
+
+Applying any fixed finite-order differential operator in the source variable `x` does not alter this conclusion: for compact `v`, shifted supports are eventually disjoint and
+
+```math
+\|K_tPv\|^2=2\|Pv\|^2
+```
+
+for every fixed differential operator `P` and large `t`.
+
+Therefore
+
+```text
+plain root storage + diagonal Critical-half amplification
+```
+
+is not a bounded global Hilbert construction.
+
+---
+
+# 4. Meaning of the obstruction
+
+The safe root transfer uses
+
+```math
+-\frac{\zeta'}{\zeta}(1+s)-\frac1s,
+```
+
+which lives in the classical zero-free half-plane after subtracting the pole.
+
+The desired Critical-half arithmetic transfer uses the same coefficients after shifting the decay exponent by `1/2`. In depth space this is exactly the unbounded multiplier `e^{L/2}`.
+
+The monotone transport displacement is
+
+```math
+L-\gamma-M_P(L)=e^{-L/2}J_\Delta(L).
+```
+
+Classical PNT gives only subexponential decay in `L`, while Critical-half amplification multiplies by `e^{L/2}`. Thus the finite-part kernel `J_Delta` is not obtained as a bounded diagonal image of the safe transport.
+
+This pinpoints the remaining architecture requirement:
+
+```text
+perform prime/continuum relative cancellation at root-square level first,
+then realize the Critical-half transfer through a genuinely non-diagonal
+boundary/state mechanism; do not amplify the two sides separately.
+```
+
+---
+
+# 5. Firewall / next gate
+
+This no-go is scoped to **diagonal** Critical-half amplification of the plain root-storage construction. It does not exclude a larger conservative colligation in which cancellation is internal before the output port.
+
+The next valid candidate must therefore:
+
+1. use the local Schur factors `Phi_h` from the Euler-cell domination theorem;
+2. use the monotone root-mass transport as the relative scale geometry;
+3. couple discrete and continuum states before the Critical-half output weighting;
+4. expose only the finite relative output corresponding to `J_Delta`;
+5. reproduce the exact Prime-2 mixed witness with no fitted parameter.
diff --git a/audits/P11_SCATTERING_POINTWISE_POSITIVITY_NOGO_2026-09-13.md b/audits/P11_SCATTERING_POINTWISE_POSITIVITY_NOGO_2026-09-13.md
new file mode 100644
index 00000000..b58afb69
--- /dev/null
+++ b/audits/P11_SCATTERING_POINTWISE_POSITIVITY_NOGO_2026-09-13.md
@@ -0,0 +1,273 @@
+# P11 Audit — Pointwise scattering-positivity No-Go
+
+**Datum:** 13. September 2026  
+**Basis:** Euler/Gamma scattering-phase realization auf dem Critical-half-Branch.  
+**Rolle:** theorem-level Architekturfilter.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein NP-GAP-/RH-Beweis.
+
+---
+
+## 0. Kurzurteil
+
+Der explizite Fenster-Streumultiplikator
+
+```math
+\tau_a(z):=-i\overline{S_a(z)}S_a'(z)
+```
+
+ist fuer **jedes** `a>0` auf der reellen Achse zwingend vorzeichenwechselnd:
+
+```math
+\boxed{\tau_a(0)<0,}
+```
+
+waehrend
+
+```math
+\boxed{\tau_a(z)\to+\infty\qquad(|z|\to\infty).}
+```
+
+Damit ist jede Object-X-/Dilationklasse ausgeschlossen, deren Positivitaetsmechanismus eine punktweise nichtnegative Wigner--Smith-Dichte
+
+```math
+\tau_a(z)\ge0
+```
+
+oder eine punktweise Faktorisierung
+
+```math
+\tau_a(z)=B_a(z)^*B_a(z)
+```
+
+auf der reellen Frequenzachse verlangen wuerde.
+
+Die gesuchte Positivitaet kann daher nur durch die **nichtlokale Fenster-/Paley--Wiener-Struktur und die NULLPOL-Range-Bedingung** entstehen.
+
+Status:
+
+```text
+sign change of finite-window scattering delay           ✓[M]
+pointwise-positive multiplier factorization              ×[M]
+pointwise-positive Wigner-Smith dilation class           ×[M]
+need for support/Paley-Wiener/nullpole compression       ✓[M]
+sharp compressed NP-GAP                                  ?[O]
+RH / full Object X                                       ?[O]
+```
+
+---
+
+# 1. Fenster-Streufaktor
+
+Aus dem Scattering-Audit:
+
+```math
+S_a(z)
+=S_\infty(z)\prod_{p\le e^{2a}}S_p(z),
+```
+
+mit
+
+```math
+S_p(z)=
+\frac{1-p^{-1/2}e^{iz\log p}}
+     {1-p^{-1/2}e^{-iz\log p}},
+```
+
+```math
+S_\infty(z)=
+\pi^{-iz}
+\frac{\Gamma(1/4+iz/2)}
+     {\Gamma(1/4-iz/2)}.
+```
+
+Fuer reelles `z` ist `|S_a(z)|=1`, und
+
+```math
+\tau_a(z)
+=
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi
++
+\sum_{p\le e^{2a}}
+(\log p)
+\left[
+1-P_{p^{-1/2}}^{\mathbb D}(z\log p)
+\right].
+```
+
+---
+
+# 2. Exakter negativer Wert bei `z=0`
+
+Archimedisch:
+
+```math
+\tau_\infty(0)
+=\psi(1/4)-\log\pi
+=-\kappa_*<0.
+```
+
+Fuer einen Prime-Faktor setze `q=p^{-1/2}`. Dann
+
+```math
+P_q^{\mathbb D}(0)
+=\frac{1+q}{1-q},
+```
+
+also
+
+```math
+1-P_q^{\mathbb D}(0)
+=-\frac{2q}{1-q}
+=-\frac{2}{\sqrt p-1}.
+```
+
+Daher exakt
+
+```math
+\boxed{
+\tau_a(0)
+=-\kappa_*
+-2\sum_{p\le e^{2a}}
+\frac{\log p}{\sqrt p-1}
+<0.
+}
+```
+
+Die Summe ist endlich. Falls `e^{2a}<2`, ist sie leer; auch dann bleibt `tau_a(0)=-kappa_*<0`.
+
+---
+
+# 3. Positive Hochfrequenz
+
+Aus der Digamma-Asymptotik
+
+```math
+\operatorname{Re}\psi\left(\frac14+\frac{iz}{2}\right)-\log\pi
+=
+\log\frac{|z|}{2\pi}+O(|z|^{-2})
+```
+
+folgt archimedisch logarithmisches Wachstum gegen `+infty`.
+
+Fuer jedes feste `p` ist dagegen
+
+```math
+(\log p)[1-P_{p^{-1/2}}^{\mathbb D}(z\log p)]
+```
+
+eine stetige beschraenkte periodische Funktion. Im Fensterprodukt treten nur endlich viele Primzahlen `p<=e^{2a}` auf. Daher existiert eine Konstante `C_a<infty` mit
+
+```math
+\left|
+\sum_{p\le e^{2a}}
+(\log p)[1-P_{p^{-1/2}}^{\mathbb D}(z\log p)]
+\right|
+\le C_a.
+```
+
+Somit
+
+```math
+\boxed{
+\tau_a(z)
+=
+\log\frac{|z|}{2\pi}+O_a(1)
+\longrightarrow+\infty.
+}
+```
+
+Zusammen mit `tau_a(0)<0` und Stetigkeit folgt zwingend ein Vorzeichenwechsel.
+
+---
+
+# 4. No-Go fuer punktweise Gram-/Time-delay-Positivitaet
+
+Angenommen, eine vorgeschlagene Object-X-Architektur wuerde auf der reellen Frequenzachse eine direkte punktweise Faktorisierung liefern
+
+```math
+\tau_a(z)=B_a(z)^*B_a(z).
+```
+
+Dann waere notwendigerweise
+
+```math
+\tau_a(z)\ge0
+```
+
+fuer alle reellen `z`, im Widerspruch zu §2.
+
+Daher
+
+```math
+\boxed{
+\text{direct pointwise multiplier Gram factorization}
+\quad\times[M].
+}
+```
+
+Dasselbe gilt fuer jede conservative scattering-Dilationklasse, fuer welche ein allgemeiner Satz die punktweise Positivitaet des Wigner--Smith-Operators auf dem vollen reellen Spektrum erzwingen wuerde.
+
+### Firewall
+
+Nicht ausgeschlossen sind positive **Kompressionen**, Shortings, Toeplitz-/Hankel-Formen oder reproduzierende-Kern-Konstruktionen auf einem echten Unterraum. Genau dort koennen die Fenster-/Paley--Wiener- und NULLPOL-Bedingungen wesentlich werden.
+
+---
+
+# 5. Warum dies mit NP-GAP vereinbar ist
+
+Auf NULLPOL gilt in Fensterquadratikform
+
+```math
+Q_W(v)
+=\int|\widehat v(z)|^2\tau_a(z)dz.
+```
+
+Die Funktionen `widehat v` sind jedoch keine beliebigen `L^2`-Gewichte:
+
+1. `supp v subset [-a,a]` zwingt eine Paley--Wiener-Struktur vom Exponentialtyp `a`;
+2. NULLPOL ist exakt
+   ```math
+   v=L_{1/2}u,
+   ```
+   also auf Fourierseite die feste Multiplikation mit `z^2+1/4` aus einem wiederum fenstergetragenen `u`;
+3. aequivalent liegen die komplexen Nullbedingungen bei `z=±i/2` vor.
+
+Daher widerspricht die punktweise Negativitaet von `tau_a` nicht einer moeglichen Positivitaet der **komprimierten Form**.
+
+Gerade dies zeigt, dass NP-GAP wesentlich ein Unschaerfe-/Hardy-/Paley--Wiener-Problem ist und nicht durch eine punktweise Symboluntergrenze geloest werden kann.
+
+---
+
+# 6. Neuer Architekturfilter
+
+Ab jetzt sind als Hauptkandidaten nur Mechanismen interessant, die die Kompression ernsthaft benutzen, zum Beispiel:
+
+```text
+Paley-Wiener Toeplitz/Hankel shorting,
+Hardy/model-space compression,
+de-Branges/reproducing-kernel geometry,
+positive harmonic extension with nontrivial boundary Schur complement,
+critical-half source lift plus Tree/Gamma coupling.
+```
+
+Nicht weiter verfolgen:
+
+```text
+pointwise tau_a >= 0,
+direct multiplication Gram tau_a=B^*B,
+"unitary scattering implies positive time delay",
+channelwise lower bounds that ignore the window subspace.
+```
+
+---
+
+# 7. Firewalls
+
+Nicht behaupten:
+
+- das No-Go widerlege NP-GAP;
+- eine vorzeichenwechselnde Wigner--Smith-Dichte verhindere positive komprimierte Quadratikformen;
+- Hardy-/de-Branges-Positivitaet sei bereits bewiesen;
+- RH sei widerlegt oder bewiesen;
+- Publikationsneuheit sei geklaert.
diff --git a/audits/P11_SPATIAL_STOPPED_OU_FILTRATION_2026-09-15.md b/audits/P11_SPATIAL_STOPPED_OU_FILTRATION_2026-09-15.md
new file mode 100644
index 00000000..195d8f57
--- /dev/null
+++ b/audits/P11_SPATIAL_STOPPED_OU_FILTRATION_2026-09-15.md
@@ -0,0 +1,444 @@
+# P11 Audit — Spatial masks as a stopped OU filtration
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Reinterpretation der bereits vorhandenen P11-Fenstermasken.  
+**Registry:** unveraendert.  
+**Nonclaim:** keine Weil-Identitaet, kein Object-X-/RH-Abschluss, keine Publikationsneuheit.
+
+---
+
+## 0. Kurzurteil
+
+Die P11-Fenstermasken sind nicht bloss technische Cutoffs. Sie implementieren punktweise eine **gestoppte AR(1)/OU-Filtration**, deren sichtbare Tiefe exakt durch den Abstand zum Fensterrand bestimmt wird.
+
+Fuer das Fenster `(-R,R)` und eine Primzahl `p` mit
+
+```math
+h_p=\log p,
+\qquad
+q_p=p^{-1/2},
+```
+
+ist der vorhandene P11-Sektor `a>=0` auf
+
+```math
+\boxed{
+\Omega_{p,a,R}
+=
+\left\{x:\ |x|\le R-\frac{(a+1)h_p}{2}\right\}
+}
+```
+
+getragen. Definiere die Randtiefe
+
+```math
+d_R(x)=R-|x|
+```
+
+und fuer fast jedes `x`
+
+```math
+\boxed{
+m_p(x)=\left\lfloor\frac{2d_R(x)}{h_p}\right\rfloor.}
+```
+
+Dann sind am Punkt `x` genau die Innovationslagen
+
+```text
+a=0,...,m_p(x)-1
+```
+
+aktiv.
+
+Addiert man den P11-Hub und diese aktiven Restlagen, so ist die lokale same-prime Kanalkovarianz
+
+```math
+\boxed{
+C_{jk}^{(p,m)}
+=(\log p)p^{-3(j+k)/4}p^{\min(j,k,m)}.
+}
+```
+
+Nach Weil-Diagonalnormalisierung ergibt sich
+
+```math
+\boxed{
+R_{q}^{(m)}(j,k)
+=q^{j+k-2\min(j,k,m)}.
+}
+```
+
+Dies ist exakt die Kovarianz eines stationaeren AR(1)-Prozesses, bei dem nur Root und Innovationen bis Tiefe `m` sichtbar sind. Fuer `j,k>m` sind die spaeteren Zustaende auf ihre bedingten Erwartungen aus Ebene `m` reduziert.
+
+Die Radius-/Tiefenentwicklung ist positiv Rang eins:
+
+```math
+\boxed{
+R_q^{(m+1)}-R_q^{(m)}
+=d_{m+1}d_{m+1}^*\succeq0,
+}
+```
+
+mit
+
+```math
+(d_{m+1})_j
+=\sqrt{1-q^2}\,q^{j-m-1}\mathbf 1_{j\ge m+1}.
+```
+
+Damit besitzt P11 bereits intern eine radiuskonsistente positive Filtration: beim Wachsen der sichtbaren Randtiefe wird jeweils genau eine neue OU-Innovation freigegeben.
+
+Status:
+
+```text
+mask = boundary-depth threshold                           ✓[M]
+local capped AR(1) covariance                            ✓[M]
+stopped-OU / conditional-expectation interpretation      ✓[M]
+rank-one positive depth increments                       ✓[M]
+P11 masks as discrete sections of common overlap cone    ✓[M]
+P11 stopped filtration = Weil/Object X                   not claimed
+```
+
+---
+
+# 1. Existing P11 mask and the boundary depth
+
+The finite-window P11 rest sector is, in the existing implementation/convention,
+
+```math
+\Phi_{p,a,R}
+=
+P_{\Omega_{p,a,R}}
+\sum_{k\ge a+1}p^{-3k/4}K_{p,k}P_R,
+```
+
+up to the already documented scalar sector factors, with
+
+```math
+K_{p,k}=K_{k\log p}.
+```
+
+For `R=1`, the committed checker implements
+
+```text
+omega = 1 - (a+1) log(p)/2,
+```
+
+and retains exactly positions with `|x|<=omega`. The general P11 mask is therefore
+
+```math
+\Omega_{p,a,R}
+=
+\left(-R+\frac{(a+1)h_p}{2},
+       R-\frac{(a+1)h_p}{2}\right)
+```
+
+up to irrelevant endpoint conventions in `L^2`.
+
+Equivalently,
+
+```math
+x\in\Omega_{p,a,R}
+\iff
+(a+1)\frac{h_p}{2}\le d_R(x).
+```
+
+Hence the number of active innovation layers at `x` is almost everywhere
+
+```math
+m_p(x)=\left\lfloor\frac{2d_R(x)}{h_p}\right\rfloor.
+```
+
+At the discrete equality surfaces the choice of closed/open endpoint does not affect the `L^2` form.
+
+---
+
+# 2. Local channel coefficient after summing active sectors
+
+Fix one prime `p`, one spatial point `x`, and write
+
+```math
+m=m_p(x).
+```
+
+The P11 hub contributes the primitive channel coefficient
+
+```math
+(\log p)p^{-3(j+k)/4}.
+```
+
+A rest sector `a` contributes to the pair `(j,k)` precisely when
+
+```text
+a+1 <= j,
+a+1 <= k,
+a < m,
+```
+
+with multiplicity factor
+
+```math
+(p-1)p^a.
+```
+
+Therefore the total local coefficient is
+
+```math
+\begin{aligned}
+C_{jk}^{(p,m)}
+&=(\log p)p^{-3(j+k)/4}
+\left[
+1+
+\sum_{a=0}^{\min(j,k,m)-1}(p-1)p^a
+\right]\\
+&=(\log p)p^{-3(j+k)/4}
+ p^{\min(j,k,m)}.
+\end{aligned}
+```
+
+This proves
+
+```math
+\boxed{
+C_{jk}^{(p,m)}
+=(\log p)p^{-3(j+k)/4}p^{\min(j,k,m)}.
+}
+```
+
+When `m>=min(j,k)` one recovers the windowless P11 ledger exactly.
+
+---
+
+# 3. Weil-normalized stopped covariance
+
+Recall
+
+```math
+w_{p,k}=(\log p)p^{-k/2}.
+```
+
+Thus
+
+```math
+\sqrt{w_{p,j}w_{p,k}}
+=(\log p)p^{-(j+k)/4}.
+```
+
+Dividing the local coefficient by the Weil weights gives
+
+```math
+\begin{aligned}
+R_q^{(m)}(j,k)
+&=
+\frac{C_{jk}^{(p,m)}}{\sqrt{w_{p,j}w_{p,k}}}\\
+&=
+p^{\min(j,k,m)-(j+k)/2}\\
+&=q^{j+k-2\min(j,k,m)}.
+\end{aligned}
+```
+
+Hence
+
+```math
+\boxed{
+R_q^{(m)}(j,k)
+=q^{j+k-2\min(j,k,m)}.
+}
+```
+
+For `min(j,k)<=m` this is `q^{|j-k|}`. If both indices lie beyond the stopping depth,
+
+```math
+R_q^{(m)}(j,k)
+=q^{j-m}q^{k-m}.
+```
+
+---
+
+# 4. Exact stopped-OU realization
+
+Let
+
+```math
+Z_0,Z_1,Z_2,...
+```
+
+be orthonormal coordinates. Define the AR(1) state
+
+```math
+X_j
+=q^jZ_0
++\sqrt{1-q^2}\sum_{r=1}^j q^{j-r}Z_r.
+```
+
+Then
+
+```math
+\langle X_j,X_k\rangle=q^{|j-k|}.
+```
+
+Define the stopped state at depth `m`
+
+```math
+\boxed{
+X_j^{(m)}
+=q^jZ_0
++\sqrt{1-q^2}
+\sum_{r=1}^{\min(j,m)}q^{j-r}Z_r.
+}
+```
+
+For `j>m`, this is exactly the conditional expectation of the full state given the filtration generated by `Z_0,...,Z_m`.
+
+Direct telescoping gives
+
+```math
+\begin{aligned}
+\langle X_j^{(m)},X_k^{(m)}\rangle
+&=
+q^{j+k}
++(1-q^2)
+\sum_{r=1}^{\min(j,k,m)}q^{j+k-2r}\\
+&=
+q^{j+k-2\min(j,k,m)}.
+\end{aligned}
+```
+
+Therefore
+
+```math
+\boxed{
+\langle X_j^{(m)},X_k^{(m)}\rangle
+=R_q^{(m)}(j,k).
+}
+```
+
+So the spatial P11 mask implements a literal stopped OU filtration, not merely an analogy.
+
+---
+
+# 5. Positive rank-one radius increments
+
+Increasing the visible depth from `m` to `m+1` adds exactly the new innovation coordinate `Z_{m+1}`. Thus
+
+```math
+R_q^{(m+1)}(j,k)-R_q^{(m)}(j,k)
+=(1-q^2)
+q^{j-m-1}q^{k-m-1}
+\mathbf 1_{j,k\ge m+1}.
+```
+
+Set
+
+```math
+(d_{m+1})_j
+=\sqrt{1-q^2}\,q^{j-m-1}\mathbf1_{j\ge m+1}.
+```
+
+Then
+
+```math
+\boxed{
+R_q^{(m+1)}-R_q^{(m)}
+=d_{m+1}d_{m+1}^*\succeq0.
+}
+```
+
+This is the exact channel-index Hamiltonian increment of the P11 boundary-depth flow.
+
+---
+
+# 6. The common overlap cone
+
+The mask condition can be written
+
+```math
+(a+1)h_p\le2d_R(x).
+```
+
+Introduce the geometric cone
+
+```math
+\boxed{
+\mathcal C_R
+=
+\{(x,t): |x|<R,\ 0<t<2(R-|x|)\}.
+}
+```
+
+For the common-jump operator
+
+```math
+K_t=T_{t/2}-T_{-t/2},
+```
+
+the inequality
+
+```math
+t<2d_R(x)
+```
+
+is precisely the condition that both half-shifted points `x+-t/2` lie inside the support window.
+
+The P11 innovation masks are the discrete prime-ray sections
+
+```math
+\boxed{
+t=(a+1)\log p}
+```
+
+of this same overlap cone.
+
+Thus a previously separate P11 spatial object and the COMMON-JUMP geometry share the exact same boundary-depth variable.
+
+---
+
+# 7. Direct-integral interpretation
+
+At a fixed spatial point `x`, each prime branch has a finite visible innovation depth `m_p(x)`, while all prime branches share the root coordinate. Hence the finite-window P11 hub+rest target can be viewed as a direct integral over `x` of prime star-tree fibres whose branch depths are
+
+```math
+m_p(x)=\lfloor2d_R(x)/\log p\rfloor.
+```
+
+As `R` increases (with `x` fixed away from the moving boundary), each branch changes only when one of the surfaces
+
+```math
+2d_R(x)=r\log p
+```
+
+is crossed; the change is the positive rank-one innovation from Section 5.
+
+This is a genuine radius-consistent positive filtration already present in P11.
+
+---
+
+# 8. Firewall from existing negative calibrations
+
+This stopped-filtration theorem does **not** imply that the native P11 Gram equals the Weil form.
+
+The already merged finite-window audits show that the native P11 Rest and native Feshbach/Schur pairing miss the exact Prime-2 Weil calibration on the documented `R=1` witness. Those results remain untouched.
+
+What changes is the structural interpretation of the remaining search space:
+
+```text
+old view:  P11 masks are window technicalities;
+new exact view: P11 masks are the positive depth filtration of the OU tree.
+```
+
+Therefore any future use of P11 for Object X should work with the **full stopped spatial filtration**, not only with the windowless AR(1) ledger.
+
+---
+
+# 9. Next gate
+
+The natural next question is whether the stopped discrete OU filtration on
+
+```math
+\mathcal C_R\cap\{t=r\log p\}
+```
+
+and the continuous Critical-half/Gamma geometry on the same overlap cone can be embedded into one positive boundary-depth system whose Schur complement reproduces the pole-subtracted discrepancy normal form.
+
+A successful construction must also overcome the existing Prime-2 native-Feshbach mismatch; merely renaming the current P11 Rest Gram does not suffice.
diff --git a/audits/P11_STOPPED_OU_STRONG_TERMINAL_RATE_2026-09-15.md b/audits/P11_STOPPED_OU_STRONG_TERMINAL_RATE_2026-09-15.md
new file mode 100644
index 00000000..a244e0fc
--- /dev/null
+++ b/audits/P11_STOPPED_OU_STRONG_TERMINAL_RATE_2026-09-15.md
@@ -0,0 +1,327 @@
+# P11 Audit — Stopped-OU interior norm rate and Strong-Terminal boundary mechanism
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Konvergenzfolge der gestoppten OU-Filtration.  
+**Registry:** unveraendert.  
+**Nonclaim:** keine neue globale P11-/Weil-Konvergenz ueber bereits dokumentierte Firewall hinaus; die Aussage betrifft den Kanal-Gram und erklaert den Strong-vs-Norm-Mechanismus.
+
+---
+
+## 0. Kurzurteil
+
+Der gestoppte normalisierte AR(1)-Kern konvergiert mit wachsender Innovationstiefe **stark, aber nicht in Operatornorm** zum vollen AR(1)-Kern. Die fehlende Kovarianz ist lediglich in immer hoehere Kanalindizes verschoben und behaelt ihre Norm.
+
+Nach Einsetzen der echten Weilgewichte aendert sich die Lage im festen Innenraum: der physikalische Prime-Kanal-Gram konvergiert dort sogar exponentiell in Operatornorm. Gleichmaessig ueber alle Primzahlen gilt fuer eine sichtbare kontinuierliche Tiefe `L>=2`
+
+```math
+\boxed{
+\sup_p
+\|C_p-C_p^{(m_p(L))}\|
+\le
+C_0 L e^{-L/2},
+}
+```
+
+mit dem expliziten universellen Faktor
+
+```math
+C_0=\frac{1+2^{-1/2}}{1-2^{-1/2}}.
+```
+
+Auf dem gesamten wachsenden Raum kann trotzdem keine solche Normkonvergenz aus diesem Argument folgen, weil die bewegte Fensterrandzone stets Punkte mit Stopptiefe `L_R(x) approx 0` enthaelt. Fuer jeden festen kompakten Source-Vektor wandert diese schlechte Zone dagegen nach aussen. Genau dies ist der geometrische Mechanismus hinter **Strong-Terminal statt globaler Operatornorm-Konvergenz**.
+
+Status:
+
+```text
+normalized stopped AR1 -> full AR1 strongly             ✓[M]
+normalized operator-norm convergence                     ×[M]
+weighted fixed-depth prime Gram norm rate                ✓[M]
+uniform-in-prime interior rate O(L e^{-L/2})             ✓[M]
+boundary layer obstructs global sup-norm mechanism       ✓[M]
+compatibility with Strong-Terminal architecture          ✓[M]_part
+```
+
+---
+
+# 1. Normalized covariance tail
+
+Recall
+
+```math
+R_q(j,k)=q^{|j-k|},
+```
+
+and the stopped covariance
+
+```math
+R_q^{(m)}(j,k)
+=q^{j+k-2\min(j,k,m)}.
+```
+
+The innovation vectors are
+
+```math
+(d_r)_j
+=\sqrt{1-q^2}q^{j-r}\mathbf1_{j>=r}.
+```
+
+Hence
+
+```math
+\boxed{
+R_q-R_q^{(m)}
+=\sum_{r>m}d_rd_r^*\succeq0.
+}
+```
+
+For `m=0`,
+
+```math
+R_q-R_q^{(0)}
+=R_q-uu^*
+=T_q^*T_q.
+```
+
+For general `m`, the nonzero tail block is obtained from the `m=0` block by the unilateral shift isometry. Therefore
+
+```math
+\boxed{
+\|R_q-R_q^{(m)}\|
+=\|T_q\|^2.
+}
+```
+
+Since
+
+```math
+T_q=\sqrt{1-q^2}(I-qS^*)^{-1},
+```
+
+we have
+
+```math
+\|T_q\|
+=\frac{\sqrt{1-q^2}}{1-q}
+=\sqrt{\frac{1+q}{1-q}}.
+```
+
+Thus
+
+```math
+\boxed{
+\|R_q-R_q^{(m)}\|
+=\frac{1+q}{1-q}
+}
+```
+
+for every `m`.
+
+So normalized operator-norm convergence is impossible.
+
+---
+
+# 2. Strong convergence
+
+Let `P_{>m}` be the projection onto channel indices `j>m`. Then
+
+```math
+R_q-R_q^{(m)}
+=P_{>m}(R_q-R_q^{(m)})P_{>m}.
+```
+
+The operators are uniformly bounded in `m`, while
+
+```math
+P_{>m}x\to0
+```
+
+for every `x in ell^2`. Hence
+
+```math
+\boxed{
+(R_q-R_q^{(m)})x\to0
+}
+```
+
+for every fixed `x`.
+
+Therefore
+
+```math
+\boxed{R_q^{(m)}\to R_q\quad\text{strongly but not in norm}.}
+```
+
+---
+
+# 3. Weil-weighted physical channel Gram
+
+For fixed prime `p`, put
+
+```math
+h=\log p,
+\qquad
+q=p^{-1/2}.
+```
+
+The Weil channel weights satisfy
+
+```math
+w_{p,j}=hq^j.
+```
+
+Let
+
+```math
+D_p=\operatorname{diag}_{j>=1}(\sqrt h\,q^{j/2}).
+```
+
+Then the physical weighted Gram is
+
+```math
+C_p=D_pR_qD_p,
+```
+
+and the stopped physical Gram is
+
+```math
+C_p^{(m)}=D_pR_q^{(m)}D_p.
+```
+
+Therefore
+
+```math
+C_p-C_p^{(m)}
+=D_p(R_q-R_q^{(m)})D_p.
+```
+
+On the tail `j=m+r`,
+
+```math
+D_p
+=\sqrt h\,q^{m/2}
+\operatorname{diag}_{r>=1}(q^{r/2}).
+```
+
+Using the normalized tail norm from Section 1,
+
+```math
+\begin{aligned}
+\|C_p-C_p^{(m)}\|
+&\le
+hq^m
+\left\|\operatorname{diag}(q^{r/2})\right\|^2
+\frac{1+q}{1-q}\\
+&=
+\boxed{
+hq^{m+1}\frac{1+q}{1-q}.}
+\end{aligned}
+```
+
+---
+
+# 4. Continuous boundary depth
+
+For continuous visible depth `L`, the P11 mask gives
+
+```math
+m_p(L)=\left\lfloor\frac Lh\right\rfloor.
+```
+
+If `h<=L`, then
+
+```math
+(m_p(L)+1)h>L,
+```
+
+so
+
+```math
+q^{m_p(L)+1}
+=e^{-(m_p(L)+1)h/2}
+<e^{-L/2}.
+```
+
+Therefore
+
+```math
+\|C_p-C_p^{(m_p(L))}\|
+\le
+h e^{-L/2}\frac{1+q}{1-q}
+\le
+L e^{-L/2}C_0,
+```
+
+where
+
+```math
+C_0
+=\frac{1+2^{-1/2}}{1-2^{-1/2}}.
+```
+
+If `h>L>=2`, then `m_p(L)=0` and
+
+```math
+hq^{m+1}=he^{-h/2}.
+```
+
+The function `h e^{-h/2}` is decreasing for `h>=2`, hence
+
+```math
+he^{-h/2}
+\le
+Le^{-L/2}.
+```
+
+Thus for every `L>=2`, uniformly over all primes,
+
+```math
+\boxed{
+\sup_p
+\|C_p-C_p^{(m_p(L))}\|
+\le
+C_0Le^{-L/2}.
+}
+```
+
+---
+
+# 5. Why this does not give global moving-window norm convergence
+
+In the spatial P11 system the depth is not a fixed `L`; it is
+
+```math
+L_R(x)=2(R-|x|).
+```
+
+For every radius `R`, points arbitrarily close to the moving boundary have
+
+```math
+L_R(x)\approx0.
+```
+
+Therefore a global multiplication/direct-integral operator norm, which takes an essential supremum over the spatial fibre, cannot use the large-`L` estimate uniformly across the whole moving window.
+
+By contrast, for a fixed compactly supported source and `R->infty`, every spatial point in the fixed source region satisfies
+
+```math
+L_R(x)\to\infty.
+```
+
+The local weighted Gram error there decays exponentially. This is precisely the geometry expected for strong convergence on fixed vectors.
+
+---
+
+# 6. Relation to the existing Strong-Terminal front
+
+Earlier P11 research had already identified strong convergence, rather than global operator-norm convergence, as the natural terminal topology for moving-window objects.
+
+The present stopped-OU calculation gives a direct channel-geometric reason:
+
+```text
+normalized covariance tail: norm is merely shifted, never shrinks;
+Weil weights: interior tail becomes exponentially small;
+moving physical boundary: always regenerates a fresh shallow-depth fibre.
+```
+
+Thus the stopped-OU interpretation is compatible with, and conceptually explains, the existing Strong-Terminal architecture. It does not by itself prove any additional full spatial intertwining theorem beyond the stated channel-Gram estimates.
diff --git a/audits/P11_STORAGE_GATE_CURRENT_2026-09-15.md b/audits/P11_STORAGE_GATE_CURRENT_2026-09-15.md
new file mode 100644
index 00000000..aadad0fa
--- /dev/null
+++ b/audits/P11_STORAGE_GATE_CURRENT_2026-09-15.md
@@ -0,0 +1,230 @@
+# P11 / Object X — Current relative-storage gate
+
+**Datum:** 15. September 2026
+
+## 0. Current verdict
+
+The hard audit has reduced the branch to one nontrivial structural question. Local prime and archimedean pieces are no longer mysterious:
+
+```text
+Prime p-power tower     = discrete Critical-half OU precision
+Gamma ground            = continuous Critical-half OU precision
+P11 Hub+Rest            = covariance/state metric of the same discrete OU state
+Riemann-R background    = positive continuum mixture of the same Euler cells
+higher Gamma layer      = positive stable first-order filter bank
+```
+
+All of these statements are exact. None closes RH.
+
+The unresolved issue is exclusively the **relative global coupling** between discrete prime spacings `h=log p` and the smooth Riemann-R continuum `rho(h)dh`.
+
+---
+
+## 1. Canonical local cell
+
+For `h>0`, `q=e^{-h/2}`, define
+
+```math
+B_h
+=\sqrt{\frac{h q(1+q)}{1-q}}
+(T_h-I)(I-qT_h)^{-1}.
+```
+
+Then
+
+```math
+B_h^*B_h
+=\sum_{k\ge1}h e^{-kh/2}K_{kh}^*K_{kh}.
+```
+
+At `h=log p` this is the complete positive p-power Weil tower. The same `q` is the P11 AR(1) transition.
+
+The cell has a passive KYP realization, but the centered Weil contribution is not positive cellwise.
+
+---
+
+## 2. P11 covariance versus Weil precision
+
+For the sampled field
+
+```math
+X_{p,k}=\sqrt h\,e^{-kh/4}K_{kh}v
+```
+
+and P11 tail state
+
+```math
+Y_{p,m}=\sqrt{1-q^2}\sum_{k\ge m}q^{k-m}X_{p,k},
+```
+
+Hub+Rest is the covariance/state form
+
+```math
+\sum_m\|Y_{p,m}\|^2
++\frac{q^2}{1-q^2}\|Y_{p,1}\|^2,
+```
+
+while the positive p-power Weil tower is the precision form
+
+```math
+\frac1{1-q^2}\sum_m\|Y_{p,m}-qY_{p,m+1}\|^2.
+```
+
+Thus the earlier search for a contraction between P11 and COMMON-JUMP was the wrong category. The canonical bridge is inverse-covariance metric duality.
+
+---
+
+## 3. Continuous limit
+
+With
+
+```math
+X_v(t)=e^{-t/4}K_tv,
+\qquad
+Y(u)=\int_u^\infty e^{-(t-u)/2}X_v(t)dt,
+```
+
+one has
+
+```math
+(1/2-\partial_u)Y=X_v
+```
+
+and
+
+```math
+E_{Gamma,0}(v)
+=\int\|X_v(t)\|^2dt
+=\int\|Y'(u)-Y(u)/2\|^2du.
+```
+
+So Gamma ground is literally the continuum precision limit of the prime/P11 geometry.
+
+---
+
+## 4. Smooth reference uses the same cells
+
+The positive density
+
+```math
+\rho(h)=\frac{d}{dh}R(e^h)
+=\sum_{m\ge1}\frac{h^{m-1}}{m!\zeta(m+1)}
+```
+
+satisfies
+
+```math
+\int_0^\infty\rho(h)
+\sum_{k\ge1}h e^{-kh/2}\delta_{kh}(dt)dh
+=(e^{t/2}-e^{-t/2})dt.
+```
+
+Its continuous Euler product is
+
+```math
+Z_0(S)=\exp\int_0^\infty\rho(h)
+\log(1-e^{-Sh})^{-1}dh
+=\frac{S}{S-1}.
+```
+
+After `S=1/2+s`, the inverse is the lossless Critical-half allpass
+
+```math
+\frac{s-1/2}{s+1/2}.
+```
+
+The state realization
+
+```math
+\dot x=-x/2+u,
+\qquad y=u-x
+```
+
+satisfies
+
+```math
+\frac{y}{u}=\frac{s-1/2}{s+1/2},
+\qquad
+\frac{d}{dt}|x|^2=|u|^2-|y|^2.
+```
+
+Hence the smooth reference itself has a forward positive/lossless realization.
+
+---
+
+## 5. Prime-only exact form
+
+For the centered cell response
+
+```math
+C_v(h)=\|B_hv\|^2
+-2\frac{h e^{-h/2}}{1-e^{-h/2}}\|v\|^2,
+```
+
+NULLPOL gives
+
+```math
+Q_W(v)
+=\sum_p C_v(\log p)
+-\frac12\int_0^\infty\rho(h)C_v(h)dh
++E_{Gamma,>=1}(v)
+-(\kappa_*-4)\|v\|^2.
+```
+
+All prime powers are compressed into one cell per prime. The transform is invertible by ordinary Moebius inversion, so no autocorrelation data have been discarded.
+
+---
+
+## 6. What is already excluded
+
+```text
+orthogonal direct sum of local passive cells          insufficient
+ordinary passive cascade of normalized Euler cells    collapses to zero
+root-only cross-prime correction                       insufficient
+cellwise centered positivity                           false
+pointwise multiplier positivity                        false
+static Gamma frame repair                              false
+```
+
+The passive cascade collapse is elementary: for fixed real `s>0`,
+
+```math
+\prod_{p\le P}
+\frac{1-p^{-1/2}}{1-p^{-(s+1/2)}}\to0.
+```
+
+Thus a nontrivial infinite-prime transfer necessarily requires a **relative discrete/continuum renormalization**.
+
+---
+
+## 7. Literature firewall
+
+The scale operator
+
+```math
+(Ag)(h)=h\sum_{k\ge1}g(kh)
+```
+
+has Mellin symbol `zeta(s+1)`. This is in the classical Muentz/Nyman-Beurling family; no novelty is claimed for that Mellin-zeta mechanism.
+
+Similarly, Euler products in the critical strip and their relation to RH have a substantial literature. The project must not claim novelty for analytic continuation of Euler products or for positive-real formulations of `xi'/xi`.
+
+---
+
+## 8. Single remaining gate
+
+A substantive Object-X mechanism now requires a **non-diagonal relative storage kernel** coupling
+
+```text
+discrete scale cells h=log p
+          to
+continuous cells rho(h) dh
+          plus
+higher Gamma reservoir,
+```
+
+with all local metrics already fixed by the OU/KYP constructions.
+
+The coupling must be derived from the existing overlap-cone/stopped-OU incidence; no coefficient may be fitted from Weil positivity or from the zero set.
+
+A PASS would produce a forward positive storage realization of the known Lagarias transfer. A FAIL for the natural overlap-cone kernel class should close this branch as an Object-X architecture while preserving the exact covariance/precision identities as independent mathematics.
diff --git a/audits/P11_UNIVERSAL_STOPPED_OU_PRECISION_2026-09-15.md b/audits/P11_UNIVERSAL_STOPPED_OU_PRECISION_2026-09-15.md
new file mode 100644
index 00000000..dde58ed7
--- /dev/null
+++ b/audits/P11_UNIVERSAL_STOPPED_OU_PRECISION_2026-09-15.md
@@ -0,0 +1,420 @@
+# P11 Audit — Universal stopped OU kernel and critical-half precision duality
+
+**Datum:** 15. September 2026  
+**Branch:** `research/critical-half-green-tree-bridge-2026-09-13`  
+**Rolle:** theorem-level Verdichtung der gestoppten P11-Filtration.  
+**Registry:** unveraendert.  
+**Nonclaim:** kein Object-X-/Weil-/RH-Abschluss; Whitening allein ist kein Fortschritt.
+
+---
+
+## 0. Kurzurteil
+
+Die gestoppte P11-Kovarianz ist die exakte Gitterabtastung eines **universellen**, primzahlunabhaengigen Critical-half-OU-Kerns
+
+```math
+\boxed{
+G_L(s,t)
+=
+\exp\left(-\frac{s+t}{2}+\min(s,t,L)\right),
+\qquad s,t,L\ge0.
+}
+```
+
+Seine Entwicklung in der Stopptiefe `L` ist positiv Rang eins:
+
+```math
+\boxed{
+\partial_LG_L(s,t)
+=
+\mathbf1_{L<\min(s,t)}
+ e^{-(s-L)/2}e^{-(t-L)/2}.
+}
+```
+
+P11 entsteht exakt durch
+
+```math
+s=j\log p,
+\qquad
+t=k\log p,
+\qquad
+L=m\log p.
+```
+
+Raeumlich ist die Stopptiefe
+
+```math
+L=2(R-|x|).
+```
+
+Damit ist die P11-Fensterentwicklung eine diskrete Abtastung eines kontinuierlichen positiven Hamiltonianflusses.
+
+Zusaetzlich ist der inverse Kovarianz-/Praezisionsoperator des ungestoppten Kerns erneut
+
+```math
+-\partial_t^2+1/4.
+```
+
+Dieselbe Critical-half-Masse `1/2` wirkt daher in zwei Variablen:
+
+```text
+source variable x:  NULLPOL = Range(-d_x^2+1/4),
+channel depth t:    OU covariance precision = -d_t^2+1/4.
+```
+
+Der diskrete P11-Innovationsoperator ist die exponentielle Gitterversion des zugehoerigen first-order Faktors.
+
+Status:
+
+```text
+universal stopped kernel G_L                          ✓[M]
+rank-one positive L-flow                              ✓[M]
+P11 exact lattice sampling                            ✓[M]
+critical-half precision in channel variable           ✓[M]
+discrete-to-continuous first-order precision bridge   ✓[M]
+precision metric as Object X                          not claimed
+```
+
+---
+
+# 1. Feature realization of the stopped kernel
+
+Let `e_0` be a root vector and let `L^2(0,L)` be the innovation line. Define
+
+```math
+\boxed{
+\Phi_t^{[L]}
+=
+e^{-t/2}e_0
+\oplus
+\left[
+\mathbf1_{[0,\min(t,L)]}(u)e^{-(t-u)/2}
+\right].
+}
+```
+
+Then
+
+```math
+\begin{aligned}
+\langle\Phi_s^{[L]},\Phi_t^{[L]}\rangle
+&=e^{-(s+t)/2}
++\int_0^{\min(s,t,L)}e^{-(s-u)/2}e^{-(t-u)/2}du\\
+&=e^{-(s+t)/2+\min(s,t,L)}.
+\end{aligned}
+```
+
+Thus
+
+```math
+\boxed{
+\langle\Phi_s^{[L]},\Phi_t^{[L]}\rangle=G_L(s,t).
+}
+```
+
+For `L>=min(s,t)`,
+
+```math
+G_L(s,t)=e^{-|s-t|/2},
+```
+
+the full critical-half OU/Green kernel.
+
+For `s,t>L`,
+
+```math
+G_L(s,t)=e^{-(s-L)/2}e^{-(t-L)/2},
+```
+
+which is exactly the covariance of the conditional expectations of the future states given the stopped state at depth `L`.
+
+---
+
+# 2. Positive Hamiltonian flow in L
+
+From the feature formula,
+
+```math
+G_{L_2}-G_{L_1}
+=
+\int_{L_1}^{L_2}
+\left[e^{-(s-u)/2}\mathbf1_{u<s}\right]
+\left[e^{-(t-u)/2}\mathbf1_{u<t}\right]du
+```
+
+for `L_2>L_1`. Hence
+
+```math
+\boxed{
+G_{L_2}-G_{L_1}\succeq0.
+}
+```
+
+Differentiating away from the null boundary surfaces gives
+
+```math
+\boxed{
+\partial_LG_L
+=g_Lg_L^*,
+\qquad
+ g_L(s)=e^{-(s-L)/2}\mathbf1_{s>L}.
+}
+```
+
+Thus the continuous depth evolution is rank-one positive at every `L`.
+
+This is the continuous counterpart of the discrete P11 identity
+
+```math
+R_q^{(m+1)}-R_q^{(m)}=d_{m+1}d_{m+1}^*.
+```
+
+---
+
+# 3. Exact P11 sampling
+
+Fix `p`, put
+
+```math
+h=\log p,
+\qquad
+q=e^{-h/2}=p^{-1/2}.
+```
+
+At lattice nodes
+
+```math
+s=jh,
+\qquad
+t=kh,
+\qquad
+L=mh,
+```
+
+one gets
+
+```math
+\begin{aligned}
+G_{mh}(jh,kh)
+&=
+\exp\left(
+-\frac{(j+k)h}{2}+h\min(j,k,m)
+\right)\\
+&=q^{j+k-2\min(j,k,m)}\\
+&=R_q^{(m)}(j,k).
+\end{aligned}
+```
+
+Therefore
+
+```math
+\boxed{
+R_q^{(m)}(j,k)=G_{m\log p}(j\log p,k\log p).
+}
+```
+
+The finite-window P11 prime branch is not merely analogous to a stopped OU process: it is its exact logarithmic lattice sample.
+
+---
+
+# 4. Spatial time parameter
+
+From the P11 masks,
+
+```math
+m_p(x)=\left\lfloor\frac{2(R-|x|)}{\log p}\right\rfloor.
+```
+
+Hence the sampled continuous stopping depth is
+
+```math
+m_p(x)\log p
+\le
+2(R-|x|)
+<
+(m_p(x)+1)\log p.
+```
+
+Thus the universal continuous depth parameter is canonically
+
+```math
+\boxed{L_R(x)=2(R-|x|).}
+```
+
+The same inequality `t<L_R(x)` is precisely the two-sided half-translation overlap condition for
+
+```math
+K_t=T_{t/2}-T_{-t/2}.
+```
+
+This identifies the stopped OU time with the COMMON-JUMP overlap-cone coordinate.
+
+---
+
+# 5. Critical-half precision on the channel ray
+
+On the half-line, the full kernel
+
+```math
+G_\infty(s,t)=e^{-|s-t|/2}
+```
+
+is the Green kernel of
+
+```math
+\boxed{
+L_t=-\partial_t^2+\frac14
+}
+```
+
+with the natural root Robin condition
+
+```math
+u'(0)=\frac12u(0).
+```
+
+The corresponding positive precision/Cameron-Martin form is
+
+```math
+\boxed{
+\mathfrak p(u)
+=
+\int_0^\infty
+\left(|u'|^2+\frac14|u|^2\right)dt
++\frac12|u(0)|^2.
+}
+```
+
+For decaying `u`, this has either first-order representation
+
+```math
+\boxed{
+\mathfrak p(u)
+=
+|u(0)|^2+
+\int_0^\infty|u'(t)+\tfrac12u(t)|^2dt
+=
+\int_0^\infty|u'(t)-\tfrac12u(t)|^2dt.
+}
+```
+
+Thus the same operator `-d^2+1/4` that characterizes NULLPOL in the physical source variable is the inverse covariance of the P11/OU channel variable.
+
+---
+
+# 6. Discrete precision is the exponential-step analogue
+
+For a stationary AR(1) branch
+
+```math
+Z_k=qZ_{k-1}+\sqrt{1-q^2}\,\varepsilon_k
+```
+
+the normalized innovations are
+
+```math
+\boxed{
+\varepsilon_k
+=
+\frac{Z_k-qZ_{k-1}}{\sqrt{1-q^2}}.
+}
+```
+
+Accordingly, the discrete precision form is local nearest-neighbor:
+
+```math
+\boxed{
+|Z_0|^2
++
+\sum_{k\ge1}
+\frac{|Z_k-qZ_{k-1}|^2}{1-q^2}.
+}
+```
+
+Set `q=e^{-h/2}` and sample a smooth channel field `Z_k=u(kh)`. Then
+
+```math
+Z_k-qZ_{k-1}
+=h\left(u'(kh)+\frac12u(kh)\right)+O(h^2),
+```
+
+while
+
+```math
+1-q^2=1-e^{-h}=h+O(h^2).
+```
+
+Therefore
+
+```math
+\frac{|Z_k-qZ_{k-1}|^2}{1-q^2}
+=h\left|u'+\frac12u\right|^2+O(h^2),
+```
+
+which is the Riemann-sum discretization of the continuous precision form.
+
+This is the precise discrete/continuous critical-half bridge behind the P11 tail operator.
+
+---
+
+# 7. Important firewall: precision whitening is not by itself Object X
+
+Any strictly positive covariance can formally be whitened by its inverse square root. Therefore the statement
+
+```text
+P11 covariance -> precision -> diagonal norm
+```
+
+would be vacuous if used only as a change of metric.
+
+The present observation is potentially useful only because the precision is **local and canonical**:
+
+```text
+continuous:  -d_t^2+1/4 with a fixed Robin root condition;
+discrete:    nearest-neighbor AR(1) innovation difference;
+spatial:     stopping time L_R(x)=2(R-|x|).
+```
+
+A genuine Object-X step would have to show that these local precision and boundary terms reproduce the Weil/prime-discrepancy coupling with no fitted coefficients and survive the existing Prime-2 calibration witness.
+
+---
+
+# 8. Failed scalar-root repair
+
+A diagnostic on the existing Prime-2 witness shows that applying the local discrete precision to the **masked** P11 Rest output moves the mixed pairing much closer to the Weil target, but not onto it:
+
+```math
+\text{masked-precision pairing}
+=-\frac{7\sqrt2}{16}\log2,
+```
+
+whereas
+
+```math
+\text{Weil target}
+=-\frac{8\sqrt2}{16}\log2.
+```
+
+For that one witness, adding `1/4` of the Hub pairing would fill the gap exactly. This must **not** be promoted: on other source directions the precision defect is not proportional to the Hub.
+
+The abstract reason is already visible at fixed stopping depth `m`. The stopped inverse-tail reconstruction compresses the exterior channel tail to one geometric boundary amplitude, so the difference from the full diagonal tail contains an operator of rank larger than one. A scalar Root correction cannot repair it on all channel data.
+
+Thus the exact `1/4` witness calibration is recorded only as a coincidence/firewall, not as a candidate theorem.
+
+---
+
+# 9. Sharpened next gate
+
+The surviving nontrivial possibility is a genuinely local **boundary-tail system** on the overlap cone, not a scalar Hub patch:
+
+```math
+\boxed{
+\mathcal C_R
+=\{(x,t):0<t<2(R-|x|)\}.
+}
+```
+
+The universal stopped kernel provides the positive depth flow; the precision operator provides the local channel derivative; the missing exterior tail is a boundary state at `t=2(R-|x|)`.
+
+The next mathematically meaningful test is whether that boundary state can be coupled canonically to the pole-subtracted prime discrepancy / Gamma continuum so that the full Prime-2 calibration is recovered without fitting.
diff --git a/scripts/check_np_prolate_1_arb.py b/scripts/check_np_prolate_1_arb.py
new file mode 100644
index 00000000..b73ab985
--- /dev/null
+++ b/scripts/check_np_prolate_1_arb.py
@@ -0,0 +1,432 @@
+# -*- coding: utf-8 -*-
+"""Rigorous NP-PROLATE-1 certificate at a=1/2.
+
+Proves, separately in even/odd parity, a sufficient bound for positivity of the
+NULLPOL-compressed Weil form using the positive trace-class defect
+
+    D_c = P_N (c - tau)_+ P_N.
+
+The proof target is
+
+    lambda_max(D_c) <= T_full - tr(R_inner) + U < c,
+
+where
+  * T_full is an Arb upper bound for tr(D_c),
+  * R_inner is a finite Ritz compression of an operator D_inner <= D_c,
+  * U is certified by Cholesky from U G - A_inner > 0.
+
+No floating-point number is used in a proof decision. Decimal constants below
+are exact rational decimal endpoints/nodes. Root brackets are not assumed to
+contain unique roots: the complement is interval-verified to have the declared
+sign, while every bracket is booked entirely into an explicit trace error.
+
+Scope: a=1/2 only. No RH/Object-X/all-window claim.
+"""
+
+import sys
+from decimal import Decimal, getcontext
+from flint import ctx, arb, acb
+
+ctx.prec = 512
+getcontext().prec = 50
+
+PI = arb.pi()
+A = arb(1) / 2
+IUNIT = acb(0, 1)
+LOGPI = PI.log()
+LOG2 = arb(2).log()
+SQRT2 = arb(2).sqrt()
+PRIMEAMP = SQRT2 * LOG2
+LAM = acb(0, arb(1) / 2)
+
+ABS_TOL = arb(2) ** (-220)
+REL_TOL = arb(2) ** (-220)
+
+# Diagnostic roots from an independent high-precision search. Each is expanded
+# to a rigorous exact decimal bracket of half-width ROOT_EPS. The proof does
+# not trust the diagnostic root itself: verify_partition() proves the sign on
+# the entire complement of these brackets.
+ROOT_EPS = Decimal("0.0000001")
+TAIL_R = Decimal("128")
+
+DATA = {
+    "even": {
+        "sign": +1,
+        "c": "1.7",
+        "U": "1.693",
+        "roots": [
+            "13.2588803604",
+            "14.2510858535", "21.1449699109",
+            "24.4124889332", "29.6782656938",
+            "33.9745980005", "38.3639912138",
+            "43.4030948831", "47.1181745799",
+            "52.7742198106", "55.9064719698",
+            "62.1208362097", "64.7083657344",
+            "71.4665404010", "73.5052148605",
+            "80.8440711318", "82.2666480154",
+            "90.4044236771", "90.8429390474",
+        ],
+        "nodes": [
+            "7.372", "13.217", "19.532", "26.176", "33.988", "38.350",
+            "45.053", "52.784", "55.897", "63.842", "71.473", "73.499",
+            "82.262", "90.406",
+        ],
+    },
+    "odd": {
+        "sign": -1,
+        "c": "1.4",
+        "U": "1.27",
+        "roots": [
+            "12.5039896611",
+            "15.0442686334", "20.7035885648",
+            "24.8936386200", "29.2563104515",
+            "34.4436084540", "37.9224685383",
+            "43.9064611818", "46.6313259144",
+            "53.3542742933", "55.3376948147",
+            "62.8775698696", "63.9599471282",
+        ],
+        "nodes": [
+            "9.578", "16.368", "24.907", "29.243", "35.609", "43.915",
+            "46.623", "54.457", "62.881",
+        ],
+    },
+}
+
+
+def qarb(s):
+    """Exact decimal -> Arb point ball."""
+    return arb(str(s))
+
+
+def acb_conj(z):
+    return z.conjugate()
+
+
+def pw_kernel(z, w):
+    """Paley-Wiener kernel K_a(z,w), entire via sinc."""
+    z, w = acb(z), acb(w)
+    d = z - acb_conj(w)
+    return acb(A / PI) * (acb(A) * d).sinc()
+
+
+def pw_kernel_no_conj(z, eta):
+    """Analytic continuation in a real-variable slot eta.
+
+    For real eta this equals K_a(z, eta). Unlike pw_kernel(z, eta), eta is not
+    conjugated; this is useful inside acb.integral where eta is the analytic
+    integration variable.
+    """
+    z, eta = acb(z), acb(eta)
+    d = z - eta
+    return acb(A / PI) * (acb(A) * d).sinc()
+
+
+def parity_kernel(z, w, sgn):
+    return pw_kernel(z, w) + sgn * pw_kernel(z, -acb(w))
+
+
+def parity_kernel_lam_real(z, sgn):
+    """Analytic z-extension of K_parity(lambda,z) for real z."""
+    z = acb(z)
+    return pw_kernel_no_conj(LAM, z) + sgn * pw_kernel_no_conj(LAM, -z)
+
+
+def parity_denom(sgn):
+    d = parity_kernel(LAM, LAM, sgn).real
+    if not (d > 0):
+        raise RuntimeError(f"parity evaluation denominator not certified >0: {d}")
+    return d
+
+
+def np_kernel_section(z, x, sgn, den):
+    """Shorted parity kernel K_NP^sgn(z,x), x real fixed."""
+    z, x = acb(z), acb(x)
+    base = parity_kernel(z, x, sgn)
+    corr = parity_kernel(z, LAM, sgn) * parity_kernel(LAM, x, sgn) / acb(den)
+    return base - corr
+
+
+def np_kernel_diag_analytic(z, sgn, den):
+    """Analytic extension of K_NP^sgn(z,z) from the real axis."""
+    z = acb(z)
+    # K(z,z)=a/pi and K(z,-z)=a/pi*sinc(2az) for real z.
+    base = acb(A / PI) * (1 + sgn * (2 * acb(A) * z).sinc())
+    left = parity_kernel(z, LAM, sgn)
+    right = parity_kernel_lam_real(z, sgn)
+    return base - left * right / acb(den)
+
+
+def tau_analytic(z):
+    """Analytic real-axis extension of tau_{1/2}."""
+    z = acb(z)
+    zp = acb(arb(1) / 4) + IUNIT * z / 2
+    zm = acb(arb(1) / 4) - IUNIT * z / 2
+    psi_sym = (zp.digamma() + zm.digamma()) / 2
+    return psi_sym - acb(LOGPI) - acb(PRIMEAMP) * (acb(LOG2) * z).cos()
+
+
+def tau_real_ball(zball):
+    return tau_analytic(acb(zball)).real
+
+
+def dball(lo, hi):
+    """Exact Decimal interval [lo,hi] as an Arb ball."""
+    mid = (lo + hi) / 2
+    rad = (hi - lo) / 2
+    return arb(f"{mid} +/- {rad}")
+
+
+def root_brackets(root_strings):
+    out = []
+    for rs in root_strings:
+        r = Decimal(rs)
+        out.append((r - ROOT_EPS, r + ROOT_EPS))
+    return out
+
+
+def declared_segments(brackets):
+    """Complement segments on [0,TAIL_R], with alternating below/above sign.
+
+    tau(0)<c, so the first segment is below. Brackets themselves are omitted.
+    """
+    segs = []
+    cur = Decimal(0)
+    below = True
+    for lo, hi in brackets:
+        if cur < lo:
+            segs.append((cur, lo, below))
+        cur = hi
+        below = not below
+    if cur < TAIL_R:
+        segs.append((cur, TAIL_R, below))
+    return segs
+
+
+def verify_sign_recursive(lo, hi, c, below, depth=0):
+    """Prove tau<c (below=True) or tau>c on exact decimal interval.
+
+    Adaptive interval subdivision is used purely for proof enclosure sharpness.
+    """
+    z = dball(lo, hi)
+    tv = tau_real_ball(z)
+    gap = c - tv if below else tv - c
+    if gap > 0:
+        return 1
+    if depth >= 36:
+        raise RuntimeError(
+            f"sign verification unresolved at [{lo},{hi}], below={below}, gap={gap}"
+        )
+    mid = (lo + hi) / 2
+    return (
+        verify_sign_recursive(lo, mid, c, below, depth + 1)
+        + verify_sign_recursive(mid, hi, c, below, depth + 1)
+    )
+
+
+def verify_partition(brackets, c):
+    leaves = 0
+    for lo, hi, below in declared_segments(brackets):
+        leaves += verify_sign_recursive(lo, hi, c, below)
+
+    # Tail firewall. Re psi(1/4+i y) is increasing in |y| termwise in its
+    # convergent real series. The prime cosine term is bounded below by
+    # -sqrt(2)log2. Thus tau(z)>=this constant for z>=TAIL_R.
+    R = qarb(TAIL_R)
+    psiR = acb(arb(1) / 4, R / 2).digamma().real
+    tail_lb = psiR - LOGPI - PRIMEAMP
+    if not (tail_lb > c):
+        raise RuntimeError(f"tail lower bound not > c: {tail_lb} <= {c}")
+    return leaves, tail_lb
+
+
+def below_inner_intervals(brackets):
+    return [(lo, hi) for lo, hi, below in declared_segments(brackets) if below]
+
+
+def cint(func, lo, hi):
+    if not lo < hi:
+        return arb(0)
+    val = acb.integral(
+        lambda z, analytic: func(z),
+        acb(qarb(lo)), acb(qarb(hi)),
+        abs_tol=ABS_TOL, rel_tol=REL_TOL,
+        eval_limit=10**7, depth_limit=500,
+    )
+    # The integrals are real by construction. Any residual imaginary enclosure
+    # is harmless; only the real enclosure is used.
+    return val.real
+
+
+def weight(z, c):
+    return acb(c) - tau_analytic(z)
+
+
+def trace_inner(intervals, c, sgn, den):
+    total = arb(0)
+    for lo, hi in intervals:
+        total += cint(
+            lambda z: weight(z, c) * np_kernel_diag_analytic(z, sgn, den),
+            lo, hi,
+        )
+    return total
+
+
+def root_edge_trace_upper(brackets, c):
+    """Upper bound all omitted root-bracket contributions.
+
+    For a=1/2, each positive-half parity kernel before shorting has diagonal
+    <= 2a/pi = 1/pi, and orthogonal shorting only decreases the diagonal.
+    """
+    total = arb(0)
+    for lo, hi in brackets:
+        z = dball(lo, hi)
+        w = c - tau_real_ball(z)
+        wu = w.upper()
+        if wu > 0:
+            total += qarb(hi - lo) * wu / PI
+    return total
+
+
+def build_gram(nodes, sgn, den):
+    xs = [qarb(Decimal(x)) for x in nodes]
+    n = len(xs)
+    G = [[arb(0) for _ in range(n)] for __ in range(n)]
+    for i in range(n):
+        for j in range(i, n):
+            v = np_kernel_section(acb(xs[i]), acb(xs[j]), sgn, den).real
+            G[i][j] = G[j][i] = v
+    return xs, G
+
+
+def build_A(xs, intervals, c, sgn, den):
+    n = len(xs)
+    M = [[arb(0) for _ in range(n)] for __ in range(n)]
+    for i in range(n):
+        for j in range(i, n):
+            total = arb(0)
+            xi, xj = xs[i], xs[j]
+            for lo, hi in intervals:
+                total += cint(
+                    lambda z, xi=xi, xj=xj: (
+                        weight(z, c)
+                        * np_kernel_section(z, acb(xi), sgn, den)
+                        * np_kernel_section(z, acb(xj), sgn, den)
+                    ),
+                    lo, hi,
+                )
+            M[i][j] = M[j][i] = total
+    return M
+
+
+def cholesky_certified(M):
+    n = len(M)
+    L = [[arb(0) for _ in range(n)] for __ in range(n)]
+    pivots = []
+    for i in range(n):
+        for j in range(i + 1):
+            s = M[i][j]
+            for k in range(j):
+                s -= L[i][k] * L[j][k]
+            if i == j:
+                if not (s > 0):
+                    return False, pivots, i, L
+                pivots.append(s)
+                L[i][i] = s.sqrt()
+            else:
+                L[i][j] = s / L[j][j]
+    return True, pivots, None, L
+
+
+def solve_spd_from_cholesky(L, B):
+    """Solve (L L^T) X = B rigorously with Arb list matrices."""
+    n = len(L)
+    m = len(B[0])
+    Y = [[arb(0) for _ in range(m)] for __ in range(n)]
+    for j in range(m):
+        for i in range(n):
+            s = B[i][j]
+            for k in range(i):
+                s -= L[i][k] * Y[k][j]
+            Y[i][j] = s / L[i][i]
+    X = [[arb(0) for _ in range(m)] for __ in range(n)]
+    for j in range(m):
+        for ii in range(n):
+            i = n - 1 - ii
+            s = Y[i][j]
+            for k in range(i + 1, n):
+                s -= L[k][i] * X[k][j]
+            X[i][j] = s / L[i][i]
+    return X
+
+
+def matrix_sub_scaled(G, Aop, U):
+    n = len(G)
+    return [[U * G[i][j] - Aop[i][j] for j in range(n)] for i in range(n)]
+
+
+def trace_matrix(M):
+    return sum((M[i][i] for i in range(len(M))), arb(0))
+
+
+def run_sector(name):
+    d = DATA[name]
+    sgn = d["sign"]
+    c = qarb(d["c"])
+    U = qarb(d["U"])
+    brackets = root_brackets(d["roots"])
+    intervals = below_inner_intervals(brackets)
+    den = parity_denom(sgn)
+
+    print(f"=== NP-PROLATE-1 {name.upper()} ===")
+    print(f"prec={ctx.prec}  a=1/2  c={c.str(20)}  U={U.str(20)}")
+    print(f"root brackets={len(brackets)}  Ritz dim={len(d['nodes'])}")
+
+    leaves, tail_lb = verify_partition(brackets, c)
+    print(f"SIGN PARTITION ZERTIFIZIERT: leaves={leaves}, tail_lb={tail_lb.str(30)}")
+
+    Tin = trace_inner(intervals, c, sgn, den)
+    Eroot = root_edge_trace_upper(brackets, c)
+    Tfull = Tin + Eroot
+    print(f"trace_inner = {Tin.str(40)}")
+    print(f"root_edge_upper = {Eroot.str(40)}")
+    print(f"T_full_upper ball = {Tfull.str(40)}")
+
+    xs, G = build_gram(d["nodes"], sgn, den)
+    okG, pivG, failG, LG = cholesky_certified(G)
+    if not okG:
+        raise RuntimeError(f"Ritz Gram Cholesky FAIL at pivot {failG}")
+    print(f"GRAM > 0 ZERTIFIZIERT; min pivot ball={min(pivG, key=lambda x: x.lower()).str(30)}")
+
+    Aop = build_A(xs, intervals, c, sgn, den)
+
+    # Certify theta_max(R_inner) < U from U G - A > 0.
+    H = matrix_sub_scaled(G, Aop, U)
+    okH, pivH, failH, _ = cholesky_certified(H)
+    if not okH:
+        raise RuntimeError(f"U G - A Cholesky FAIL at pivot {failH}")
+    print(f"RITZ TOP < U ZERTIFIZIERT; min pivot ball={min(pivH, key=lambda x: x.lower()).str(30)}")
+
+    # Rigorous Ritz trace tr(G^{-1} A).
+    X = solve_spd_from_cholesky(LG, Aop)
+    trR = trace_matrix(X)
+    print(f"Ritz trace ball = {trR.str(40)}")
+
+    cert = Tfull - trR + U
+    gap = c - cert
+    print(f"lambda_max(D) <= T-trR+U = {cert.str(40)}")
+    print(f"CERTIFICATE GAP c-(T-trR+U) = {gap.str(40)}")
+    if not (gap > 0):
+        raise RuntimeError(f"NP-PROLATE-1 {name}: certificate gap not rigorously positive")
+
+    print(f"NP-PROLATE-1 {name.upper()} PASS ZERTIFIZIERT")
+    return gap
+
+
+if __name__ == "__main__":
+    which = sys.argv[1].lower() if len(sys.argv) > 1 else "all"
+    if which not in ("even", "odd", "all"):
+        raise SystemExit("usage: check_np_prolate_1_arb.py [even|odd|all]")
+    sectors = ["even", "odd"] if which == "all" else [which]
+    for s in sectors:
+        run_sector(s)
+    print("ALL REQUESTED NP-PROLATE-1 SECTORS PASS ZERTIFIZIERT")