R43: full-rest local resolvent decay and exact frozen-U COND limit - #89
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Leitfaden für ReviewerFührt einen neuen, ausschließlich beweisorientierten analytischen Kandidaten ein, der den Abfall O_r(V^-1) des vollständigen Residuenresolventen im festen Fenster sowie den exakten COND-Grenzwert bei festem U herleitet, mit expliziten Argumenten zu Restgliedern, Bandzählung und Variationsprinzipien; außerdem wird ein qualitativer Strong-Collapse-Gegencheck bereitgestellt, während der simultane globale Grenzwert und damit verbundene zentrale Behauptungen weiterhin offen bleiben. Sequenzdiagramm für den COND-Grenzwert bei festem UsequenceDiagram
participant g as Old graph input
participant K as K_X,U
participant B as Full ambient resolvent
participant G as G_X,cond^(U,V)
participant W as W_cond(U,V)
g->>K: K_X,U g
K->>B: Apply Bhat_V
B-->>G: K_X,U^* Bhat_V K_X,U
G-->>W: Form normalized COND transfer
B-->>G: Fixed-U correction <= eta(U,V) K_X,U^* K_X,U
G-->>W: W_cond(U,V) -> W_Gamma in norm
Zustandsdiagramm zum Geltungsbereich des neuen ErgebnissesstateDiagram-v2
[*] --> Candidate
Candidate --> FixedWindowDecay: RL1/RL17 established
Candidate --> FixedUCond: RL22-RL27 established
Candidate --> StrongCollapse: RL20/RL31 crosscheck
FixedWindowDecay --> GlobalGap: r=U, U<V<=2U remains uncontrolled
FixedUCond --> GlobalGap: simultaneous U,V limit not implied
StrongCollapse --> GlobalGap: no global flag or terminal conclusion
GlobalGap --> [*]
Flussdiagramm für den Abfall des Residuenresolventen im festen Fensterflowchart LR
A[Full residual rows] --> B[Coherent higher-power tails]
B --> C["Uniform tail bound < 1/2"]
C --> D[Disjoint log-prime bands]
D --> E[All-row energy inequality]
E --> F[Resolvent variational identity]
F --> G["Compressed resolvent norm <= eta(r,V)"]
G --> H["O_r(V^-1) decay"]
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Original review guide in EnglishReviewer's GuideIntroduces a new proof-only analytical candidate establishing O_r(V^-1) decay of the complete fixed-window residual resolvent and the exact fixed-U COND limit, with explicit tail, band-counting, and variational arguments; it also supplies a qualitative strong-collapse crosscheck while leaving the simultaneous global limit and related flagship claims open. Sequence diagram for the fixed-U COND limitsequenceDiagram
participant g as Old graph input
participant K as K_X,U
participant B as Full ambient resolvent
participant G as G_X,cond^(U,V)
participant W as W_cond(U,V)
g->>K: K_X,U g
K->>B: Apply Bhat_V
B-->>G: K_X,U^* Bhat_V K_X,U
G-->>W: Form normalized COND transfer
B-->>G: Fixed-U correction <= eta(U,V) K_X,U^* K_X,U
G-->>W: W_cond(U,V) -> W_Gamma in norm
State diagram for the scope of the new resultstateDiagram-v2
[*] --> Candidate
Candidate --> FixedWindowDecay: RL1/RL17 established
Candidate --> FixedUCond: RL22-RL27 established
Candidate --> StrongCollapse: RL20/RL31 crosscheck
FixedWindowDecay --> GlobalGap: r=U, U<V<=2U remains uncontrolled
FixedUCond --> GlobalGap: simultaneous U,V limit not implied
StrongCollapse --> GlobalGap: no global flag or terminal conclusion
GlobalGap --> [*]
Flow diagram for fixed-window residual resolvent decayflowchart LR
A[Full residual rows] --> B[Coherent higher-power tails]
B --> C["Uniform tail bound < 1/2"]
C --> D[Disjoint log-prime bands]
D --> E[All-row energy inequality]
E --> F[Resolvent variational identity]
F --> G["Compressed resolvent norm <= eta(r,V)"]
G --> H["O_r(V^-1) decay"]
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Exact scope and provenance
One new analytical proof document on exact main
55a3a1617513cc5d82c47d0cfd606c6b0894c984.New head:
112f611727dbd3386bff5e11b9a5d5d143cdee6b.Proof blob:
3ff0d0d696c9dc455a6fefec14f7856b7ded5f3e, equal to the locally written proof's Git hash.File:
audits/P11_R43_FULL_REST_LOCAL_RESOLVENT_DECAY_AND_COND_LIMIT_2026-09-07.md(642 lines).No existing files, registry, workflows or main changed. No merge. The integrated LOCAL-O1, conditional orientation and fixed-U COND results are retained as inputs, not reopened.
Status: new analytical candidate; author crosscheck only. Independent exact-head mathematical review OPEN. Neither earlier independent reviews nor the externally reported local patch
528153ec...are attributed to this new proof.New quantitative statement on the actual full residual
For every fixed old spatial window r>0 and the complete P11 residual resolvent,
This is operator norm on the entire fixed L2 window, not merely compactly sandwiched graph data, a finite-dimensional proxy, or a hand-picked source.
A universal threshold a_* exists by the ordinary unconditional prime number theorem. With
the proof obtains
||E* B_V E|| <= eta(r,V). ForV>=2(a_*+3r+2), this is at most16(r+1)/V. No numerical value of a_* or effective PNT error constant is claimed.Main proof mechanism and full-rest firewall
Use the full orthogonal martingale rows
(p,0), retaining the coherent higher-prime-power sum inside every row. Its tail satisfiesuniformly in V. This is a genuine tail-error bound, not the forbidden inequality
R*R >= primitive*primitive.Select disjoint log-prime bands of width 1 with graph mass at least 1. Normalize test weights within each band; the actual operator is not renormalized. Their translated old-window observation regions lie in disjoint spatial hulls. Actual half-shift preimages and the martingale masks are checked explicitly. For arbitrary ambient x, including non-old-supported resolvent outputs, derive
Insert the exact resolvent energy identity with
x=Bhat_V E_r g. This proves the compression norm bound. The middle operator remains the true full residual, including every other martingale row.Strong collapse, exact COND endpoint, canonical source
The ambient resolvent
Bhat_V=E_V B_V P_V+(I-M_V)tends strongly to zero. It does NOT converge to zero in operator norm: its norm is exactly 1 for every finite V due to the outside identity.For fixed old U, the existing exact sandwich now gives an actual norm limit and rate:
The genuine inverse-square-root reverse source converges to
f_Gamma=Gamma_R^(-1/2)epsilon_R, andNo positivity away from zero, growth or decay of this final expression as U grows is asserted. Small transported vectors alone do not control the varying Schur form.
A further corollary treats the truly moving canonical normalization: any fixed old hub component tends to zero even in the full normalized Schur energy. Its complementary moving hub/geometry contribution is retained, not relabeled as the pure NEW channel.
Separate qualitative crosscheck
Section 9 reproves strong collapse without the prime number theorem. Fixed full
(p,0)rows converge strongly-*; their primitive energies diverge on every nonzero fixed L2 vector by translation decorrelation and divergence of prime reciprocals. The complete higher-power tails are square-summable. Weak compactness plus the resolvent energy identity forces the strong zero limit. This is a different author derivation, not an independent external review.Exact remaining global gap
This does not close BR39 or the global actual flag budget. For
r=UandU<V<=2U, the construction givesM(U,V)=0; it provides no dyadic smallness. The fixed-U outside factor grows with U. Do not substitute source radius R for the actual old support U ofH_U^*E_R f.No global B-FLAGTIGHT, C6, Object-X or RH result. The already conditional orientation implication stays available; the true quantitative no-escape question remains open. This result determines the frozen-old/far-future endpoint, not the simultaneous terminal limit.
Review request and technical checks
Please check RL3 against full P11 (3.4)/(3.5), RL6/RL7 tail constants, the all-row energy bound with test weights, exact mask activation and translated hull disjointness, ordinary-PNT band counting, the resolvent variational step, actual graph adjoints and inverse-square-root source, and all fixed-U versus simultaneous-limit quantifiers.
Technical checks only: UTF-8/no illegal control bytes; paired Markdown fences; RL1–RL31 tags; exact rational tail inequality; locally computed proof blob matches GitHub. These are not mathematical certificates. No CI result is asserted for this proof-only PR.
Keep Draft. No merge or registry promotion without explicit authorization and fresh exact-head mathematical review.
Zusammenfassung von Sourcery
Dokumentiert den lokalen Resolventenzerfall des vollständigen Rests sowie den exakten COND-Endpunkt für festes (U), während das offene Problem des globalen wachsenden Fensters erhalten bleibt.
Neue Funktionen:
Verbesserungen:
Dokumentation:
Aufgaben:
Original summary in English
Summary by Sourcery
Document full-rest local resolvent decay and the exact frozen-U COND endpoint while preserving the open global growing-window problem.
New Features:
Enhancements:
Documentation:
Chores: