Independent written audit of coupled far-endpoint rank leakage (no κ)

research/p104-far-endpoint-leakage-review/ADVERSARIAL-REVIEW.md

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+# Adversarial static review of the far-endpoint leakage audit
+
+**Verdict: PASS for the stated conditional section6 audit; no unresolved
+mathematical finding.** One scope qualification was raised and incorporated
+before this final review: the characteristic-two control concerns the
+displayed matrix hypotheses after deletion of the field restriction, not
+a separately imposed genuine shortest-word condition.
+
+This review checks the algebra directly. It does not treat agreement with
+another agent as proof. I read the coordinator derivation and the three
+frozen source texts, but did not read the separate MATH-REVIEW, execute or
+import mathematical code, run a proof kernel or contact the network.
+Only this review file is authored by this reviewer.
+
+## Exact objects reviewed
+
+The source is thu-nmrc PR6717, current head
+`3c9072b8738323dda9f9c7dfb421d578ed3fc17a`, particularly PROOF section6.
+
+| Object | SHA256 |
+| --- | --- |
+| Coordinator derivation, after characteristic-two qualification | `22e5e1fa64b2aba2a5d8d8df4f5f06c8f6313493778f33ac07f3a8d4222c6d6c` |
+| 6717-PROOF.md | `9e108fdf827ffe92215be5e782f763a18968ce7613b66dfdd25a38f94958ce18` |
+| 6717-README.md | `02fc8aa47e0f3b9ff042caad5f9049325f6b32dc4fff9d816136037b0402e188` |
+| 6717-TASK.json | `67bce2c4caa1ff3fce659c5f7033ad1fad08c42f786efbfc749c6a3cedc20ca0` |
+
+The original coordinator draft is separately retained as
+`COORDINATOR-DERIVATION-BEFORE-CHAR2-QUALIFICATION.md`. The correction is
+scope wording, not a changed rank calculation or new computational result.
+
+## Kernel lemma and ordered leakage identities
+
+For an idempotent Q, `ker(I-Q)=im(Q)`. Therefore the retained column
+subspace A0 has dimension `a-alpha`. Likewise `c=cQ` characterizes the
+row space of Q, so the retained row subspace C0 has dimension `c-beta`.
+The restriction of C0 to im(Q) is injective: vanishing on that image gives
+`cQ=0`, and `c=cQ` then gives c=0. Consequently the map from im(Q) into
+C0* has rank dim(C0). Its kernel contains A0 because every original row
+in C annihilates every original column in A. Rank-nullity gives exactly
+
+`rank(Q) >= a+c-alpha-beta`.
+
+This uses actual kernels, so there is no unjustified simultaneous choice
+of separately retained columns and rows. It works over arbitrary fields
+and needs neither positive inner products nor simultaneous diagonalization.
+
+The leakage derivation also preserves matrix order. From `QU=U` and
+`PQ=QP`, `PU=PQU=QPU`, hence `(I-Q)PU=0`. From `VQ=V`,
+`VP=VQP=VPQ`, hence `VP(I-Q)=0`. Substituting P=E+F yields
+
+`(I-Q)EU=-(I-Q)FU`, `VE(I-Q)=-VF(I-Q)`.
+
+The two right sides each have rank at most rank(F)=e. Since the map
+already vanishes on U, its image on span(U,EU) is precisely its image
+on EU; there is no second q-dimensional leakage contribution. The
+same argument applies to the row closure. Thus alpha,beta<=e is valid
+without assuming any internal block commutation or e<=q.
+
+The decomposition rank(F)=rank(P)-2q remains an explicitly inherited
+forced-projector premise. Given that premise, this argument makes no
+additional rank-minimality assumption on P or Q.
+
+## Endpoint formulas, minors and all four pairings
+
+For the left endpoint, `v0=-tail0`. For odd i>=3,
+`E_i e0=e_i+a_i` and `v0 E_i=-(e_i^T+tail_i)`. These follow directly
+from `h_i e0=1`, `v_i e0=1`, `v0 e_i=-1` and `v0 a_i=1`.
+The column minor on coordinates0,1 is
+`[[1,1],[0,-1]]`, determinant -1. The row minor on coordinates1,i is
+`[[-1,-1],[0,-1]]`, determinant1. Both are units over every field.
+
+The four pairings vanish separately: `v0 e0=0`,
+`v0(e_i+a_i)=-1+1=0`, `(e_i^T+tail_i)e0=0`, and
+`(e_i^T+tail_i)(e_i+a_i)=1-1=0`. The last row's overall minus sign
+does not change zero pairing.
+
+For the right endpoint, `u_n=(-1,1,...,-1,1)^T` and `v_n=h`.
+For odd i<=n-3, `h_i u_n=0`, `v_i u_n=-1` and `h a_i=0`,
+so `E_i u_n=-a_i` and `h E_i=h_i`. The column minor of
+`[u_n,-a_i]` on coordinates0,i+1 is
+`[[-1,-1],[-1,0]]`, determinant -1. The row minor of `[h;h_i]`
+on those coordinates is `[[1,1],[1,0]]`, also determinant -1.
+
+The four right pairings are `h u_n=0`, `h a_i=0`,
+`h_i u_n=0`, `h_i a_i=0`: every relevant alternating sum has even
+length. The dimensions and annihilation claims therefore hold with
+no characteristic-zero counting assumption. Tensoring these fixed scalar
+closure matrices with I_q gives dimensions2q and2q and zero pairing.
+This does not tensor a scalar classification of arbitrary P or Q.
+
+A far endpoint exists at every allowed boundary. Odd i>=3 permits the
+left endpoint; the remaining possibility i=1 permits the right because
+n>=4 implies i<=n-3. In particular n=4,i=1 uses the right endpoint,
+while n=4,i=3 and i=n-1 use the left.
+
+## Rank debt and written countercontrols
+
+Applying the kernel lemma gives `rank(Q)>=4q-2e`. Independently,
+compatibility with a rank-q square-zero query gives rank(Q)>=2q:
+its rank-q image and rank-q supported row space have zero pairing.
+Writing rank(Q)=2q+f therefore gives `f>=max(0,2q-2e)`.
+For e>=q, e+f>=q follows from f>=0. For 0<=e<=q,
+e+f>=2q-e>=q. The remaining endpoint contributes another2q.
+The sole-odd selected triple consequently has total rank at least7q.
+Large e creates no sign or negative-dimension gap.
+
+All three written controls are valid at their stated scopes:
+
+1. At n=4,i=1, the left near endpoint satisfies `E_i e0=e0`
+   and `v0 E_i=v0`. Its two closures each have dimension1, so using
+   the left endpoint's supposed dimension2 here would be false.
+2. Without CA=0, Q=I on K^2, A=K^2 and C=(K^2)* give
+   alpha=beta=0 but rank(Q)=2<4=a+c. Independent row/column
+   dimensions cannot simply be added without annihilation.
+3. In characteristic two, five copies of `e0 e1^T` have rank1,
+   square zero and factors `I-2R_j=I=-I`; three copies of
+   `diag(1,1,0,0)` at distinct sites satisfy the displayed compatibility,
+   prefix and pairwise commutation conditions but have total rank6<7.
+   However, this five-query word is not genuinely shortest there:
+   the endpoint I has a zero-query realization. The amended coordinator
+   text correctly limits this control to deleting the field restriction
+   from the displayed matrix hypotheses. It is not a counterexample
+   to the field-independent leakage lemma or to a different theorem
+   imposing genuine minimality separately.
+
+## Scope, credit and remaining acceptance
+
+The coordinator accurately treats the complete shortest-query normal
+form, canonical prefix formula, forced odd projector and compatibility
+floor as inherited premises. This review independently checks the kernel
+argument, ordered leakage application, endpoint closures and sole-odd
+rank conclusion conditional on those premises. It does not certify the
+source's even-interior branch, two-odd branch, normal-form derivation,
+all-n attaining word, physical Gaussian/native implementation or invoice.
+
+The source's rank threshold and application remain thu-nmrc's work with
+substantial OpenAI Codex assistance. Ali Romman's PR6398 and retained
+alternating-caller-closure tools, complete coupled normal form, Cinna
+Davis's query/controller lineage and all earlier notices retain credit.
+The review is a classical linear-algebra audit, not invention of a new
+general rank lemma. This separate AI review gives no independent human
+acceptance, proof-kernel certification, κ improvement, selected-frontier
+change or discharge of the five framework assumptions.
+
+No further correction is requested for the reviewed final derivation.

research/p104-far-endpoint-leakage-review/AUDIT-RECEIPT.json

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+{
+  "schema_version": 1,
+  "status": "CONDITIONAL_WRITTEN_REVIEW_NO_GAP_FOUND",
+  "scope": "Source6717 PROOFsection6 only; inherited canonicalnormalform/prefix/forcedprojector/floor, otherbranches/attaineroutside",
+  "source_head": "3c9072b8738323dda9f9c7dfb421d578ed3fc17a",
+  "proof_review_sha256": "22e5e1fa64b2aba2a5d8d8df4f5f06c8f6313493778f33ac07f3a8d4222c6d6c",
+  "adversarial_review_sha256": "4c144fea94d0b51ffe3e371b092f450b621b80af17490c6c316bc20452394cc1",
+  "separate_source_aware_written_review_sha256": "c9d7d0e7c1bd239d84567472d729a112a6fcb5193f0a83ca5f4df83ef86401f9",
+  "scientific_commands": [],
+  "source_or_new_program_execution": false,
+  "proof_kernel": false,
+  "finite_grid_claim": false,
+  "written_scope_controls": [
+    "nearendpointclosurecollapses",
+    "zeropairingnecessary",
+    "char2displayedmatrixhypothesesextensionfails; notgenuinelyshortestthere"
+  ],
+  "pre_freeze_correction": "char2minimalityqualification; earlierlocaldraftretained",
+  "source_author_REVIEW_seen_by_scout": true,
+  "no_blind_or_human_acceptance": true,
+  "no_new_kappa": true,
+  "all_five_selected_assumptions_retained": true,
+  "frozen_at": "2026-11-04T20:30:05.241607+00:00"
+}

research/p104-far-endpoint-leakage-review/NOTICE

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+New external review: ToikY with substantial OpenAI Codex assistance. Apache-2.0.
+Original uniform mixed-rank7q theorem, leakage application and attaining family: thu-nmrc PR6717 with substantial OpenAI Codex assistance.
+Retained scalar obstruction/forced projector/closure/image-row methods and all-length alternating-caller tools: Ali Romman PR6398 and original contributors. Complete coupled normalform and Cinna Davis query/controller lineage retain their credit.
+The source inputs are exact unchanged copies with their original attribution and notices; SOURCE-PINS.json binds them to source6717 head3c9072b8738323dda9f9c7dfb421d578ed3fc17a.
+Kernel/annihilator/rank-nullity methods are classical. Separate AI review is not independent human or proof-kernel acceptance.

research/p104-far-endpoint-leakage-review/README.md

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+# Independent review of the far-endpoint leakage argument
+
+**No gap found in PR6717 PROOF §6 under its enumerated canonical-source premises.** This external written audit answers the source README's independent-review request. It verifies only the sole-odd-interior-site plus both-endpoints case of thu-nmrc's uniform coupled stage-rank theorem at exact head `3c9072b8738323dda9f9c7dfb421d578ed3fc17a`. The theorem/construction remain predecessor credit. No new kappa or selected-frontier change.
+
+[REVIEW.md](REVIEW.md) supplies a direct kernel/rank-nullity proof of the zero-paired leakage inequality, including injectivity of row restriction to im(Q). Applying the two actual ordered leakage equations gives rank(Q)>=4q-2e for arbitrary coupled P,Q, without tensoring the unknown projectors or assuming rank minimality. Unit minors and all four pairings verify the far closures for every even n>=4, q>=1 over the allowed fields. The n4/i1 and n4/i3 endpoint choices are explicit; e+f>=q covers every possible extra rank.
+
+Three **written, unexecuted** scope controls explain why the near endpoint, omitted zero pairing and a characteristic-two extension of the displayed source matrix hypotheses cannot be silently accepted. The char2 control is not a genuinely shortest word in that characteristic, so it does not refute a different statement with an additional genuine-minimality hypothesis or the abstract leakage lemma.
+
+The complete coupled normalform, canonical prefix and forced odd-projector facts remain inherited premises. The even-interior and two-odd branches, all-n attaining word, executable receipts and physical/native/price/owned-return/stock/precision/tape/compiler/recursive-outer applications are outside this review. All five selected framework assumptions remain. A stage-dimension inventory is not a complete supplier or paid-call lower bound.
+
+## Evidence and provenance
+
+Exact source PROOF/README/TASK copies and Git/SHA256 bindings are included. The coordinator's derivation was followed by a separate adversarial AI algebra check. A separately prepared AI written review corroborated the kernels/minors; that reviewer had read the author's REVIEW during scouting, so this is source-aware external review, not a blind or human review. The adversarial check corrected the char2 minimality qualification before the proof was frozen; the earlier local draft is retained in session receipts.
+
+No scientific code, original or new checker, matrix enumeration, numerical solver, compiler or proof kernel ran. There is no VERIFY.json and no registered scientific replay. File integrity, Git byte/mode equality and an inert manual archive/readback are provenance checks only. Repository CI, if present, is likewise not a proof replay. Read [SOURCE-PINS.json](SOURCE-PINS.json), the unchanged [source proof](inputs/6717-PROOF.md), [REVIEW.md](REVIEW.md) and [ADVERSARIAL-REVIEW.md](ADVERSARIAL-REVIEW.md) to inspect the result.
+
+Canonical main `7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision8 and authoritative selected6030 were refreshed; the priority prose still names an older record. TASK.json retains the picker card and declares the change from formal-interface-lemma to a narrower written source-bound proof review. Main/overlap/source status are refreshed again before submission. Neither absence of a matching body nor a DONE label establishes novelty or admission.
+
+## Open next questions
+
+Independently audit the normalform and prefix-to-forced-projector binding, then the other two rank-bound branches and all-n sharp attaining word at this same head. This matters because a valid §6 does not establish the complete theorem. Start with PROOF §§2–5 and7 and the retained original sources, after a new overlap checkpoint and budget. That work is unreserved. Physical supplier and complete invoice applications need their own exact caller/cost contracts; their unknown costs are not zero.
+
+## Credit
+
+thu-nmrc PR6717 supplies the uniform mixed-rank theorem, far-endpoint application and attaining family with substantial OpenAI Codex assistance. Ali Romman PR6398/alternating-caller-closure, complete coupled normalform authors, Cinna Davis's query/controller lineage and all retained contributors remain credited. The rank/kernel/annihilator tools are classical. External review: ToikY with substantial OpenAI Codex and separate AI review roles, Apache-2.0. This adds review evidence only; no human or proof-kernel acceptance.