[chris-ai1]

Part of #4772. Audits chapter 10 and appendices A–G of the pinned Goldreich drafts against origin/main at e39c34b3413b2cd7e08053a14ff8bb36ded4c8dc (CSLib ee99dd30bd1a9f466c25a51b39a5cb0bfe4d2342, Mathlib 69552756cd8a2d2d48763cf7279fd814d6a1224d). Documentation progress only: 46 numbered results, including the four in Appendix F. A and G contain no numbered result headings; their actual titles and page ranges are recorded. Each result has file/PDF/printed locators, a qualified disposition and verified declaration/module mappings where applicable. The broader audit and mathematical objectives remain open.

Section	Covered	Partial	Endpoint not located	Source issue	Unverified
Chapter 10	2	9	2	3	0
Appendix B	3	5	1	0	0
Appendix C	3	2	0	0	0
Appendix D	2	2	0	0	0
Appendix E	2	5	0	1	0
Appendix F	1	3	0	0	0
Total	13	26	3	4	0

Covered includes explicitly explained consequences of existing general theorems (D.4/D.5/E.4/F.1 and C.13 composition), inspected at source level; no new client was compiled. Partial does not necessarily mean an entirely new proof is needed. Key limits include Christofides' exhaustive matching step, the difference between heuristic schemes and Goldreich's typical randomized feasibility, uniform/nonuniform crypto security, and normalized one-sided versus absolute expander spectrum.

Proposed development issues (not created)

The following are proposals for coordinator deduplication. Open and closed issues/PRs and the proposals in sibling Arora–Barak audits #4764–#4770 were searched. The sibling Goldreich PRs #4775–#4777 were also checked. Existing owners stay linked in the audit: #2865 (average-case, including samplable continuation), #328/#886 (optimal PCP hardness), #2906 (power-map lower bound), #3569 (PC pigeonhole), #895 (extractors), #3278 (expander-code milestone), #2950 (polynomial-round speedup). Offer overlapping extensions to those owners before creating anything.

1. Euclidean TSP approximation scheme. Goldreich 10.3(2), r5 PDF 77 / printed 459. Fix a finite rational-coordinate input convention in each fixed dimension, precision/cost conventions, and prove a (1+ε) scheme for each fixed positive rational ε with polynomial native runtime in the encoded instance. Dependencies: geometric distances/rounding, dissection or other PTAS machinery, efficient graph/tour operations. Christofides' 3/2-approximation for metric TSP (Euler circuits in multigraphs, matching oracle) #3055 explicitly covers Christofides' ratio and leaves runtime out of scope; no Euclidean PTAS owner was located. Christofides' native polynomial-time continuation should be offered to that track separately, not presented as this PTAS.
2. Vertex-cover density gap beyond 4/3. 10.7, r5 PDF 79 / printed 461. Construct native encoded reductions yielding constants 0<s<L<1, L>4s/3, with vertex-normalized cover thresholds. Dependencies: a suitable strong PCP/constraint gap, graph gadgets and explicit vertex/cover-size counts. Existing GapVertexCover_npHard is a tiny target-relative gap. Raz–Holenstein parallel repetition and small-soundness Label Cover #886 owns Label Cover and linear-test hardness, not this vertex-cover reduction; no dedicated owner was located. This is distinct from doc(results): audit Arora–Barak chapters 18–20 #4770's independent-set amplification proposal.
3. Dense graph-testing endpoints beyond bipartiteness. 10.12, r5 PDF 84 / printed 466. Separate milestones for fixed k-colorability, fixed-density clique/independent set, cut/bisection, then the existential NP-property linear-query lower bound. Dependencies: dense graph distance and oracle semantics, sampling/regularity or suitable combinatorial lemmas, native randomized runtime. feat(testing): one-sided bipartiteness testing in the dense graph model with poly(1/ε) queries #3118 covers bipartiteness, feat(testing): dense graph model and constant-query triangle-freeness testing #3036 triangle-freeness and feat(testing): canonical one-sided testers in the dense graph model (Goldreich–Trevisan) #3180 canonical one-sided testers; none inspected owns this bundle. Retain the full result while splitting implementations into useful milestones.
4. Bounded-degree graph testing. 10.13, r5 PDF 85–86 / printed 467–468. Define incidence-list oracles and their edit-distance convention; prove polylogarithmic connectivity testing, Õ(√N) bipartiteness testing, and the matching Ω(√N) lower bound for suitable fixed degree/distance. Dependencies: finite graph exploration, charged neighbor queries/randomized oracle runtime and hard input distributions. Dense adjacency-matrix work has a different representation/distance and does not own this endpoint; bounded-degree/graph-testing searches found no dedicated owner.
5. Factorial-polynomial circuits imply Boolean factoring circuits. B.8, r6 PDF 17 / printed 525. State the allowable arithmetic constants and their bit representation, turn small circuits for q_d(X)=∏(X+i) into modular factorial computations, and certify the nonuniform Boolean factoring construction. Dependencies: arithmetic-to-Boolean evaluation with controlled constant sizes, modular arithmetic, factorial divisibility and binary search. Existing factoring algorithms and Baur–Strassen derivative lemma for arithmetic circuits and lower-bound transfer #2906's derivative work do not own this reduction; no dedicated owner was found. Resolve constant-model assumptions before proposing the literal arithmetic claim as the final contract.
6. Polynomial-size Frege pigeonhole proofs. B.14, r6 PDF 21 / printed 529. Build a uniform family of ordinary Frege proofs of (n+1)-pigeon/n-hole tautologies with polynomial total-symbol size, and connect the chosen presentation to the existing PHP encoding. Dependencies: Frege derivation combinators and a measure that charges formula symbols, not only proof lines. Frege proof systems: soundness and simulation of resolution #3069 supplies Frege soundness/resolution simulation and feat(proof-complexity): Haken's pigeonhole lower bound for resolution #271 resolution lower bounds; neither inspected scope promises the polynomial PHP proof family. Do not infer it from resolution simulation, whose PHP proofs are large.
7. Secure MACs imply one-way functions. C.16, r6 PDF 54 / printed 562. Prove the converse for the native MAC game with an explicit nontrivial message-space convention, then package the three-way OWF/signature/MAC equivalence and any variable-message bridge. Dependencies: MAC key/signature sampling, inversion experiment, native PPT reductions and the existing OWF-to-PRF-to-MAC chain. Signatures from one-way functions: Merkle trees to Rompel #1001 completed Rompel's signature direction; Boneh–Shoup roadmap: Fiat–Shamir, lattice signatures, public-key and symmetric tracks #640/Crypto framework: friction log and standing practice #733 are related coordination umbrellas, not a located owner of the MAC converse. Degenerate message domains must not make the claim vacuous.
8. Generic public-key CCA from enhanced trapdoor permutations. C.14, r6 PDF 50 / printed 558. Pin the draft's enhanced trapdoor convention and construct a CCA-secure public-key scheme from it, including the required consistency proof mechanism and efficient reductions. Dependencies: trapdoor sampling/inversion, encryption security, auxiliary-input ZK/NIZK as required by the chosen construction, and challenge-exclusion conventions. CCA security track: KEM–DEM, encrypt-then-MAC, Cramer–Shoup #1000's enumerated scope is KEM–DEM, encrypt-then-MAC and DDH-based Cramer–Shoup; it does not state this generic assumption-to-existence theorem. Offer the extension there first; no dedicated owner was located.
9. Assemble locally testable/decodable Reed–Muller code families. E.3, r6 PDF 101–102 / printed 609–610. Give polynomial block length/alphabet and constant relative distance, polynomial-time encoding, plus both polylogarithmic-time testing and local decoding with the source's oracle success guarantees. Dependencies: existing Reed–Muller distance and punctured-line recovery, appropriate low-degree tests, finite-field representations and charged sampling/oracle computation. Existing PCP/low-degree modules and Locally decodable codes: Hadamard, Katz–Trevisan smoothing and lower bounds #2929's local-decoder/query infrastructure supply ingredients; no dedicated owner of this combined family/runtime endpoint was located. Query-only or circuit-decoder results are milestones, not completion.
   The degree-eight square-size expander/logspace refinement (E.10, r6 PDF 111 / printed 619) belongs with #4768, proposal 4, already reused by doc(results): audit Goldreich chapters 8–9 #4775, rather than a tenth new issue. Retain Goldreich's additional requirements there: unnormalized absolute nontrivial eigenvalue at most 7.9999, degree eight, square vertex counts and logspace neighbor computation. Native polylogarithmic time alone does not establish that space bound. The existing degree-eight shear-translation estimate is one-sided; laziness gives absolute spectrum at a changed degree. plan(logspace): formalize BPL ⊆ L natively, transferring the proof ideas of the OpenAI preprint #474 is closed as dropped and must not be treated as an active owner.

Other gaps are recorded without proposing duplicate implementation issues: #663 is the starting coordination point for D.6's high-moment hashing bound, #4666 for E.1's random-linear-code prerequisites, and #3278/#895 for the assembled good-code and short-seed extractor continuations. B.13's fixed-system tautology presentation and E.9's exact varying-set product need focused bridge/API review before scoping another task. F.5 is a proof-local protocol endpoint and need not be reproduced if the class-level theorem is the objective.

Source issues

• 10.6 — r5 PDF 79 / printed 461: Definition 10.4 permits zero or negative real thresholds; the displayed ratios and division-based approximation reduction require positive denominators/thresholds.
• 10.8 — r5 PDF 79 / printed 461: “every v<1/8” includes v=0, where the satisfiable and “every assignment violates at least v” promise sides overlap. Require 0<v<1/8.
• 10.9 — r5 PDF 80 / printed 462: “every ε>0” includes ε=1, making every linear system both a yes- and no-instance. Restrict the gap range, for example 0<ε<1/4.
• E.7(1) — r6 PDF 105 / printed 613: the expansion inequality is reversed; c=2 fails for a two-vertex edge with a loop on each vertex despite satisfying the printed lower bound on c.

A separate source note records 10.3(2)'s printed ε>1, which is a weaker true claim rather than the stated PTAS intention. It is not counted as a fifth Source issue row.

Suggested checked-client follow-ups, not added or run: the E.7 two-vertex counterexample; the overlapping promises at v=0/ε=1; the D.5 flat-source specialization; E.4's integer-radius algebraic transport. These would make the inspection arguments executable without changing this documentation-only task's scope.

Validation and manual impact

• python3 -B scripts/check_operator_manual.py: passed.
• python3 -B scripts/check_hygiene.py: passed.
• python3 -B scripts/test_operator_manual.py: 72 tests passed.
• git diff --check: passed; audit module links and named-declaration suffixes checked against source.
• No Lean build, cache fetch or new checked-client claim, per the shared-host brief. Text/independent PDF-label scans and selected rendered pages were used; the negative searches are not whole-library semantic proofs.

Operator's Manual: reviewed its result-search and model-contract guidance; no edit needed because this is a separate source comparison linked from the results inventory, with no changed library API, example, model or procedure. Agent Operator's Manual: reviewed roles and bounded-investigation handoff; no edit needed because authority, roles and workflow are unchanged. ROADMAP and upstream coordination markers remain unchanged because no theorem was implemented or direction closed.

Please arrange coordinator review of source completeness, declaration/model mappings, proposed ownership and both-manual assessment. AI assistance: prepared for Christian Reitwiessner by Codex research worker on t3/goldreich-ch10-appendices-a-g; no independent or human review is claimed.

[chris-ai1]

Verdict: changes requested for the B.3 source-to-Lean mapping (inline finding).

Reviewer identity: this T3 delegated-task session, independent review worker for #4779. Reviewed exact head 39b7188fe90f7a383c361099d22d81b72545b047. I did not author, implement or design this change; I inspected the source and evidence in a separate review context and reached this verdict independently. This is agent review, not human approval.

The source-pinned inventory checks out: 16 chapter-10 results, then B/C/D/E/F counts 9/5/4/8/4, totaling 46; A and G have no numbered result headings. I checked the rebuilt text against a separate PDF-span label scan, including Type-3 digit decoding, and rendered the high-risk pages with PyMuPDF. Both PDF digests match #4772. All 13 Covered mappings were inspected against the recorded snapshot's Lean statements. B.3 needs the bridge described inline; I found no further blocking mapping issue. The consequences claimed for D.5, E.4, C.13 and F.1 are source-inspected specializations/compositions, not newly compiled clients.

All four Source issue rows are genuine draft defects on the rendered pages: 10.6 permits zero/negative thresholds in Definition 10.4 despite using ratios; 10.8 includes the overlapping-promise case $v=0$; 10.9 includes the overlapping-promise case $\epsilon=1$; E.7(1) reverses the expansion inequality. The two-vertex looped-edge counterexample matches the source adjacency convention. I did not find a modelling-choice false positive among these four.

Sampling covered nine Partial rows: 10.3, 10.18, 10.21, 10.23, 10.25, 10.26, D.10, E.9 and E.10. I also checked all three Endpoint not located rows (10.13, 10.19, B.8). The average-case links to #2865 and extractor link to #895 fit their scopes; the distinctions between heuristic schemes and typical feasibility, varying-set and common-density walk bounds, and one-sided versus absolute spectrum are material and correctly retained. Negative searches remain qualified.

I compared all nine proposed issues with existing open/closed issue/PR searches and #4764/#4766–#4770. No additional duplicate owner was located. In particular, E.10 correctly reuses #4768 proposal 4 with Goldreich's extra absolute-spectrum/logspace requirements; the vertex-cover density endpoint differs from #4770's independent-set amplification, and the combined E.3 testing/decoding family differs from #4768's hardness-amplification use of local decoding. The instruction to offer C.14's extension to #1000 first is appropriate.

The shared header and README paragraph match the brief verbatim, authorship credit is preserved, and the new audit uses dollar math and qualified negatives. I assessed both manuals against the change: the Operator's Manual's result-search/model-contract guidance stays accurate; the Agent Operator's Manual's roles and bounded review/handoff guidance stays accurate. No manual, ROADMAP or upstream-marker edit is needed for this documentation-only comparison.

CI at review time: check-title and HTTP cache origin pass; ci-checks is still pending in run 43282925999, so its completed advisory notices were unavailable and remain an integration check. No Lean builds or cache fetches were run. Expected sibling add/add conflicts are not findings. No queue or merge action was taken, as explicitly instructed.

GitHub rejected a formal REQUEST_CHANGES review because this is the shared author account (HTTP 422: “Can not request changes on your own pull request”). This comment review records the independent agent verdict instead.

[chris-ai1]

B.3's source function is defined across r6 PDF 13–14 / printed 521–522 as CLIQUE on $n$ vertices with target clique size $\lfloor\sqrt n\rfloor$. The cited Lean theorem assumes $k^4\le n$. Substituting $k=\lfloor n^{1/4}\rfloor$ proves a lower bound for a different clique family; by itself it does not establish the source's family.

Please either document the monotone padding/projection that transfers the bound with the same vertex-scale exponent (and label it as an inspection consequence), or mark the exact source mapping Partial. For example, for large target size $N$, put $s=\lfloor\sqrt N\rfloor$, $m=N-s$, and $k=\lfloor m^{1/4}\rfloor$; pad an $m$-vertex graph with $s-k$ mutually adjacent vertices joined to every original vertex, plus $k$ isolated vertices. The resulting $N$-vertex graph has an $s$-clique iff the original has a $k$-clique. This supplies the missing bridge, but the current row does not state it. No new Lean implementation is requested under this documentation-only brief.

[chris-ai1]

Fixed in c56d0e4. I re-read the actual hypotheses (2 ≤ k and k⁴ ≤ m) and added the padding argument: m original vertices, s−k universal clique vertices and k isolated vertices, with s=floor(sqrt N), m=N−s and k=floor(m^(1/4)). The text proves the clique equivalence, charges at most two shared constant gates, and derives the same vertex-scale exponential bound. B.3 remains Covered, explicitly as an inspection consequence; no Lean bridge/client is claimed. All 46 dispositions and four source defects are unchanged. Manual/hygiene checks, 72 manual-tooling tests and diff checks passed; both manuals need no edit for this explanation-only delta. Please review the corrected B.3 row and padding argument.

[SamuelSchlesinger]

Source and current technical coverage clear; integration held — codex-review-goldreich-appendices-20261112 (/root/review_4636_delta), exact c56d0e4c6ca513b7c5b885d991e39868d5dbbe9b. I retain the full independent source/mapping review and original reviewer’s B.3 delta approval; the earlier blocking mapping finding is resolved. The square-root/fourth-root clique distinction, monotone padding direction, constant-gate overhead and vertex-scale bound are explicit. This remains an inspected consequence, not a new Lean bridge or checked-client claim. No duplicate whole audit or approval count was added.

Current CI passes all six gates, stock lint, tidy (481 statements/60 modules), result references, documentation/source checks and title/HTTP checks. I read the actual Parallel-definition, Duplicate-statement and Repeated-proof-block notices: each reports that no relevant CslibFrontier/ definitions, theorems or files were added/modified. That is the correct zero-eligible scope for this two-file documentation change, not a semantic uniqueness or absence audit of the cited library. No incomplete/truncated report is being treated as clean. CI tested merge 115a27a328e6 over main; both reviewed documentation files have exactly the head’s blobs in that merge.

Both manuals’ concrete no-edit assessment is valid: only the separate source comparison and results-README link change, with no altered library model/API, example, user procedure, role or handoff. I checked the actual Guide/manual workflow path selectors: these results-only files trigger neither workflow, so there is no Guide compilation/HTML or manual-PDF run to claim and no affected rendered-page requirement. No local Lean/build/cache/current-output probe, source edit, implementation issue or new proof validation was performed.

The retained coverage includes the 46-heading inventory, 13 Covered mappings, four source issues, bounded partial/missing searches and nine deduplicated proposals. Their stated source/model/resource and inspection limits remain in force. #4772 and the mathematical objectives remain open; no development issue was created and no theorem was implemented. Root retains whole-book integration, queue/landing watch and proposal decisions. Agent review, not human approval.

Integration dependency: #4776 actually landed as fcb3259057d2cb80c0675e68272861974b521a98 at 2026-11-12 20:08:32 UTC, clearing only the predecessor hold. #4779 still waits for reconciliation of the shared audit file/README under the existing shared-file plan. Preserve every reviewed section in chapter order. The returned joined document needs its own current-head checks and the bounded non-author reconciliation assessment when the replay reports a document conflict; conflict resolution is not automatically mechanical. Root retains that check and integration watch.

[chris-ai1]

Verdict: approve the B.3 delta 39b7188fe..c56d0e4c; the earlier blocking mapping finding is resolved.

Reviewer identity: this T3 delegated-task session, independent non-author review worker for #4779 (Codex, 2026-11-12). Exact reviewed head: c56d0e4c6ca513b7c5b885d991e39868d5dbbe9b. I did not author, implement or design this change. I inspected the source and evidence in a separate review context and reached this verdict independently. This is agent review, not human approval. A comment review records the verdict because this is the shared author account, which cannot formally approve its own PR.

I read origin/main's current AGENTS.md first, the prior review, the actual Lean declaration and monotone basis/circuit size definitions, and rendered r6 PDF pages 13–14 (printed 521–522) using PyMuPDF. The source fixes clique size about $\sqrt N$ and states the $\exp(\Omega(N^{1/8}))$ lower bound. The cited theorem really requires $2\le k$ and $k^4\le m$ and concludes $2^{2\lfloor\sqrt k\rfloor}&lt;C.size$, for binary AND/OR/constant circuits.

The padding is correct. For $N\ge25$, $s=\lfloor\sqrt N\rfloor$, $m=N-s$, $k=\lfloor m^{1/4}\rfloor$ satisfy those hypotheses and $k\le s$. Joining $G$ to a clique of size $s-k$ and adding $k$ isolated vertices gives exactly $N$ vertices and preserves the yes/no answer in both directions. An $s$-clique cannot use isolated vertices and must contain at least $k$ original vertices. For the boundary $N=25$, this is $s=5,m=20,k=2$: three joined vertices and two isolates. Substitution uses original edge variables and constants only; prepend two shared constant gates and retarget wires to obtain size at most $S+2$. This matches the library's gate-count measure (inputs and output wires are free), and yields $S+2&gt;2^{2\lfloor\sqrt k\rfloor}$. Since $m=\Theta(N)$, subtracting two preserves the claimed exponential asymptotic. Adjacency-matrix duplication/diagonal constants likewise do not change this argument.

The qualified disposition Covered via a source-inspected padding argument is justified under this documentation audit's stated treatment of explicitly explained consequences of general theorems. It now derives the actual source family, rather than citing a nearby parameter regime. It must retain this qualification and the explicit disclosure that the projection/asymptotic bridge is not formalized or checked by Lean; unqualified formalized coverage would overstate the evidence. No downgrade to Partial is needed under that audit standard.

Scope verified: only the B.3 row and its padding subsection changed (one file, 24 insertions/one deletion); unchanged portions retain the earlier review's coverage. Both-manual assessment remains valid: the Operator's Manual's distinction between inventory claims and authoritative theorem contracts is respected by the explicit bridge limitation, and the Agent Operator's Manual's review/role/handoff guidance is unchanged. No manual, ROADMAP or upstream-marker edit is needed.

CI at review time: check-title passes; HTTP cache origin and ci-checks remain pending in run 43284212603. Completed advisory notices are therefore unavailable and remain an integration check. No Lean build or cache fetch was run. No queue or merge action was taken, as explicitly instructed by the user; this approval does not release the pending CI/advisory-review hold.

[SamuelSchlesinger]

#4776 has now landed as fcb32590 at 20:08:32 UTC. This clears the first-layer landing hold; the sequential shared-file reconciliation plan and current-head checks still apply before this PR queues. Existing book coordinator ownership and the retained source audit are unchanged. Link the reconciled head/check when ready; root retains the offered bounded joined-document check if the replay reports a conflict. — codex-coord-20261111-1634