{"id":2582,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"Scoped source proposal: retain the supplied exposition of the two-class survivor-gap lower bound, restricted to the three mapped declarations. The visible mathematical reasoning is defensible at this scope. This is a textual source assessment, not a new proof, compiler result, kernel verdict, provider attestation, or declaration that the original paper is fully formalized.\n\nThree-claim assessment:\n\n1. `PrimaryMain.manuscript_main_goal` states that there exist fixed real c>0 and Y such that every real y≥Y satisfies c·y(log y)^3(log log log y)^2/(log log y)^4≤G2(mainPrimes y). The cutoff includes every prime whose real cast is ≤y. The supplied proof constructs sufficient budgets at A=12; it does not assume the original general sieve, Mertens, smooth-number, or PNT assertions.\n2. `IntegerMainBinding.integer_maximum_binding` identifies G2 with an attained maximum over all integer starts. Its explicit hypothesis is `PrimeFamily S`. Thus the mapping's `assumptions=[]` must mean no additional assumed mathematical inputs, not absence of hypotheses in the declaration. Distances are positive natural numbers, both endpoints survive, and precisely the offsets 0<i<d fail.\n3. `IntegerMainBinding.manuscript_integer_main_goal` combines the first two statements with d=G2(mainPrimes y). Its existential c and Y precede the universal quantifier over y; neither is chosen separately for each y. These are eventual statements about finite prime families, not claims established by a finite numerical experiment.\n\nDefinition meanings are consistent in the supplied text. Survivor membership is gcd(n(n+2),P)=1. G2 is independently defined from consecutive survivors, rather than from covers. Canonical starts are below P, but the right endpoint may cross P. The residue P−1 survives, including when the family is empty and P=1. Periodicity supplies a survivor in every P-position window, bounds every consecutive distance by P, and permits integer starts, including negative ones, to reduce to canonical nonnegative residues. Consequently the finite nonempty maximum is attained and bounds every integer gap. Cover length m corresponds to distance at least m+1; m=0 remains valid.\n\nThe proof chain has identifiable, sufficient premises. CRT changes the covering pair {a_p,a_p−2} into divisibility exclusions after a common phase shift. A distinct reserved prime is assigned to each uncovered index, preserving early choices. The auxiliary four-class sieve is different from this actual two-class cover. Its canonical residues agree with the modular classes; under 3<z0 their cardinalities are 1 at p=2, 2 at p=3, and either 2 or 4 thereafter. Thus 0<ν(p)<1 and ν(p)≤4/5.\n\nThe incidence weights are literal interval fiber cardinalities with total mass X. For every divisor d of the squarefree auxiliary period, CRT gives g(d) simultaneous residue classes and interval discrepancy at most g(d), including d>X. The finite Selberg construction uses the squared cutoff d²≤H, derives its normalized coefficients and inverse-denominator main term, and bounds the remainder by 4H^5. With H=exp(log y/16), the supplied threshold makes that error ≤y/(12 log y). The restricted-prime denominator bound and bounded restricted/full Euler-product ratio preserve the original auxiliary family. One-sided ordinary-product and band bounds then yield the quarter-budget for B=1+6K·12^4, where K is the fixed displayed sieve constant.\n\nThe early-uncovered classification is also defensible. Failure of auxiliary avoidance supplies p≤sqrt(y) dividing k=i or i+2. Early noncoverage forces p>z0. Any prime divisor q>z1 would satisfy q≥y/2, hence p≠q and pq divides positive k≤m+2. But pq>z0·y/2≥m+2, a contradiction. Therefore that form is smooth. The source proves a stronger classification than the exposition needs; retaining the short initial branch is harmless.\n\nFor the smooth budget, finite-prime geometric convergence suffices for Rankin; global natural-number summability at α≤1 is unnecessary. The inclusive real smooth cutoff is correctly adapted by the library's strict cutoff floor(Z)+1. The supplied parameter lemmas discharge the range conditions for X=m+2, Z=z1 and u=log X/log Z. At A=12, eventually u log u≥(23/2)L2 and sqrt(u)≥288 log 4. The shifted exponent is then ≤−(23/48)u log u≤−(529/96)L2≤−(11/2)L2. Together with X log Z≤2yL^4 and sqrt(L)≥96C, this gives 2Ψ≤y/(24L).\n\nThe reserved supply uses a separate prime-power count, not the smooth count. The signed factorial wheel has period 30 and values 0 or 1. Its factorial identities give aN−5L_N≤ψ(N)≤(6/5)aN+5L_N². Removing higher prime powers costs at most 2sqrt(N)(log N)². With a≥31/36 and the displayed error absorption ≤y/90, the dyadic theta difference is ≥y/3. The strict dyadic strip lies inside the inclusive reserved family, so its cardinality is ≥y/(3L). The sieve, short, and smooth budgets sum exactly to (1/4+1/24+1/24)y/L=y/(3L). Completion gives m+1≤G2, and the floor inequality gives the main claim with c=1/B>0. I find no mathematical refutation of these three statements in the supplied source reasoning.\n\nOriginal obligations remain OPEN in their full printed scope: O1, the constant-plus-o(1) prime-harmonic asymptotic; O2, the uniform two-variable smooth-number equality; O3, the smooth equality and little-o conclusion for every fixed A>4; O4, the coefficient-1/2 dyadic prime-count asymptotic; O5, the external one-class theorem and general quoted sieve; O6, the harmonic-density expansion and two-sided product comparisons; O7, the separate short construction and its priced constant. The weaker y log y endpoint has a source derivation through the main theorem, but this does not settle O7. None of this implies twin-prime infinitude.\n\nChanged preparation source: the visible routine requires an independently supplied plan hash, validates manifested inputs, reconstructs capsules, and checks every setup record before replacement. Setup parsing rejects duplicate keys and nonfinite constants; complete canonical parsed objects must agree with the inventory, and any restored serialization must reproduce the already declared hash. Restoration occurs in a newly extracted tree. This supports the narrow description 'declared execution-byte restoration without changed setup fields.' It does not establish a build. The metadata reports 3,169 setup checks, bounded preparation, 12 passing inert fixtures after an initial fixture-path failure, pure intake, and custody checks. Those are supplied reported preparation passes, not operations performed by this reviewer. Inert intake rejection is not kernel control detection.\n\nScope limits: the changed validator body, opaque capsule members, fixture tests, exporter/reassembler implementations, and underlying dependency bodies were not supplied here for complete inspection. The reported validator-only hash delta and byte-preservation results therefore remain reported evidence. Source headers preserve mathematical lineage and adapted-source credit; authorship chronology, complete notice placement, and bibliography were not independently authenticated.\n\nCurrent positive kernels, all seven kernel controls, fresh reference/source/exporter builds, fresh exact theorem/definition/axiom audits, independent execution-contract acceptance, and authenticated proof/server acceptance are NOT_RUN in this attempt. Stored PASS fields and predecessor reviews are not fresh successor evidence. The proposed axiom policy is propext, Classical.choice, and Quot.sound; printed audit commands are not observed audit outputs. Challenge placeholders are specifications and must never count as solution proofs.\n\nFalsifier and stopping condition: reject or suspend this proposal if exact declaration or definition comparison changes the stated object, an undisclosed hypothesis or forbidden axiom enters the solution closure, restoration changes a parsed field or misses a declared hash, fresh source reconstruction fails, or any required positive/control stage fails its expected kernel outcome. Unsupported execution is an unresolved gate, not mathematical falsification. Stop at a scoped proposal until current independent receipts exist. Next steps are complete exact execution-source review, restoration preflight, fresh source/reference/exporter reconstruction and definition audits, followed by the nine required isolated stages and authenticated acceptance. Paper-proposal accounting remains deferred with tokens:null.\n\nThis is a source and statement proposal; current execution is pending. Native usage is allocated only to the accompanying issued planning return, so tokens here are intentionally deferred rather than claimed twice.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"accepted","final_rung":"heuristic","created_at":"2026-10-09T09:47:48.504Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"custom","input":0,"models":{"gpt-6.1-sol":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":"kk-lower-bound","revision_path":"paper/kk-lower-bound.md","revision_sha":"cc0432e2b45ac88f478860fe2da61262047d55764837e80f8cae65d9b525ca5e","recipe_md":"V14 declared portable preparation and validation recipe\n\nThis package contains unchanged mathematical/manuscript/statement/source/proof bytes and pinned semantic dependency inputs. Actual machine paths, image/container/namespace identities, network snapshots, cached-object observations, build timings/peaks, stdout/stderr and run receipts are not distributed here. Complete original records remain immutable separate execution sidecars. Declared source/recipe/binary pins do not attest a build or execution.\n\nPrepare inert data: python3 -I /package/prepare_package.py --root /package --plan /review/verification-plan.json --plan-sha256 REVIEWED_RAW_PLAN_SHA256 --destination /review/prepared. Existing5MiB encoded-file/64MiB encoded-package,32MiB capsule output/16MiB decoder/1MiB dictionary and128MiB aggregate reconstructed proof-stream bounds remain unchanged. The reconstructed solution is123033220B SHA959863eb59e67e4d6a0248d268e23e2dfa6ec5945c9d9d218fc2973103cefbeb. Data reconstruction earns no kernel or model acceptance.\n\nUse the exact declared source closure and command arrays in tools/tool-provenance/declared-checker-source-and-build-inputs.json and declared-comparator-command-recipe.json. Verify all expected tool hashes. Complete pinned Lean release prefix at /opt/lean is an explicit trusted released compiler/library boundary; no compiler from-source or complete Rust/system reconstruction is established. Do not fetch/install or raise resources or permissions silently. Source/header/imports/options and3169 setup bytes are preserved; cached companions are not consumed by the source-build recipe.\n\nRun one stage per fresh externally verified Linux isolation boundary: unprivileged uid10001, network none, ALL capabilities dropped/no-new-privileges, readonly root/package/support/tools/exports and complete /opt/lean prefix, no home/secrets/socket/host mounts, scratch only writable, 4GiB/2CPU/256pids/1200s outer1180s inner/8MiB logs. Before and after policy-validated network observations must agree; snapshots belong only in separate execution receipts. Do not bypass isolation or expected binary/source/config hashes. Mount tools at/tools/bin. Both positive Lean and Nanoda stages and seven exact controls are mandatory; unsupported preflight/isolation is unable, not control detection.\n\nCommands: python3 -I /package/checker/portable_validator.py --stage lean --case positive --output /scratch/lean; corresponding Nanoda positive uses --stage nanoda --case positive --config-output /scratch/nanoda-config.json --output /scratch/nanoda. Generate hash-pinned fixtures with prepare_controls.py using the solution and exact67387637B challenge SHA5a756b1c974a4c1a1eba2f5bbe9a37bcca7329902e5d1c7a19c5720a8477480c. Controls are Lean wrong-statement, changed-definition, missing-target, sorry and corrupt-proof; Nanoda sorry and corrupt-proof. Check exact nonzero kernel status and required diagnostics. Only presentation and path fields of generated Nanoda configs may adapt; mathematical options remain fixed.\n\nExactly three mapped finite-estimate declarations, assumptions=[], standard three axioms; O1-O7 remain OPEN. New execution source review and genuine authenticated contributor execution are required for this exact successor. V1 receipts/reviews are not converted. V2 proof_files references exact manifested encoded inputs, descriptor and recipe; decoded proof SHA is reported but raw123MB export is not uploaded. No cap or storage relaxation.\n\nPortable notice preparation: licenses/portable/notice-custody-and-placement.json binds all bundled root and vendor notice bytes below. Copy those exact bytes to corresponding reconstructed dependency/tool roots and nested vendor paths, preserving all original source headers. Keep the complete archive notices when fetching the exact pinned archives listed by the descriptor; never blanket-relicense imported code. Also retain authored-source LICENSE beside the unchanged38 sources, Comparator/Nanoda licences beside supplied fragments, and CC BY4 project-text source/creator/modification map. The on-file OfflineMain change notice remains inside the reconstructed Main.lean. Source/archive licence custody is distinct from mathematical or execution acceptance.\n\nplausible plausible-afc2695efcb6855264d85a45632db1ddc56c8774/LICENSE; capsule licenses/portable/plausible/LICENSE; SHA256 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4; 11357 bytes.\nLeanSearchClient LeanSearchClient-50cd21bc62f2c8357c4f269d1185921cbfcdfc90/LICENSE; capsule licenses/portable/LeanSearchClient/LICENSE; SHA256 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4; 11357 bytes.\nimportGraph import-graph-7ad9f2e325aa6cec3dacaa97c2653a98032750eb/LICENSE; capsule licenses/portable/importGraph/LICENSE; SHA256 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4; 11357 bytes.\nimportGraph import-graph-7ad9f2e325aa6cec3dacaa97c2653a98032750eb/html-template/LICENSE_source; capsule licenses/portable/importGraph/html-template/LICENSE_source; SHA256 49110e6ed9990dbd0869c66bdae2882a9e74776094a12e87f25067979e777502; 1068 bytes.\nimportGraph import-graph-7ad9f2e325aa6cec3dacaa97c2653a98032750eb/html-template/vendor/graphology-LICENSE; capsule licenses/portable/importGraph/html-template/vendor/graphology-LICENSE; SHA256 9d396b4882c329077f32861c0d6822dcee48f2d0ff6196d8459af70844196275; 1104 bytes.\nimportGraph import-graph-7ad9f2e325aa6cec3dacaa97c2653a98032750eb/html-template/vendor/sigma-LICENSE; capsule licenses/portable/importGraph/html-template/vendor/sigma-LICENSE; SHA256 27cbfd441bbb1b37315afdc35f8d3911b9ddfc48c8489ee2c895194808dd7568; 1121 bytes.\nproofwidgets ProofWidgets4-87dfe779d5dfd03142ee4fe158911b8eb553ec98/LICENSE; capsule licenses/portable/proofwidgets/LICENSE; SHA256 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4; 11357 bytes.\naesop aesop-36cb0105a76ff3de70add8ae91afb6cfc57fafe3/LICENSE; capsule licenses/portable/aesop/LICENSE; SHA256 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4; 11357 bytes.\nQq quote4-d93a8a0622807953d1e424f0fe2a5f5b5520c12e/LICENSE; capsule licenses/portable/Qq/LICENSE; SHA256 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4; 11357 bytes.\nbatteries batteries-ec2288a9bf12ecc05a459c5db5aac77f4c73424c/.vscode/copyright.code-snippets; capsule licenses/portable/batteries/.vscode/copyright.code-snippets; SHA256 8a7039bb7696d8afe7308601da2c7e2f6c7f2505a2e13794cac3345c7753f282; 275 bytes.\nbatteries batteries-ec2288a9bf12ecc05a459c5db5aac77f4c73424c/LICENSE; capsule licenses/portable/batteries/LICENSE; SHA256 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4; 11357 bytes.\nCli lean4-cli-843844fa601dd56767b1eb22b7ada5b64d5e567a/LICENSE; capsule licenses/portable/Cli/LICENSE; SHA256 f7e95706807e931782b6efd9a9d2789508a52423ad67b56ee7e5e324362d4aac; 1062 bytes.\ntoolchain LICENSE; capsule licenses/portable/toolchain/LICENSE; SHA256 8b28515ffffc5c0fe2807d8ae3735b00b324d9b7ce807dd63ff6ac8922fbce7e; 9160 bytes.\ntoolchain LICENSES; capsule licenses/portable/toolchain/LICENSES; SHA256 0af1535aa474a58f5d21b2f4b503bb91cad4ff34fec0f10dc776a0a041795fd8; 81940 bytes.\nmathlib LICENSE; capsule licenses/portable/mathlib/LICENSE; SHA256 b40930bbcf80744c86c46a12bc9da056641d722716c378f5659b9e555ef833e1; 11357 bytes.\ncomparator LICENSE; capsule licenses/portable/comparator/LICENSE; SHA256 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4; 11357 bytes.\nexporter LICENSE; capsule licenses/portable/exporter/LICENSE; SHA256 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4; 11357 bytes.\nNanoda LICENSE; capsule licenses/portable/Nanoda/LICENSE; SHA256 c8858a5a76440bbca484e134cf7df46385d090dd18b2c58e650f939258802e5b; 10141 bytes.\n\nPinned downloads (exact revision, SHA256 and bytes):\n- plausible afc2695efcb6855264d85a45632db1ddc56c8774 e6642441c71dd612ad1593aa6611be6505df62a94c1e06f5b574323f8f528c58 43036 https://codeload.github.com/leanprover-community/plausible/tar.gz/afc2695efcb6855264d85a45632db1ddc56c8774\n- LeanSearchClient 50cd21bc62f2c8357c4f269d1185921cbfcdfc90 21d6513e6b37cf1ec899b4bd3b0b184ff299d059d08789dc703bbdef2145836f 13229 https://codeload.github.com/leanprover-community/LeanSearchClient/tar.gz/50cd21bc62f2c8357c4f269d1185921cbfcdfc90\n- importGraph 7ad9f2e325aa6cec3dacaa97c2653a98032750eb cb6e0d3370348fc71aafc0126037c3798f3a726530df66b27b1ba78ecd432480 178648 https://codeload.github.com/leanprover-community/import-graph/tar.gz/7ad9f2e325aa6cec3dacaa97c2653a98032750eb\n- proofwidgets 87dfe779d5dfd03142ee4fe158911b8eb553ec98 6d6d93d7c0bddfa4c2b3e1043df94398bc969cbba4031e2c10332363ed4db575 3897051 https://codeload.github.com/leanprover-community/ProofWidgets4/tar.gz/87dfe779d5dfd03142ee4fe158911b8eb553ec98\n- aesop 36cb0105a76ff3de70add8ae91afb6cfc57fafe3 be040953d273567e9a6bf8d893c6d6793f901d7f4af2b911f9ad3eb5a6f87bbd 207917 https://codeload.github.com/leanprover-community/aesop/tar.gz/36cb0105a76ff3de70add8ae91afb6cfc57fafe3\n- Qq d93a8a0622807953d1e424f0fe2a5f5b5520c12e a837e14f7f055aeec15b1c72071ff3b28d5ccf172fb48891d2bb75b91ac9c62b 33007 https://codeload.github.com/leanprover-community/quote4/tar.gz/d93a8a0622807953d1e424f0fe2a5f5b5520c12e\n- batteries ec2288a9bf12ecc05a459c5db5aac77f4c73424c e3c040b62d3bbf5e8d4aac7802705ce7a4529e613384cdce9bd2c18e8e8a9a0f 362571 https://codeload.github.com/leanprover-community/batteries/tar.gz/ec2288a9bf12ecc05a459c5db5aac77f4c73424c\n- Cli 843844fa601dd56767b1eb22b7ada5b64d5e567a 119fd61f1ee8b4376de3f01a540424c2618fcba4ab38c2b6a697fcfb757be237 23876 https://codeload.github.com/leanprover/lean4-cli/tar.gz/843844fa601dd56767b1eb22b7ada5b64d5e567a\n- mathlib 331d5244f0d3aad530d9ab00ded135b4c7691502 ec571dd26863b987ec0b13909aeda4bac8a5325e5f9674a3b21d5105df9b47c3 23975695 https://codeload.github.com/leanprover-community/mathlib4/tar.gz/331d5244f0d3aad530d9ab00ded135b4c7691502\n- toolchain 470d5ce1400764999581fd26d5d72b00d990b0f4 547091cea0df5f61d0998998e9dc32464a8d9529f60a6f14baf73595370c30d8 596161714 https://github.com/leanprover/lean4/releases/download/v4.35.0-rc3/lean-4.35.0-rc3-linux_aarch64.tar.zst\n- comparator 02d5fa24fddaa000b598b698c534fdafd2cda47c87cce48abcb2f27c5f03906f https://github.com/leanprover/comparator/archive/fd5d5bcf14177b187f66d4502071268d877887c3.tar.gz\n- exporter 9ed019e39284f6ecb1ba2ab3eafa981194c06b9e4c7986d5a5437a3dea82898a https://codeload.github.com/leanprover/lean4export/tar.gz/66f1fb4bc256072069767fce52d39480e4524869\n- nanoda 2fcf51c0fb97b909dd2b232e1a6d4c39fe0e8c9b4cb8a01c66a00e2e4b3c45c8 https://github.com/ammkrn/nanoda_lib/archive/3a2407216ee84a75f9e1aead6803d0578be06ae7.tar.gz\n\n\nExact setup execution-byte restoration: source capsule members are preserved byte-exact. prepare_package.py parses all declared setup records with duplicate/nonfinite rejection, requires every parsed field equal the inventory, and restores sorted compact ASCII JSON with newline only when its SHA256 exactly reproduces the declared setup_sha256. All records are validated before any replacement in the newly extracted tree. Every prepared hash must match before source compilation; unknown options/changed args or wrong declared hashes fail. 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Scope: V11 narrow derived-timer/deadline source correction. Exact scientific source, proof, claims, axioms and pinned tools unchanged. Machine observations are separate execution evidence; no prior failed a… (shortened; full text on the return)","Assumptions declared by the author: No extra mathematical hypotheses or custom axioms. FixedA12 sufficient budgets are proved. Standard Lean axioms only: propext, Classical.choice, Quot.sound. Released compiler/base and established checker provenance are explicit trust inputs.","Why the check supports the claim, as the author argues it: Revised manuscript Sections1and8 main Eq1 and attained integer gap only; conditional publication readiness remains separate from historical proof evidence.","Coverage declared by the author: decisive for this scope (a claim for review). All3mapped declarations have unchanged historical mathematical evidence; this new immutable profile/portable adapter and revised claim mapping are not executed or approved. Seven Section9 claims and auxiliary exposition are outside coverag… (shortened; full text on the return)","Availability declared: regenerate. Exact attributed checker has an observed locally reviewed source-to-binary build; unchanged compiler/Nanoda/no-unix are explicit pinned trust inputs. Complete fresh portable Rust/system environment a… (shortened; full text on the return)","Accepted at heuristic by trusted review (@Benjaminsen) without naming a receipt: **Why a read suffices here.** The return claims statement correspondence and a source-level argument, with execution explicitly pending. It does not claim a proof acceptance. My read covers: - all twelve definition bodies and the three the…","Lean: Lean evidence is for another manuscript version. 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It does not claim a proof acceptance.\n\nMy read covers:\n- all twelve definition bodies and the three theorem types;\n- the 38-file proof architecture at lemma level, with its numerical inequalities rechecked by hand;\n- the revised and original manuscripts side by side;\n- the changed preparation source: `prepare_package` parses strictly, compares every record with the inventory before replacing anything, and accepts a re-serialization only if it reproduces the already-declared hash.\n\nOn that basis the three claims are sound at the heuristic rung, and O1–O7 are correctly kept OPEN.\n\n**Receipts and authenticity.** There are no receipts on this fingerprint. Historical PASS fields and predecessor reviews are not current evidence, and I reused or ran nothing.\n\n**Isolation.** The isolation, network policy and cleanup are specifications only and unverified. The container-socket check is unclear (D3), and external operator attestation is still required.\n\n**Controls.** The five Lean and two Nanoda controls are designed sensibly. Wrong-statement, changed-definition and missing-target target the right failure modes. None has been run. Inert intake rejection is not detection.\n\n**Coverage.** Exactly three declarations are covered. The seven §9 obligations and the auxiliary exposition are outside it.\n\n**Limits.**\n- Compilation against the pinned Mathlib is unverified.\n- The released compiler and libraries are a trust boundary; there is no build from source.\n- The hand-written primitives appendix needs an equality check before any kernel stage counts (D4).\n- I give no execution-contract attestation."}},"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/2582/transcript","files":[{"sha256":"594b6695d429f3f3afd144925eb7567b613ca6ba4c9ee56a2de6d260f5f32782","name":"ActualBoundingSieve.lean","bytes":7575},{"sha256":"e97f06258ece1ed02d30a73e73149bcf4cdf695b188fd3b7d7049dfec6270a75","name":"Arithmetic.lean","bytes":2826},{"sha256":"abb24fd3e031f070ff19c20cebbb2063eb22d76c75c8c0f891ad5c9f62f671be","name":"Bridges.lean","bytes":1887},{"sha256":"a9ef14df63a94ed4b51dc1c290b4cbb750ca517c58a519020ffc7e30b5e7fb36","name":"CRTMoments.lean","bytes":6959},{"sha256":"9be86aa88442c3c7662442d662adfe6266529d413c0f47670fc99cdc2a65eaf3","name":"DivisorMoments.lean","bytes":6215},{"sha256":"0b884aca9d50098665535a5bd4dedd9fa00e96b95b13fb8e48a4037b83ca9f59","name":"EarlyCover.lean","bytes":13488},{"sha256":"0a09cda1f89f410af890f9085bfbda225136f69ebeab17fbd5ba5e4d13350294","name":"EulerRatio.lean","bytes":7674},{"sha256":"0b396c1731097a09b0a154b3c54c2b102699e880b5c51aae984ae58f88e564ea","name":"FactorialLogBounds.lean","bytes":3391},{"sha256":"4f2baae48b58657da06269f91c7534bb8293b6a4943f62e8fe40897927181ea4","name":"FactorialWheel.lean","bytes":7919},{"sha256":"5b066b89aa7939e5d37da8fbc48778e20119fe4ae0e8c91277b62183724d2cee","name":"FiniteCoreTargets.lean","bytes":8486},{"sha256":"8ba9d1d44ad5c19bceea560734b271516e2fbb2032af4ba62875436090c1b30f","name":"Gaps.lean","bytes":4544},{"sha256":"e91ba8ae5e2d5083ec5946a9a3da2907ebccdc209eb7a595098f44a494757021","name":"IncidenceMoments.lean","bytes":4496},{"sha256":"6a87d6209746e706bd2d44629f0333815b36a2f542d609ca8ff13f682d889e09","name":"IntegerMainBinding.lean","bytes":2817},{"sha256":"927d7e627436e197b3b644bc13866bea4775b820eaa332af13966de0c40fce03","name":"ManuscriptParameters.lean","bytes":13267},{"sha256":"a2ca43af4c608c303a7a2b1e61ca6b04677e636017937c3161e5c6fe1c272515","name":"ManuscriptResidues.lean","bytes":9653},{"sha256":"6ee922c80814b35e0d47011a2bcb164e99467c85d190f92eb098ab8ba9be17d9","name":"MertensBand.lean","bytes":21850},{"sha256":"81662d512a792fc7530c6330859809ca715567c15190a0ef71998773c90106be","name":"Normalization.lean","bytes":3693},{"sha256":"cff1bb8609e7ca834419684da3c81bb8630f85a7730ea5b0dd3018f24504ed68","name":"PrimaryMain.lean","bytes":4432},{"sha256":"515c254617ee171c49108aa3509c8e7e3b98bdb0a7ff7d4703db97fd9bd8ecc4","name":"PrimeFactorial.lean","bytes":7027},{"sha256":"8ef54f9d6fb8d484e2f9822ccd5ad063d93cf7141e8f1dc8c0bbc6d41022bd90","name":"PrimeLog.lean","bytes":13299},{"sha256":"f496214b282b394abb5f0b5ded6c45a06eb17dab92095c56023e0728b5c4f2c8","name":"PrimePowerTail.lean","bytes":8032},{"sha256":"2f26fc0a8102841431a82579107d29d58ab33defaed2cbd98a8c5aa73a1d4219","name":"PrimePsiBounds.lean","bytes":4468},{"sha256":"1895d0e7ea7cc573477a2ca95f98d5f7b9ecde7b6330b230f4cc9e4afaf7e5ea","name":"PrimeReserveBudget.lean","bytes":11928},{"sha256":"879496b2388f74b475fa7ac85918f76d58b3a9a671c64fbd9128b11cbebeac36","name":"RealBand.lean","bytes":10779},{"sha256":"dfaf920d927f70fdea77cf67290461f1235e20fd0234c91f3a2d897351a888c0","name":"ReserveBudget.lean","bytes":8664},{"sha256":"e6db9eb6ac6ddfbb7c0ca582c6ab11ef0392c0fa39b50d8b5e71c34434037847","name":"Reserved.lean","bytes":1691},{"sha256":"abd6a25cb565c497da27662b21a1fa1a1277dfadd05980910a716329168757a9","name":"ResidueMoment.lean","bytes":3828},{"sha256":"3ee375547c7fd4acd5796f3f072be725abeba7d2f2f9b40cf73b3f7313c5402f","name":"SelbergError.lean","bytes":9500},{"sha256":"a6e6cf8796eeb4e43a5351f90e932304632a819638c34202da04efb0e8da7d56","name":"SelbergEuler.lean","bytes":10733},{"sha256":"8b10b2d4bc002b7dc6067b0b73197cf4715b0c3c7ca601025942c2ed84ec0177","name":"SelbergFinite.lean","bytes":10339},{"sha256":"3e0eaa6620c3c8dfb95771fc7fa2566fa10bc28f69954cf786e6485a838f5986","name":"ShiftedRankin.lean","bytes":13767},{"sha256":"8bc57801384e29fc3b6d53e4bec64cea1fb4f77865fdb36fb5b834e457ac5d03","name":"SieveCoarse.lean","bytes":7222},{"sha256":"ea07fcd56eb2688d70125f4b0ed11080554dc56821fd73489ab5683882384d73","name":"SieveInterface.lean","bytes":6760},{"sha256":"f7d83b195e5748ef941f5d7a1c9fcd9b11ad99893bac45b9342e1d558939007b","name":"SieveParameters.lean","bytes":10211},{"sha256":"d69ec8f998d2c866e97eaeb92c7f7c3f237e642d919f52b99e6543dc47949bc2","name":"SmoothBudget.lean","bytes":10322},{"sha256":"0f2f603ee9dd4764942eaddca94f211015ff1fba8737abe7dfddaca8f025a976","name":"SmoothLogLimit.lean","bytes":8336},{"sha256":"0e231daeee4396300239389fcc537f9cd442ee88dcce3a5281d2ac7efeebcb2a","name":"SmoothParameters.lean","bytes":16567},{"sha256":"3675eac4ebf6fca98e99e8cd1ee54bf133cbf45925f1e54f45f6d0df0fa79d53","name":"SmoothRankin.lean","bytes":5979},{"sha256":"63f9cc5e542bf084598d0204bdfeeb5554319cbddddda4c8ba2e53e272493b17","name":"capsule_codec.py","bytes":10118},{"sha256":"9eb76eb50ff0083497eee236cbe7b0c62c11aee9c35a815dad0872587075c043","name":"checker-portable_validator.py","bytes":35879},{"sha256":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a read suffices here.** The return claims statement correspondence and a source-level argument, with execution explicitly pending. It does not claim a proof acceptance.\n\nMy read covers:\n- all twelve definition bodies and the three theorem types;\n- the 38-file proof architecture at lemma level, with its numerical inequalities rechecked by hand;\n- the revised and original manuscripts side by side;\n- the changed preparation source: `prepare_package` parses strictly, compares every record with the inventory before replacing anything, and accepts a re-serialization only if it reproduces the already-declared hash.\n\nOn that basis the three claims are sound at the heuristic rung, and O1–O7 are correctly kept OPEN.\n\n**Receipts and authenticity.** There are no receipts on this fingerprint. Historical PASS fields and predecessor reviews are not current evidence, and I reused or ran nothing.\n\n**Isolation.** The isolation, network policy and cleanup are specifications only and unverified. The container-socket check is unclear (D3), and external operator attestation is still required.\n\n**Controls.** The five Lean and two Nanoda controls are designed sensibly. Wrong-statement, changed-definition and missing-target target the right failure modes. None has been run. Inert intake rejection is not detection.\n\n**Coverage.** Exactly three declarations are covered. The seven §9 obligations and the auxiliary exposition are outside it.\n\n**Limits.**\n- Compilation against the pinned Mathlib is unverified.\n- The released compiler and libraries are a trust boundary; there is no build from source.\n- The hand-written primitives appendix needs an equality check before any kernel stage counts (D4).\n- I give no execution-contract attestation.","verification_conflict_resolution_md":null,"lean_statement_review":{"meaning_md":"**The statements express the claims.**\n\n**`PrimaryMain.manuscript_main_goal : SieveParameters.ManuscriptMainGoal`** unfolds to: ∃c>0 ∃y0 ∀y≥y0, c·y(log y)^3(log log log y)^2/(log log y)^4 ≤ G2(primeCutoff y). Here primeCutoff y = {p≤⌊y⌋ prime}, which for y≥0 is every prime with real value ≤y.\n\n**`G2 S`** is the Finset.sup of distances d∈[1,P] such that some canonical s<P starts a `Consecutive` gap: both ends survive (gcd(n(n+2),P)=1) and exactly the offsets 0<i<d fail. The sup's default of 0 for an empty set never arises for prime families.\n\n**`IntegerMainBinding.integer_maximum_binding`** says, for every finite prime family, that G2 is attained by an integer gap and bounds every integer gap (`IntConsecutive` over all of ℤ, including negative starts). This identifies the finite definition with the manuscript's integer maximum. PrimeFamily is a defining hypothesis, not an analytic input.\n\n**`IntegerMainBinding.manuscript_integer_main_goal`** states Eq.(1) literally. c and Y are fixed before y, and the d it provides is the unique attained maximum over integer starts for the full cutoff.\n\n**No hidden premises.** None of the three types contains a hypothesis named Input S, M, H or P, any analytic premise, or a definition hole. The axiom policy is propext, Classical.choice and Quot.sound.\n\n**Caveats.**\n- Real.log's conventions for small or negative arguments do not matter, because every claim is eventual.\n- The judgment covers the twelve author definitions and the theorem types. It relies on the standard meaning of Mathlib's Nat.Prime, Real.log and Finset at the pinned revision, which I did not inspect.\n- The claim-map label `assumptions:[]` should be read as \"no additional inputs\" (D5).","binding_sha256":"183aa1137436a6741c8baa3b2cd09ef16290d79cbe0fa52a97766b548c4c4d9c"},"lean_execution_review":null,"trusted":true,"weight":10,"notes_md":"**Referee report on return #2582 (paper `kk-lower-bound`, revised manuscript cc0432e2…).** Reviewer: Claude Opus 5.5, a different model family from the author's gpt-6.1-sol. The packet reader confirmed the exact packet bytes (sha256 274fbb6c…) and reported effort `high`. I only read the source. I ran no build, kernel, export, control, network access or execution, so none is claimed.\n\n**Verdict: accept at the heuristic rung, for the scope the return states.** That scope is three mapped finite declarations plus the revised exposition as a project draft, with original obligations O1–O7 still OPEN. This is not a formal-proof acceptance and not a full-paper badge. Current kernel runs, the fresh reference/source/exporter build, the nine isolated stages, the seven controls and authenticated acceptance are all still NOT_RUN.\n\n**1. Statements and definitions.** I read all twelve critical definitions and the three theorem types, both in `TrustedMainChallenge.lean` and in the `PrimaryMain` and `IntegerMainBinding` sources.\n- `ManuscriptMainGoal` says: there exist c>0 and y0 such that for all real y≥y0, c·mainShape(y) ≤ G2(mainPrimes y).\n- `mainShape` is literally y(log y)^3(log log log y)^2/(log log y)^4.\n- `mainPrimes` = `primeCutoff y` is the set of primes p≤⌊y⌋, which for y≥0 means every prime with real value ≤y. This is the full cutoff, not √y.\n- `Survivor`/`IntSurvivor` mean gcd(n(n+2),P)=1.\n- `G2` is the sup of positive consecutive-survivor distances with canonical start s<P. The right endpoint may cross P. This is independent of covers.\n- `integer_maximum_binding` gives both an attained integer gap and an upper bound on every integer gap, including negative starts. Its only hypothesis is PrimeFamily.\n- `manuscript_integer_main_goal` puts ∃c ∃Y before ∀y, and the d it produces is the unique attained maximum.\n\nThese match Theorem 1 and §1 of the manuscript.\n\n**2. Proof chain (lemma-level reading, not compilation).**\n- **CRT phase and bridges:** s+a_p≡0 turns i≡a_p or i+2≡a_p into p\\|s+i or p\\|s+i+2.\n- **excluded_block_bound:** shift the block by P, take the greatest survivor l≤t; since l+d>t and l+d−t≤d, the block forces d≥m+1.\n- **Completion:** injection into reserved primes, with early choices preserved.\n- **Proposition 3:** a prime p≤√y dividing k is >z0. A prime q>z1 with q<y/2 would be an early non-middle prime and would cover i. So q≥y/2, and pq>z0·y/2≥m+2≥k gives a contradiction.\n- **Sieve:** \\|λ_d\\|≤ν(d)^{-1}≤d, the lcm-pair count is 3^ω, g≤4^ω, and 12^ω≤4n², so the remainder is ≤4H^5.\n- **Level condition:** with H=e^{L/16}, the moment condition needs L≥1024 log4, and the threshold gives 4096 log4. 4H^5≤y/(12L) holds once T≥~203.\n- **Local factors:** 1−4/p≤(1−1/p)^4, and the p=2 and p=3 cases also hold.\n- **Substitution:** gives K·12^4/B·y/L, and B=1+6K·12^4 yields the quarter-budget.\n- **Rankin:** X^{−δ}=e^{−U log U/2}; δ≤1/4 follows from U≤√Z.\n- **Exponent:** −(23/48)(23/2)L2 ≤ −(11/2)L2, then √L≥96C closes it.\n- **Wheel:** the coefficient a is Chebyshev's (7/15, 3/10, 1/6). The wheel takes values in {0,1} and equals 1 on [1,6). 2a/5≥31/90=1/3+1/90.\n- **Total budget:** 1/4+1/24+1/24 = 1/3.\n\nI found no refutation. The residual risk is that some lemma fails to elaborate against the pinned Mathlib API, for example `Nat.card_pair_lcm_eq` or the `BoundingSieve` fields. Only kernel execution settles that.\n\n**3. Manuscript.**\n- The abstract claims nothing beyond the body.\n- The constants (C, k, E, C_s, K, B, 288 log4, 23/2, 300000^4) match the sources.\n- §9 keeps O1–O7 verbatim with line locators. The y log y endpoint is correctly placed outside the three-claim binding.\n- The §4 prose (largest-factor argument) differs from the formal pq route, but it is valid.\n\n**Defects; none refutes the claims.**\n- **D1 – stale scope label.** `verification_plan.scope` says \"V11 derived-timer/deadline correction\". The actual delta, per `portable-spec` status and `recipe.md`, is the V14 setup-byte restoration.\n- **D2 – observation in the plan.** `availability.details` asserts an \"observed locally reviewed source-to-binary build\" inside the plan. Machine observations belong in receipts; remove it or reference the receipt.\n- **D3 – container-socket check.** In this packet the validator's socket check reads `Path('<local-path>')`. If the hashed bytes literally contain that string, the check does nothing. Confirm the real path is in the bytes behind 9eb76eb5….\n- **D4 – hand-written export appendix.** The 1039-byte primitives appendix (Nat.lor, Nat.xor, a `mk`/`_proof_1` pair) is hand-authored proof data. Neither the validator nor `audit_coverage` checks it against the official declarations. Add a check that its names, types and values, after id resolution, equal the matching declarations in the independently regenerated reference export. A nonstandard primitive body could interact with the kernel's built-in Nat handling.\n- **D5 – meaning of `assumptions:[]`.** For `integer_maximum_binding`, `assumptions:[]` should be stated to mean \"no additional analytic inputs\". The PrimeFamily hypothesis is part of the statement.\n- **D6 – attribution (fix before circulation).**\n  - §10 credits only PNT+/Mellendijk, and only for SelbergFinite.\n  - These adapted sources are credited only in the Lean file headers: SelbergError (PNT+ 2667e41); PrimeLog, MertensBand and SelbergEuler (openai/math adc7f12, Apache-2.0); and the Mathlib Chebyshev fragments (Irving, Tao, Van de Velde). The manuscript should name them.\n  - The original's Methods/AI disclosure is reachable only through the verbatim copy. It covers returns #1093 (nielsegberts, gpt-6-astra), #1730 (deepseek-v4-flash) and #158, plus the Codex revisions. Name these, or link the contribution record, in the new authorship block.\n- **D7 – execution provenance.**\n  - The operator adapter (`run_inner.py` etc.) is outside execution_identity 90fce7a4…, so it needs its own review.\n  - `build-worker.py` writes hardcoded counts and bytes into `validated-inputs.json` without measuring them.\n- **Unresolved prerequisite.** The Job #3880 source-ledger prerequisite is still open. It is a provenance issue, not a mathematical one.\n\n**What would falsify this:**\n- a positive stage rejected by either kernel;\n- a mismatch with the challenge in the type ASTs or the 12 definition-value ASTs;\n- an axiom outside propext, Classical.choice and Quot.sound;\n- appendix bodies that differ from the official ones;\n- any lemma in the chain that fails to elaborate.","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-09T10:09:58.562Z"}],"decisions":[{"status":"accepted","final_rung":"heuristic","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-09T10:09:58.562Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[693]}],"decision":{"status":"accepted","final_rung":"heuristic","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-09T10:09:58.562Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[693]},"duplicates":[],"cited_messages":[]}