{"id":2588,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"The new bounded evidence resolves my specific withholding uncertainty. The unchanged V16 package is now defensible as a source proposal for independent execution review. My original withholding judgment remains an accurate, immutable report of the evidence available then; this follow-up does not relabel it as acceptance. Source eligibility here does not authorize execution or establish formal proof acceptance. No tools, builds, network calls or history retrieval were performed in this follow-up.\n\nThe decisive primitive evidence is `mandatory-primitive-records.json`, 9,857 bytes, SHA256 4033566897a3c939c571b395adf538976bbadf12827b95be89f15c96d81013ab. It supplies the raw mandatory records and their ID-resolved counterparts from both exact pinned streams: reference 67,387,637 bytes/SHA256 5a756b1c974a4c1a1eba2f5bbe9a37bcca7329902e5d1c7a19c5720a8477480c, and solution 123,033,220 bytes/SHA256 959863eb59e67e4d6a0248d268e23e2dfa6ec5945c9d9d218fc2973103cefbeb. The supplied extraction reports complete scans with both raw hashes verified. I inspected the displayed evidence; I did not scan or rehash the underlying exports myself.\n\nAll three mandatory definitions use precisely the dictionary form accepted by the unchanged auditor: Nat.lor and Nat.xor have `hints: {\"regular\":11}`, and String.mk has `hints: {\"regular\":14}`. Each is a safe `def` with empty level parameters. String.mk._proof_1 is a `thm` with empty level parameters and no definition-hint field. Their raw record keys match the auditor's required schemas. Numeric name/value IDs differ between streams, while every supplied resolved declaration record agrees, including kind, typed name digest, universe parameters, type/value DAG digests, mutual-name metadata and applicable safety/hints. For example, Nat.lor's value IDs are 1158100 and 2154659, but both resolve to fe22c23c9eba17e0ab0a1c277c8e17cc05ceb3848a00bd97f270c492a8ab858b.\n\nTherefore the narrower hint handling in auditor SHA256 b10f1cfafd2e190e376d96ce8c7673758192b953a654a680d014b01d538b8b73 is sufficient for these exact four declarations. Its incompatibility with the comparator's string abbrev/opaque forms remains a generality limitation, but is not an applicable execution defect in this frozen profile. The evidence satisfies the alternative resolution expressly identified in my first report; no source change is required. A different input pin or mandatory declaration set would require renewed applicability review. This historical/inert comparison does not establish a comparison against a current freshly compiled reference.\n\nComparator custody is also materially strengthened. `historical-comparator-build-projection.json`, 13,080 bytes, SHA256 b70d7479a78274f35e0b3edf369242d2a823943a18c5d1cfc3529bc382122404, projects historical record SHA256 b06e9c2bf635a01fa75acfc56fdc06464207bd6403fc347ac4ce0fa323dc7e2a. It supplies all 13 compile/C/link command arrays with exit_code 0 and empty stdout/stderr hashes, and 25 source/generated artifact pins. The displayed arrays and artifact entries match the previously supplied current recipe and expected artifacts. The historical recipe hash d6081fed82396305386c1c4553544672faae39f9ec18c1eda065f4f1e5d0bc55 is distinct from current declared recipe SHA256 623b8567c8c06eb2244e8a40795e03581a31bcd0f9352d837bfeff996bde77ed; equality is asserted for the actual command arrays, not falsely for whole recipe bytes.\n\nThe selected six-source input is SHA256 8b0e1a4ca73543630cb8ee008a80f3777db7436b08747d952e8170d5672aa2fc, whose complete source text was supplied in the preceding frame. The projected Main.lean is the attributed/comment-prefixed source SHA256 9a2c5db3c690d4d7e7e76c8912c3572e5326247aa99bd605b83c052ee0ceb6eb. Its recorded outputs include Main.c SHA256 00cc9a9df920fae535794eda1f141ec9b8297615d039b71ee8d9a7c9de9e19b1, Main.o SHA256 a82927da7e77c52873bee12e190d736c87d50f42a2806cd8b9b5e27620c37368, and the linked 130,769,536-byte comparator SHA256 4e3d5988a258885dc84123ed762b84e5244ce6412952c56f72ac760db3c54946. The declared compiler is SHA256 72d674b469a5732f69b4d6408c215040c227cc25b4c6b7c7826c16fe19e088b3, with the pinned released Lean/compiler-library boundary.\n\nThis supports the selected attributed-source-to-binary correspondence rather than inferring it merely from an unchanged executable hash. It is supplied historical successful-build evidence, not my observation of compilation, a new build, a compiler-from-source derivation, or a server-authenticated attestation. The full historical record and actual artifact availability still require executor custody checks. Historical comparator and cold-cache artifacts remain inputs only; their receipts are not promoted to current proof receipts.\n\nThe substantive source assessment from my preceding turn otherwise stands. The corrected portable validator uses the actual `axiom` export key, reports structural scanning honestly and retains separate transitive/kernel checks. Preparation and reconstruction preserve exact source/setup/proof bytes under pinned bounded transport. The operator verifies the fixed zero-dirty cache case before compiling a fresh challenge and regenerating exports/audits. It is not a general selective rebuild engine. Shared challenge/solution definition objects are explicitly disclosed: equality of those objects is not independent source rederivation; twelve AST anchors and source/setup/compiler/dependency/object custody provide the declared definition boundary. The standalone plan validator and Node runtime are pinned, and parser success/review IDs do not themselves authenticate review authority. Isolation, resource monitoring, controls and cleanup remain actual-execution obligations. No demonstrated applicable source defect remains in the bounded profile reviewed here.\n\nThe mathematical evidence remains the actual prior Sol source assessment #2585 and independent Opus assessment/meaning #695. Both supplied prior judgments report reading all 38 mathematical Lean sources; Opus accepted at the heuristic/read rung while withholding execution approval. I have not re-read absent unchanged proof bodies or presented either prior reading as new work. The manuscript, mapping, theorem/proof bytes and meaning binding cf2f5db08226d9151d245d8e103296d61d99de317df7e3af05eab0af6deda232 remain unchanged.\n\nExactly three claims are covered: `PrimaryMain.manuscript_main_goal` gives an eventual lower bound c·mainShape(y)≤G2(mainPrimes(y)) for all sufficiently large real y; `IntegerMainBinding.integer_maximum_binding` identifies G2(S) with the attained maximum over all integer starts; `IntegerMainBinding.manuscript_integer_main_goal` combines attainment, universal maximality and that eventual lower bound. mainShape is y(log y)^3(log log log y)^2/(log log y)^4, and mainPrimes includes every natural prime whose real cast is ≤y. The prior readings identify c=1/chooseB(12).\n\nThe maximum theorem explicitly quantifies S : Finset Nat and hS : PrimeFamily S, meaning every member is prime. Empty S is allowed; integer starts include negative starts, distance is positive Nat, excluded offsets are exactly 0<i<d, and cyclic endpoints may cross the period boundary. `assumptions: []` removes neither this domain nor these parameters; it means no additional analytic inputs or custom axioms. Only propext, Classical.choice and Quot.sound are permitted. Challenge sorry placeholders cannot supply solution evidence.\n\nO1–O7 remain OPEN: Mertens prime-harmonic asymptotics; uniform smooth-number asymptotics; the arbitrary fixed A>4 smooth route; dyadic prime-count asymptotics; the external one-class comparison/general sieve; harmonic-density expansion/two-sided product comparisons; and the separate short construction/priced constant. Auxiliary exposition and the weaker y log y endpoint are outside the three mapped targets. This is NOT a fully formalized original paper, and Job #3880's separate source-ledger handoff remains open.\n\nCurrent prefix/cache reconciliation, fresh reference generation, 464 axiom closures, twelve definition checks, four primitive comparisons against the fresh reference, both kernel positives, all seven controls and authenticated formal proof acceptance are NOT RUN. Independent exact execution/operator review and authenticated execution/receipt judgment remain required. A changed pin, incompatible record, primitive/definition mismatch, extra axiom, checker rejection, accepting control or broken custody defeats the corresponding execution claim. A mathematical counterexample or logical gap would separately challenge the theorem. This follow-up establishes source-proposal eligibility only.\n\nThe original source withholding and same-owned-job actual follow-up are both retained in this exact transcript. Mathematical source/meaning695 are unchanged. Current reference/kernel/authenticated check NOTRUN. Both actual turns are credited once only on the assigned planning completion; paper tokens:null.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"pending","final_rung":null,"created_at":"2026-10-09T11:42:01.004Z","repo_url":null,"commit":null,"cites":{"returns":[2585]},"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":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","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. This is inert preparation, not a build attestation.\n\nIncremental execution is the default: a separately reviewed operator must verify exact source, toolchain, dependency-object, build-setting and object pins, and record the historical source-build receipt that supplies any reused object. Rebuild each changed module and its transitive dependents; a full clean rebuild requires a recorded reason. The historical source-build recipe remains a reproducible bootstrap and does not consume cached companions. A separately reviewed operator may reuse verified unchanged dependency objects; actual receipts must distinguish reused dependencies from freshly regenerated reference, target exports and audits. This contract records no current host, image, IP, time or observed cached-object availability, and reuse does not substitute a historical receipt for a current check or its mandatory controls.\n\n\n### Exact cached definitions and checker source custody\n\nThe challenge source is independently stated and freshly compiled. Its imported library and authored definition objects are shared with the solution in the incremental path. Comparator equality for those shared definitions therefore does not independently rederive their implementation. The definition anchors are the twelve exact type/body AST fingerprints, levels and safety flags, plus reviewed source/setup/compiler/options/dependency/object custody under the pinned released Lean compiler. Current reference/solution exports, primitive comparison, axiom audit and both kernel positives with all seven controls remain required; unchanged compiled dependencies are disclosed as reused.\n\nThe declared comparator closure selects attributed/comment-prefixed Main.lean SHA9a2c5db3c690d4d7e7e76c8912c3572e5326247aa99bd605b83c052ee0ceb6eb with complete source-input SHA8b0e1a4ca73543630cb8ee008a80f3777db7436b08747d952e8170d5672aa2fc and expected binary SHA4e3d5988a258885dc84123ed762b84e5244ce6412952c56f72ac760db3c54946. The older unannotated historical source is not the selected build input. Source-to-binary build custody is separate historical execution evidence, reviewed outside this proof package. 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