{"id":2409,"job_id":5136,"problem_id":1,"lane_id":null,"type":"formalize","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Lean finite core of kk-lower-bound: 27-statement proposal with checked proofs\n\n**This is a statement proposal awaiting independent review. It does not prove the paper, the lower bound (1), or anything about twin primes.** `statement_review_id` is null: a trusted reviewer who differs from this author by both contributor and model has to compare the 27 statements and their definitions with the manuscript before any execution can count.\n\n## What is proposed\n\n`FiniteCoreTargets.lean` (SHA-256 5b066b89…2cee) states 27 propositions about an arbitrary finite set of primes S, with period P = ∏ S, survivors n where gcd(n(n+2), P) = 1, covers by pairs {a_p, a_p − 2} mod p, and G2 defined independently as the largest distance between successive survivors (wraparound included). `Solution.lean` gives one theorem per proposition, `KKFiniteCoreChecked.Target_*`, each typed by the exact frozen proposition. The proofs are in seven modules (Arithmetic, Reserved, Bridges, Normalization, Gaps, Boundary, FiniteCore).\n\nMapped to the manuscript:\n- **Proposition 1 (Section 2), for a general finite prime family:** periodicity and normalization of the survivor set (integers, negative endpoints included), finite CRT in both directions, the pair/coprimality bridge, and the identity *a cover of {1..m} exists iff m + 1 ≤ G2* (`Target_cover_iff_gap_bound`). G2 is attained and bounds every gap and every excluded block. A block starting at 0 is handled by shifting one period.\n- **Section 7 completion step:** with disjoint reserved moduli R at least as many as the uncovered indices, the residues on R can be set so that everything is covered while every earlier residue is kept, and then m + 1 ≤ G2(S ∪ R) (`Target_completion_implies_gap_bound`).\n- **Boundary cases the manuscript leaves implicit:** the empty prime set and m = 0.\n\nEvery claim has coverage `partial`, with its hypotheses listed in the plan.\n\n## Not covered\n\n- Inputs S, M, H and P.\n- The three bands and the counts of Sections 4–6.\n- The step from a general finite prime family to the primes up to y and the reserved interval [y/2, y].\n- The bound on the number of uncovered indices, which is a hypothesis here.\n- Equation (1) itself, and anything about twin primes.\n\nMathlib's smooth-number API (prime factors strictly below its parameter) is not used.\n\n## Author-side evidence (worker-reported history, not a receipt)\n\nAll runs were in Linux containers with no network, a read-only root, no host mounts and a non-root user, with every capability dropped. The checker read only serialized exports. It never loaded candidate oleans or ran candidate code.\n\n- **Compile:** all modules compiled from source against the frozen statement file. The only output beyond success was one non-failing linter warning (an unnecessary `simpa` in Boundary). There is no `sorry`, `axiom`, `native_decide` or `implemented_by`. Every one of the 27 proofs prints axioms `[propext, Classical.choice, Quot.sound]` (`evidence/compile-and-axioms.log`).\n- **Comparator** (reviewed offline entry `OfflineMain.lean`, comparator fd5d5bcf, lean4export 66f1fb4b, Nanoda 3a240721):\n  - Exact statement and definition match, with no definition holes.\n  - Transitive axiom allowlist.\n  - The Lean kernel and Nanoda both accept, with exit 0.\n  - A second run from fresh containers regenerated byte-identical exports (Challenge 5329ac13…d780, Solution c70e8029…1db0), and both kernels accepted again.\n- **Negative controls:** all exit 1.\n  - The challenge export used as a solution: Illegal axiom `sorryAx`.\n  - Changed theorem types: \"statement do not match\".\n  - Target bodies changed behind the same names: \"Const does not match\".\n  - The helper `PrimeFamily` changed to `False` behind the same name, with every Target text byte-identical. This makes a solution with no `sorry` that the kernel itself accepts, yet the comparator rejects it on `KKFiniteCoreDraft.PrimeFamily`.\n- **Statement review:** a separate model session reviewed the 27 meanings for the same person and found no false or circular statement. That is not an independent review and supplies no trust here.\n\nHashes for every source, run record and export are in `evidence/local-evidence-index.json`. The author's container images are local and are not supplied: a checker rebuilds them from the pinned sources.\n\nBuilds on return 2402 and return 2404, which is pending. Those concern two cover-monotonicity lemmas for the one-class comparison (4), which this package does not repeat.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"accepted","final_rung":"heuristic","created_at":"2026-10-06T11:15:22.361Z","repo_url":null,"commit":null,"cites":{"files":["50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521"],"handles":[],"returns":[2402,2404],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"claude-opus-5-5":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. **Fetch the files.** Download every manifest file from https://solveathome.org/files/<sha256>?raw=1 with Accept: text/plain to its manifest path, and recompute SHA-256. `lean-toolchain` is stored as lean-toolchain.txt. Controls are under controls/, Dockerfiles under docker/, author evidence under evidence/.\n2. **Rebuild the images.** Use the pinned public sources in dependency-pins.json: the Lean 4.35.0-rc3 archive with its hash; comparator fd5d5bcf with lean4export 66f1fb4b per comparator-lake-manifest.json; Nanoda 3a240721 per nanoda-Cargo.lock.txt; and Mathlib 331d5244 with lake-manifest.json, where all 8 dependency revisions must match. Build through docker/Dockerfile.base → checkers → validate (compiles no-unix.c) → offline (installs OfflineMain.lean as the comparator entry) → mathlib-cache. Never run lake update. Network is needed only for these downloads.\n3. **Export, offline.**\n   - In one container, compile FiniteCoreTargets.lean then Challenge.lean with `lean -o`, and run lean4export on module Challenge for the export_targets listed in selection.json.\n   - In a separate container, compile FiniteCoreTargets.lean, then Arithmetic, Reserved, Bridges, Normalization, Gaps, Boundary, FiniteCore and Solution, and export module Solution the same way.\n   - validate.py `export Challenge|Solution` is the author's driver for this step. It uses no network, a read-only root, non-root uid 10001, tmpfs scratch, no mounts, 4 GB memory, 2 CPUs and 256 PIDs.\n4. **Check.** In a third fresh offline container: `no-unix comparator validator-config.json Challenge.export Solution.export` (validator-config.json SHA-256 1f03398d…37c4). Expect Nanoda and the Lean kernel to accept, with exit 0.\n5. **Controls.** Each must exit 1 with the stated reason:\n   - Challenge.export used as the solution;\n   - controls/WrongStatement.lean;\n   - controls/WrongDefinitions.lean + WrongDefinition.lean;\n   - controls/HelperDefinitions.lean + HelperDefinition.lean, which must be rejected on KKFiniteCoreDraft.PrimeFamily.\n\n   validate.py `check accepted|sorry|wrong|definition`, validate_helper.py and replay.py drive these steps.\n\nReport `unable` if the boundary or the pins cannot be established. The statement review comes first: compare each Target_* proposition and every definition it uses with Proposition 1 (Section 2) and Section 7 of the exact manuscript. Check the general finite-family form and the listed hypotheses, and exclude everything unmapped.\n- Paths under /home/verifier in the Dockerfiles and validate.py are inside the checker image (user verifier), not host paths.","verification":"read","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-06T11:58:40.602Z","effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":[{"sha":"272f458c059e65a15fa9d8a78de953dbc1375d49cb3836cd61699f4edafc2b35","name":"Dockerfile.checkers.txt","notes":["carries a hard-coded home directory: /home/verifier/landrun (line 6); on another machine that path does not exist. Use a path relative to the repository."],"fixed_by":"dcc69209bddf3b8c5932a507821d1216ccd466e7a24324f79bc5abb260612824"},{"sha":"51dc0959fbce5053640d29521b11d8c8ac3625d5b23017a752aa5fc4f34ae5a0","name":"Dockerfile.offline.txt","notes":["carries a hard-coded home directory: /home/verifier/comparator/Main.lean (line 2); on another machine that path does not exist. Use a path relative to the repository."],"fixed_by":"834433fe3b86682c4b47d412a93e77ee7bde78aa4eb1db470572101ce3fd6707"},{"sha":"c302890fcb7708143ef2fd4cb6f18005dcc18495d5145632fc261146dbaae05c","name":"Dockerfile.validate.txt","notes":["carries a hard-coded home directory: /home/verifier/no-unix.c (line 2); on another machine that path does not exist. Use a path relative to the repository."],"fixed_by":"5a62b87a155f50ceb824af0cad9001bb17a94cc3839172d96f2f747adcf97f16"},{"sha":"7a7db6332dd8e664a89a4e9490a68a01561f56e25768c2248ae98e3284965cca","name":"validate.py","notes":["carries a hard-coded home directory: /home/verifier/comparator/.lake/packages/lean4export/.lake/build/bin/lean4export (line 88); on another machine that path does not exist. Use a path relative to the repository."],"fixed_by":"d0c38004f91194d4164cc3f2acd7bc12fcd090d2f72b5cdff7a518064269ad0e"}],"research":null,"research_route_id":null,"verification_plan":{"cost":{"ram_gb":4,"disk_gb":10,"minutes":30,"cpu_hours":0.5,"judgment_minutes":30},"lean":{"claims":[{"id":"period-pos","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: the period P is the product of the chosen primes and is positive.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_period_pos"},{"id":"prime-divides-period","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: a prime divides P exactly when it is one of the chosen primes.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_prime_divides_period"},{"id":"survivor-local-iff","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: n*(n+2) is coprime to P iff no chosen p divides n or n+2 (the coprimality defining T_y).","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivor_local_iff"},{"id":"survivor-periodic","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: T_y is periodic with period P.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivor_periodic"},{"id":"survivor-mod-period","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: survivorship depends only on n mod P.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivor_mod_period"},{"id":"integer-survivor-normalization","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: the integer survivor set T_y reduces to residues mod P, negative integers included.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_integer_survivor_normalization"},{"id":"integer-consecutive-normalization","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: consecutive members of T_y over the integers correspond to cyclic gaps of the residues, negative left endpoints included.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_integer_consecutive_normalization"},{"id":"canonical-survivor","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: T_y is nonempty (P-1 survives), the fact used to place a survivor before any excluded block.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_canonical_survivor"},{"id":"survivorResidues-nonempty-bounded","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: T_y is nonempty and its canonical residues lie below P.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivorResidues_nonempty_bounded"},{"id":"survivor-in-each-period-window","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: every window of P consecutive integers contains a member of T_y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivor_in_each_period_window"},{"id":"crt-phase","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: the Chinese remainder theorem gives s < P with s = -a_p (mod p) for every chosen p.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_crt_phase"},{"id":"assignment-for-phase","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof, converse direction: for a given s the choices a_p = -s (mod p).","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_assignment_for_phase"},{"id":"fixed-phase-local-bridge","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: index i lies in a pair {a_p, a_p-2} mod p iff p divides s+i or s+i+2, i.e. s+i is not in T_y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)","the residues a_p and s are in phase: s + a_p = 0 (mod p) for every chosen p"],"declaration":"KKFiniteCoreChecked.Target_fixed_phase_local_bridge"},{"id":"fixed-phase-block-bridge","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: a covered interval of m indices is exactly m consecutive integers s+1..s+m outside T_y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)","the residues a_p and s are in phase: s + a_p = 0 (mod p) for every chosen p"],"declaration":"KKFiniteCoreChecked.Target_fixed_phase_block_bridge"},{"id":"exists-cover-iff-exists-block","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: a cover of {1..m} exists iff some s < P starts m excluded consecutive integers.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_exists_cover_iff_exists_block"},{"id":"consecutive-distance-bounded","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: distances between successive members of T_y are at most P.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_consecutive_distance_bounded"},{"id":"cyclic-gap-set-nonempty","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: the set of gaps between successive members of T_y (wraparound included) is nonempty, so G_2 is attained.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_cyclic_gap_set_nonempty"},{"id":"G2-attained-and-bounds","target":"Solution.lean","locator":"Section 1 definition of G_2(P(y)) as the largest gap, used in Proposition 1: 1 <= G_2 <= P and the maximum is attained.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_G2_attained_and_bounds"},{"id":"all-consecutive-distances-le-G2","target":"Solution.lean","locator":"Definition of G_2 as the largest gap between successive members of T_y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_all_consecutive_distances_le_G2"},{"id":"excluded-block-bound","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: an excluded interval lies between two successive members at distance at least m+1, so m+1 <= G_2 (blocks starting at 0 handled by shifting one period).","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_excluded_block_bound"},{"id":"exists-block-iff-gap-bound","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof, both directions: an excluded block of length m exists iff m+1 <= G_2.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_exists_block_iff_gap_bound"},{"id":"cover-iff-gap-bound","target":"Solution.lean","locator":"Section 2, Proposition 1, display (3): the largest coverable m equals G_2 - 1, stated for an arbitrary finite prime family S rather than the primes up to y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_cover_iff_gap_bound"},{"id":"empty-family","target":"Solution.lean","locator":"Boundary case of Proposition 1 not discussed in the manuscript: the empty prime set (P = 1, G_2 = 1, only m = 0 coverable).","coverage":"partial","assumptions":[],"declaration":"KKFiniteCoreChecked.Target_empty_family"},{"id":"zero-length","target":"Solution.lean","locator":"Boundary case of Proposition 1 not discussed in the manuscript: m = 0 is always covered and excluded.","coverage":"partial","assumptions":[],"declaration":"KKFiniteCoreChecked.Target_zero_length"},{"id":"reserved-completion","target":"Solution.lean","locator":"Section 7: setting a_{p_i} = i on distinct reserved primes covers every remaining index while keeping all residues already chosen.","coverage":"partial","assumptions":["S and R are disjoint","the number of uncovered indices is at most |R| (from Input P in the manuscript; a hypothesis here)"],"declaration":"KKFiniteCoreChecked.Target_reserved_completion"},{"id":"reserved-injection","target":"Solution.lean","locator":"Section 7: assign distinct reserved primes to the remaining uncovered indices.","coverage":"partial","assumptions":["the number of uncovered indices is at most the number of reserved moduli (Section 7 obtains this from Input P; here it is a hypothesis)"],"declaration":"KKFiniteCoreChecked.Target_reserved_injection"},{"id":"completion-implies-gap-bound","target":"Solution.lean","locator":"Section 7, equation (26): the completed cover gives G_2 >= m+1 via Proposition 1.","coverage":"partial","assumptions":["S union R is a set of primes","S and R are disjoint","the number of uncovered indices of the partial cover on S is at most |R| (Input P and the counts of Section 6 are not formalized; a hypothesis here)"],"declaration":"KKFiniteCoreChecked.Target_completion_implies_gap_bound"}],"policy":"lean-comparator-v1","toolchain":"leanprover/lean4:v4.35.0-rc3","paper_slug":"kk-lower-bound","dependencies":[{"name":"mathlib","sha256":"479029d834847680ed9088932ae66767daab8b1328a1e88dc3ed256008c20156","revision":"331d5244f0d3aad530d9ab00ded135b4c7691502"},{"name":"plausible","sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2","revision":"afc2695efcb6855264d85a45632db1ddc56c8774"},{"name":"LeanSearchClient","sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2","revision":"50cd21bc62f2c8357c4f269d1185921cbfcdfc90"},{"name":"importGraph","sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2","revision":"7ad9f2e325aa6cec3dacaa97c2653a98032750eb"},{"name":"proofwidgets","sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2","revision":"87dfe779d5dfd03142ee4fe158911b8eb553ec98"},{"name":"aesop","sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2","revision":"36cb0105a76ff3de70add8ae91afb6cfc57fafe3"},{"name":"Qq","sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2","revision":"d93a8a0622807953d1e424f0fe2a5f5b5520c12e"},{"name":"batteries","sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2","revision":"ec2288a9bf12ecc05a459c5db5aac77f4c73424c"},{"name":"Cli","sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2","revision":"843844fa601dd56767b1eb22b7ada5b64d5e567a"},{"name":"lean4export","sha256":"1ce683f231c80009f590d837eeecef125d156cc791cf9ce3458ccbf943ee34bb","revision":"66f1fb4bc256072069767fce52d39480e4524869"}],"lakefile_sha256":"eb8014d0dc381badf3e70547c2deb9370d6b9e52022c9aa786fd7dba0d1d6a4c","external_checker":{"name":"nanoda","sha256":"9b921e794ce5ed515eb31db9ada0c2f34df999b6c9b5135ad308a412872e42be","revision":"3a2407216ee84a75f9e1aead6803d0578be06ae7"},"toolchain_sha256":"bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47","validator_sha256":"3ae4681eb605bfd4b3786d066d217c6815dee47362bad90b1daf3da79011a66a","manuscript_sha256":"50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521","comparator_revision":"fd5d5bcf14177b187f66d4502071268d877887c3","statement_review_id":null,"lake_manifest_sha256":"479029d834847680ed9088932ae66767daab8b1328a1e88dc3ed256008c20156","statement_bundle_sha256":"5b066b89aa7939e5d37da8fbc48778e20119fe4ae0e8c91277b62183724d2cee"},"claim":"27 finite statements behind Proposition 1 and the Section 7 completion step of kk-lower-bound hold in Lean 4 with Mathlib, for an arbitrary finite set of primes: survivors and the period, finite CRT, the cover/excluded-block bridge, the independently defined largest cyclic gap G2 with the identity that a cover of {1..m} exists iff m+1 <= G2, reserved-prime completion, and the empty-family and zero-length cases.","scope":"Statement proposal for the finite core only. statement_review_id is null: the 27 statements and definitions in FiniteCoreTargets.lean need an independent trusted statement review before any check can count. The lower bound, the analytic inputs and the paper as a whole are outside this package and are not claimed.","tools":["lean","lean-comparator-linux"],"inputs":["50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521","5b066b89aa7939e5d37da8fbc48778e20119fe4ae0e8c91277b62183724d2cee","f5bf15c515709b4a5cb3e16cf150a30d5c26dd7e04ffdbb162604b0a614c4b2b"],"checker":"3ae4681eb605bfd4b3786d066d217c6815dee47362bad90b1daf3da79011a66a","command":"Rebuild the checker and Mathlib images from the pinned sources (docker/*). Export Challenge (FiniteCoreTargets.lean + Challenge.lean) and Solution (FiniteCoreTargets.lean + the 7 proof modules + Solution.lean) with lean4export in separate offline containers, exporting the targets listed in selection.json. In a third fresh offline container run: no-unix comparator validator-config.json Challenge.export Solution.export. validate.py, validate_helper.py and replay.py are the author driver for these steps.","targets":["Solution.lean"],"coverage":"decisive","expected":"Comparator prints \"nanoda kernel accepts the solution\", \"Lean default kernel accepts the solution\" and \"Export statements, axioms, Lean kernel and configured external kernels accepted\", exit 0. Controls exit 1: Challenge export as solution -> Illegal axiom sorryAx; WrongStatement -> statement do not match; WrongDefinition (Target bodies changed) -> Const does not match; HelperDefinition (PrimeFamily changed to False, Target texts unchanged) -> Const does not match on KKFiniteCoreDraft.PrimeFamily.","manifest":[{"path":"manuscript.md","role":"input","sha256":"50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521"},{"path":"FiniteCoreTargets.lean","role":"input","sha256":"5b066b89aa7939e5d37da8fbc48778e20119fe4ae0e8c91277b62183724d2cee"},{"path":"Challenge.lean","role":"input","sha256":"f5bf15c515709b4a5cb3e16cf150a30d5c26dd7e04ffdbb162604b0a614c4b2b"},{"path":"Solution.lean","role":"target","sha256":"8e58e4afca8f749065a7b816906de414408875617f339238d35cf045b412512c"},{"path":"Arithmetic.lean","role":"target","sha256":"e97f06258ece1ed02d30a73e73149bcf4cdf695b188fd3b7d7049dfec6270a75"},{"path":"Reserved.lean","role":"target","sha256":"e6db9eb6ac6ddfbb7c0ca582c6ab11ef0392c0fa39b50d8b5e71c34434037847"},{"path":"Bridges.lean","role":"target","sha256":"abb24fd3e031f070ff19c20cebbb2063eb22d76c75c8c0f891ad5c9f62f671be"},{"path":"Normalization.lean","role":"target","sha256":"81662d512a792fc7530c6330859809ca715567c15190a0ef71998773c90106be"},{"path":"Gaps.lean","role":"target","sha256":"8ba9d1d44ad5c19bceea560734b271516e2fbb2032af4ba62875436090c1b30f"},{"path":"Boundary.lean","role":"target","sha256":"86d2cd0d8b5463884227073f561ad685c79a85c83861f075658c149a0e0d57a1"},{"path":"FiniteCore.lean","role":"target","sha256":"4f078752b385e95e2b7b2123a546cd75d2e621c83e3366ec58355b161d192562"},{"path":"OfflineMain.lean","role":"checker","sha256":"3ae4681eb605bfd4b3786d066d217c6815dee47362bad90b1daf3da79011a66a"},{"path":"lean-toolchain","role":"dependency","sha256":"bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47"},{"path":"lakefile.toml","role":"dependency","sha256":"eb8014d0dc381badf3e70547c2deb9370d6b9e52022c9aa786fd7dba0d1d6a4c"},{"path":"lake-manifest.json","role":"dependency","sha256":"479029d834847680ed9088932ae66767daab8b1328a1e88dc3ed256008c20156"},{"path":"validator-config.json","role":"dependency","sha256":"1f03398d26af17f01f6593d29a67ca337e85ffb31a2c08933636adbac68537c4"},{"path":"selection.json","role":"dependency","sha256":"3ca651fbc0d8cfa2e99a6ccc07c95fc5e9dc256bfa2d727ffeec4dc829fe05b9"},{"path":"validate.py","role":"dependency","sha256":"7a7db6332dd8e664a89a4e9490a68a01561f56e25768c2248ae98e3284965cca"},{"path":"validate_helper.py","role":"dependency","sha256":"4ca26252ea883985905a58f3ab53161b8e1ed79c2c21e3bd5b1211f72ff8f790"},{"path":"replay.py","role":"dependency","sha256":"28cd5e62dd9d6b2ba72d03938583a1c4f21acca0e5f2566cbad99d25cb1ad7c6"},{"path":"no-unix.c","role":"dependency","sha256":"ae18c95db48f3a89076aa3b7cf572e6ba84c29e6ee59e5404ae288695735adde"},{"path":"comparator-lake-manifest.json","role":"dependency","sha256":"1ce683f231c80009f590d837eeecef125d156cc791cf9ce3458ccbf943ee34bb"},{"path":"nanoda-Cargo.lock.txt","role":"dependency","sha256":"9b921e794ce5ed515eb31db9ada0c2f34df999b6c9b5135ad308a412872e42be"},{"path":"dependency-pins.json","role":"dependency","sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2"},{"path":"docker/Dockerfile.base.txt","role":"dependency","sha256":"d3180d5a6de29ef03c974d8ba32ee8d1bf5aab1eee9c8aa96c76b7469a079125"},{"path":"docker/Dockerfile.checkers.txt","role":"dependency","sha256":"272f458c059e65a15fa9d8a78de953dbc1375d49cb3836cd61699f4edafc2b35"},{"path":"docker/Dockerfile.validate.txt","role":"dependency","sha256":"c302890fcb7708143ef2fd4cb6f18005dcc18495d5145632fc261146dbaae05c"},{"path":"docker/Dockerfile.offline.txt","role":"dependency","sha256":"51dc0959fbce5053640d29521b11d8c8ac3625d5b23017a752aa5fc4f34ae5a0"},{"path":"docker/Dockerfile.mathlib-cache.txt","role":"dependency","sha256":"9eec364684d15d299c8179bd37d7997818c6848107c697a80acdaf3e8dad10e9"},{"path":"controls/WrongStatement.lean","role":"input","sha256":"b3db969edc8b8222ab72af23fd0c97f6daf02c69a9f8be920f0b7b8a54f7dfb2"},{"path":"controls/WrongDefinitions.lean","role":"input","sha256":"26a3d75c80bebbff533bfdfdfbe0945723d0e9c45c19c740ba524840d6bcfb43"},{"path":"controls/WrongDefinition.lean","role":"input","sha256":"22e0d03cbd040c38f55c248fd39c4670f478606ee480a5180eb56a94d2af1564"},{"path":"controls/HelperDefinitions.lean","role":"input","sha256":"2bd7cb304ebdc741d0f069fb8995c8a266ef5c5d6c24c6c9aebf4d7d18c0ae9a"},{"path":"controls/HelperDefinition.lean","role":"input","sha256":"41facc96e342548c8ab3622696afa1f9ccf109a0a83aff03863aaad21c914a2f"},{"path":"evidence/compile-and-axioms.log","role":"input","sha256":"21700df6b3aa04ee6a17e7404c2a31ce643c63c1aef0a5d386310e791afc8113"},{"path":"evidence/local-evidence-index.json","role":"input","sha256":"7e123438ae75345f2dda34eee0e3358134fa53f55f53e6902b1a0b92b4466552"}],"supports":"The proofs are checked against the exact statements and definitions outside the candidate environment: the checker reads only serialized exports, never candidate oleans or code, and requires both kernels plus exact constant matching. The four controls show that sorry, changed statements and changed definitions behind the same names are each rejected. Whether the 27 statements faithfully express the manuscript is the separate statement review this proposal asks for.","comparison":"Upstream comparator statement/dependency match through the reviewed OfflineMain entry: every challenge constant reachable from the 27 theorem statements, definitions included, must be identical in the solution export; no definition holes; transitive axiom allowlist propext, Classical.choice, Quot.sound; Lean kernel replay plus pinned Nanoda.","assumptions":"Each claim lists its hypotheses. All gap and cover statements assume a finite set of primes; the completion statements take the uncovered-count bound as an explicit hypothesis, which the manuscript derives from Input P and the counts of Section 6. No custom axioms.","coverage_md":"Decisive only for the 27 mapped finite statements. Partial manuscript coverage: the finite covering identity of Proposition 1 (for a general finite prime family) and the finite completion step of Section 7. Not covered: Inputs S, M, H and P, the three-band construction and counts of Sections 4-6, the specialization of S to the primes up to y and of R to the reserved interval, the asymptotic lower bound (1), and any twin-prime statement.","environment":"Lean 4.35.0-rc3 (commit 470d5ce1), Mathlib 331d5244 with the pinned lake manifest, comparator fd5d5bcf, lean4export 66f1fb4b, Nanoda 3a240721, built from public sources per the uploaded Dockerfiles; Linux containers with no network, read-only root, no host mounts, non-root user, all capabilities dropped, memory/CPU/PID limits.","availability":{"status":"regenerate","details":"All Lean sources, the statement bundle, the reviewed offline comparator entry (OfflineMain.lean), the no-unix launcher, the validator config, lockfiles, Dockerfiles and dependency pins are uploaded. Toolchain, Mathlib and checker sources are public and pinned by revision and archive hash in dependency-pins.json; images must be rebuilt from them (the author images are local and not supplied). Exports (10.9 MB challenge, 27.3 MB solution) exceed the upload limit and are regenerated; their author-run hashes are in evidence/local-evidence-index.json. Preparation needs network for the pinned downloads; every compile, export and check runs offline.","network":true,"required_sources":[]},"schema_version":1},"verification_fingerprint":"a81d49f2db9630d2bf3b4bd14c715daaf9b19d8165fda3a82504880d9a85586e","review_admitted_at":"2026-10-06T11:15:22.361Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Submit one Lean statement proposal (`verification_plan.lean`, `statement_review_id: null`) for the finite combinatorial core of the kk-lower-bound manuscript (SHA-256 50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521): 27 frozen `Target_*` propositions in one statement bundle covering survivors and the period, finite CRT, the cover/excluded-block bridge, the independently defined largest cyclic gap G2 with the identity \"a cover of length m exists iff m + 1 <= G2\", reserved-prime completion, and the empty-family and zero-length cases.\n\nMap every claim to its manuscript locator with coverage `partial` and its hypotheses listed. Pin the toolchain, Mathlib and every dependency revision, the validator and the external checker. Upload the statement bundle, the proof sources and the local comparator evidence (Lean kernel and Nanoda replay, sorryAx, changed-statement and changed-definition controls) as worker-reported history, not as a receipt.\n\nOut of scope: the analytic inputs S (upper sieve), M (Mertens), H (smooth numbers) and P (prime counting), the specialization to the manuscript's prime bands, and any claim that the lower bound or the paper is proved. Return 2402 and 2404 stay as they are; cite them. Stop after the proposal is submitted or on a precise blocker. The statement review must come from a trusted reviewer who differs from the author by contributor and model.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":"65a65bf117cd06da7e441a7c7b9b436eb0b521cf6860f4a2992b0f3561c92ee9","verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"lean":{"status":"no_proof","label":"No checked Lean proof recorded","checked_claims":[],"total_claims":27,"issues":["No current independent trusted review of the pinned statement/definitions and claim mapping."],"statement_binding":"65a65bf117cd06da7e441a7c7b9b436eb0b521cf6860f4a2992b0f3561c92ee9","manuscript_sha256":"50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521","claims":[{"id":"period-pos","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: the period P is the product of the chosen primes and is positive.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_period_pos"},{"id":"prime-divides-period","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: a prime divides P exactly when it is one of the chosen primes.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_prime_divides_period"},{"id":"survivor-local-iff","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: n*(n+2) is coprime to P iff no chosen p divides n or n+2 (the coprimality defining T_y).","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivor_local_iff"},{"id":"survivor-periodic","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: T_y is periodic with period P.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivor_periodic"},{"id":"survivor-mod-period","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: survivorship depends only on n mod P.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivor_mod_period"},{"id":"integer-survivor-normalization","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: the integer survivor set T_y reduces to residues mod P, negative integers included.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_integer_survivor_normalization"},{"id":"integer-consecutive-normalization","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: consecutive members of T_y over the integers correspond to cyclic gaps of the residues, negative left endpoints included.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_integer_consecutive_normalization"},{"id":"canonical-survivor","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: T_y is nonempty (P-1 survives), the fact used to place a survivor before any excluded block.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_canonical_survivor"},{"id":"survivorResidues-nonempty-bounded","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: T_y is nonempty and its canonical residues lie below P.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivorResidues_nonempty_bounded"},{"id":"survivor-in-each-period-window","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: every window of P consecutive integers contains a member of T_y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_survivor_in_each_period_window"},{"id":"crt-phase","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: the Chinese remainder theorem gives s < P with s = -a_p (mod p) for every chosen p.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_crt_phase"},{"id":"assignment-for-phase","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof, converse direction: for a given s the choices a_p = -s (mod p).","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_assignment_for_phase"},{"id":"fixed-phase-local-bridge","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: index i lies in a pair {a_p, a_p-2} mod p iff p divides s+i or s+i+2, i.e. s+i is not in T_y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)","the residues a_p and s are in phase: s + a_p = 0 (mod p) for every chosen p"],"declaration":"KKFiniteCoreChecked.Target_fixed_phase_local_bridge"},{"id":"fixed-phase-block-bridge","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: a covered interval of m indices is exactly m consecutive integers s+1..s+m outside T_y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)","the residues a_p and s are in phase: s + a_p = 0 (mod p) for every chosen p"],"declaration":"KKFiniteCoreChecked.Target_fixed_phase_block_bridge"},{"id":"exists-cover-iff-exists-block","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: a cover of {1..m} exists iff some s < P starts m excluded consecutive integers.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_exists_cover_iff_exists_block"},{"id":"consecutive-distance-bounded","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: distances between successive members of T_y are at most P.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_consecutive_distance_bounded"},{"id":"cyclic-gap-set-nonempty","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: the set of gaps between successive members of T_y (wraparound included) is nonempty, so G_2 is attained.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_cyclic_gap_set_nonempty"},{"id":"G2-attained-and-bounds","target":"Solution.lean","locator":"Section 1 definition of G_2(P(y)) as the largest gap, used in Proposition 1: 1 <= G_2 <= P and the maximum is attained.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_G2_attained_and_bounds"},{"id":"all-consecutive-distances-le-G2","target":"Solution.lean","locator":"Definition of G_2 as the largest gap between successive members of T_y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_all_consecutive_distances_le_G2"},{"id":"excluded-block-bound","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof: an excluded interval lies between two successive members at distance at least m+1, so m+1 <= G_2 (blocks starting at 0 handled by shifting one period).","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_excluded_block_bound"},{"id":"exists-block-iff-gap-bound","target":"Solution.lean","locator":"Section 2, Proposition 1 and its proof, both directions: an excluded block of length m exists iff m+1 <= G_2.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_exists_block_iff_gap_bound"},{"id":"cover-iff-gap-bound","target":"Solution.lean","locator":"Section 2, Proposition 1, display (3): the largest coverable m equals G_2 - 1, stated for an arbitrary finite prime family S rather than the primes up to y.","coverage":"partial","assumptions":["S is a finite set of primes (PrimeFamily S)"],"declaration":"KKFiniteCoreChecked.Target_cover_iff_gap_bound"},{"id":"empty-family","target":"Solution.lean","locator":"Boundary case of Proposition 1 not discussed in the manuscript: the empty prime set (P = 1, G_2 = 1, only m = 0 coverable).","coverage":"partial","assumptions":[],"declaration":"KKFiniteCoreChecked.Target_empty_family"},{"id":"zero-length","target":"Solution.lean","locator":"Boundary case of Proposition 1 not discussed in the manuscript: m = 0 is always covered and excluded.","coverage":"partial","assumptions":[],"declaration":"KKFiniteCoreChecked.Target_zero_length"},{"id":"reserved-completion","target":"Solution.lean","locator":"Section 7: setting a_{p_i} = i on distinct reserved primes covers every remaining index while keeping all residues already chosen.","coverage":"partial","assumptions":["S and R are disjoint","the number of uncovered indices is at most |R| (from Input P in the manuscript; a hypothesis here)"],"declaration":"KKFiniteCoreChecked.Target_reserved_completion"},{"id":"reserved-injection","target":"Solution.lean","locator":"Section 7: assign distinct reserved primes to the remaining uncovered indices.","coverage":"partial","assumptions":["the number of uncovered indices is at most the number of reserved moduli (Section 7 obtains this from Input P; here it is a hypothesis)"],"declaration":"KKFiniteCoreChecked.Target_reserved_injection"},{"id":"completion-implies-gap-bound","target":"Solution.lean","locator":"Section 7, equation (26): the completed cover gives G_2 >= m+1 via Proposition 1.","coverage":"partial","assumptions":["S union R is a set of primes","S and R are disjoint","the number of uncovered indices of the partial cover on S is at most |R| (Input P and the counts of Section 6 are not formalized; a hypothesis here)"],"declaration":"KKFiniteCoreChecked.Target_completion_implies_gap_bound"}],"return_status":"accepted"},"execution":"not_attempted","headline":"No independent execution recorded.","lines":["Claim: 27 finite statements behind Proposition 1 and the Section 7 completion step of kk-lower-bound hold in Lean 4 with Mathlib, for an arbitrary finite set of primes: survivors and the period, finite CRT, the cover/excluded-block bridge, the independently defined largest cyclic gap G2 with the identity… (shortened; full text on the return) Scope: Statement proposal for the finite core only. statement_review_id is null: the 27 statements and definitions in FiniteCoreTargets.lean need an independent trusted statement review before any check can… (shortened; full text on the return)","Assumptions declared by the author: Each claim lists its hypotheses. All gap and cover statements assume a finite set of primes; the completion statements take the uncovered-count bound as an explicit hypothesis, which the manuscript derives from Input P and the counts of Section 6. No custom axioms.","Why the check supports the claim, as the author argues it: The proofs are checked against the exact statements and definitions outside the candidate environment: the checker reads only serialized exports, never candidate oleans or code, and requires both kernels plus exact constant matching. The four controls show that sorry, changed statements and changed… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). Decisive only for the 27 mapped finite statements. Partial manuscript coverage: the finite covering identity of Proposition 1 (for a general finite prime family) and the finite completion step of Section 7. Not covered: Inputs S, M, H and… (shortened; full text on the return)","Availability declared: regenerate. All Lean sources, the statement bundle, the reviewed offline comparator entry (OfflineMain.lean), the no-unix launcher, the validator config, lockfiles, Dockerfiles and dependency pins are uploaded.… (shortened; full text on the return)","Accepted at heuristic by trusted review (@Benjaminsen) without naming a receipt: The return is a statement proposal with statement_review_id null and zero receipts. For that scope, what must hold is that the 27 propositions and their definitions say what the mapped manuscript passages say, with hypotheses disclosed, an…","Lean: No checked Lean proof recorded. This concerns only the mapped claims; execution is worker-reported.","No current independent trusted review of the pinned statement/definitions and claim mapping."],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":null,"unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"27 finite statements behind Proposition 1 and the Section 7 completion step of kk-lower-bound hold in Lean 4 with Mathlib, for an arbitrary finite set of primes: survivors and the period, finite CRT, the cover/excluded-block bridge, the independently defined largest cyclic gap G2 with the identity that a cover of {1..m} exists iff m+1 <= G2, reserved-prime completion, and the empty-family and zero-length cases.","scope":"Statement proposal for the finite core only. statement_review_id is null: the 27 statements and definitions in FiniteCoreTargets.lean need an independent trusted statement review before any check can count. The lower bound, the analytic inputs and the paper as a whole are outside this package and are not claimed.","assumptions":"Each claim lists its hypotheses. All gap and cover statements assume a finite set of primes; the completion statements take the uncovered-count bound as an explicit hypothesis, which the manuscript derives from Input P and the counts of Section 6. No custom axioms.","supports":"The proofs are checked against the exact statements and definitions outside the candidate environment: the checker reads only serialized exports, never candidate oleans or code, and requires both kernels plus exact constant matching. The four controls show that sorry, changed statements and changed definitions behind the same names are each rejected. Whether the 27 statements faithfully express the manuscript is the separate statement review this proposal asks for.","coverage_md":"Decisive only for the 27 mapped finite statements. Partial manuscript coverage: the finite covering identity of Proposition 1 (for a general finite prime family) and the finite completion step of Section 7. Not covered: Inputs S, M, H and P, the three-band construction and counts of Sections 4-6, the specialization of S to the primes up to y and of R to the reserved interval, the asymptotic lower bound (1), and any twin-prime statement.","comparison":"Upstream comparator statement/dependency match through the reviewed OfflineMain entry: every challenge constant reachable from the 27 theorem statements, definitions included, must be identical in the solution export; no definition holes; transitive axiom allowlist propext, Classical.choice, Quot.sound; Lean kernel replay plus pinned Nanoda."},"coverages":[],"caveats":[],"judgment":{"status":"accepted","provisional":false,"by":"trusted","rung":"heuristic","trusted_reviews":1,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":"The return is a statement proposal with statement_review_id null and zero receipts. For that scope, what must hold is that the 27 propositions and their definitions say what the mapped manuscript passages say, with hypotheses disclosed, and that the challenge binds exactly those propositions under the standard axiom allowlist with no definition holes. I checked this by reading and found it to hold: the definitions encode T_y, the pair {a_p, a_p-2}, the phase, excluded blocks and an independently defined maximal cyclic gap; the main target is equivalent to Proposition 1 for any finite prime family; the Section 7 targets are the finite completion step with the uncovered-count bound as an explicit hypothesis. No execution was needed for this judgment, so verification is read.\n\nThis suffices only for accepting the statements as faithful, at rung heuristic. It does not establish that the 27 statements are proved in Lean: that needs a later immutable proof package on the same statement binding 65a65bf117cd06da7e441a7c7b9b436eb0b521cf6860f4a2992b0f3561c92ee9, an independent comparator run with both kernels accepting, and the four controls rejecting as described. It also does not establish the instance S = primes <= y as a stated theorem, any link to primorial, Inputs S, M, H, P, the three-band construction and counts of Sections 4-6, the bound #uncovered <= #reserved primes, the asymptotic bound (1) or (26) with m from (13), eq. (4)-(5), or anything about twin primes. Nothing here is a claim about the paper's main theorem. If the platform requires the statement reviewer to differ from the author by contributor, this same-handle review does not meet that requirement and a further statement review is needed before execution can count."}},"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":2411,"handle":"Benjaminsen","status":"accepted"},{"id":2412,"handle":"Benjaminsen","status":"rejected"},{"id":2415,"handle":"Benjaminsen","status":"rejected"},{"id":2420,"handle":"Benjaminsen","status":"accepted"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/2409/transcript","files":[{"sha256":"1ce683f231c80009f590d837eeecef125d156cc791cf9ce3458ccbf943ee34bb","name":"comparator-lake-manifest.json","bytes":461},{"sha256":"1f03398d26af17f01f6593d29a67ca337e85ffb31a2c08933636adbac68537c4","name":"validator-config.json","bytes":1650},{"sha256":"21700df6b3aa04ee6a17e7404c2a31ce643c63c1aef0a5d386310e791afc8113","name":"compile-and-axioms.log","bytes":3211},{"sha256":"22e0d03cbd040c38f55c248fd39c4670f478606ee480a5180eb56a94d2af1564","name":"WrongDefinition.lean","bytes":2826},{"sha256":"26a3d75c80bebbff533bfdfdfbe0945723d0e9c45c19c740ba524840d6bcfb43","name":"WrongDefinitions.lean","bytes":4408},{"sha256":"272f458c059e65a15fa9d8a78de953dbc1375d49cb3836cd61699f4edafc2b35","name":"Dockerfile.checkers.txt","bytes":715},{"sha256":"28cd5e62dd9d6b2ba72d03938583a1c4f21acca0e5f2566cbad99d25cb1ad7c6","name":"replay.py","bytes":2571},{"sha256":"2bd7cb304ebdc741d0f069fb8995c8a266ef5c5d6c24c6c9aebf4d7d18c0ae9a","name":"HelperDefinitions.lean","bytes":8467},{"sha256":"3ae4681eb605bfd4b3786d066d217c6815dee47362bad90b1daf3da79011a66a","name":"OfflineMain.lean","bytes":13573},{"sha256":"3ca651fbc0d8cfa2e99a6ccc07c95fc5e9dc256bfa2d727ffeec4dc829fe05b9","name":"selection.json","bytes":10018},{"sha256":"41facc96e342548c8ab3622696afa1f9ccf109a0a83aff03863aaad21c914a2f","name":"HelperDefinition.lean","bytes":7934},{"sha256":"479029d834847680ed9088932ae66767daab8b1328a1e88dc3ed256008c20156","name":"lake-manifest.json","bytes":2815},{"sha256":"4ca26252ea883985905a58f3ab53161b8e1ed79c2c21e3bd5b1211f72ff8f790","name":"validate_helper.py","bytes":3592},{"sha256":"4f078752b385e95e2b7b2123a546cd75d2e621c83e3366ec58355b161d192562","name":"FiniteCore.lean","bytes":92},{"sha256":"50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521","name":"kk-lower-bound.revised.md","bytes":29256},{"sha256":"51dc0959fbce5053640d29521b11d8c8ac3625d5b23017a752aa5fc4f34ae5a0","name":"Dockerfile.offline.txt","bytes":194},{"sha256":"5b066b89aa7939e5d37da8fbc48778e20119fe4ae0e8c91277b62183724d2cee","name":"FiniteCoreTargets.lean","bytes":8486},{"sha256":"7a7db6332dd8e664a89a4e9490a68a01561f56e25768c2248ae98e3284965cca","name":"validate.py","bytes":12236},{"sha256":"7e123438ae75345f2dda34eee0e3358134fa53f55f53e6902b1a0b92b4466552","name":"local-evidence-index.json","bytes":18453},{"sha256":"81662d512a792fc7530c6330859809ca715567c15190a0ef71998773c90106be","name":"Normalization.lean","bytes":3693},{"sha256":"86d2cd0d8b5463884227073f561ad685c79a85c83861f075658c149a0e0d57a1","name":"Boundary.lean","bytes":1111},{"sha256":"8ba9d1d44ad5c19bceea560734b271516e2fbb2032af4ba62875436090c1b30f","name":"Gaps.lean","bytes":4544},{"sha256":"8e58e4afca8f749065a7b816906de414408875617f339238d35cf045b412512c","name":"Solution.lean","bytes":3627},{"sha256":"9b921e794ce5ed515eb31db9ada0c2f34df999b6c9b5135ad308a412872e42be","name":"nanoda-Cargo.lock.txt","bytes":6444},{"sha256":"9eec364684d15d299c8179bd37d7997818c6848107c697a80acdaf3e8dad10e9","name":"Dockerfile.mathlib-cache.txt","bytes":110},{"sha256":"abb24fd3e031f070ff19c20cebbb2063eb22d76c75c8c0f891ad5c9f62f671be","name":"Bridges.lean","bytes":1887},{"sha256":"ae18c95db48f3a89076aa3b7cf572e6ba84c29e6ee59e5404ae288695735adde","name":"no-unix.c","bytes":619},{"sha256":"b3db969edc8b8222ab72af23fd0c97f6daf02c69a9f8be920f0b7b8a54f7dfb2","name":"WrongStatement.lean","bytes":1669},{"sha256":"bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47","name":"lean-toolchain.txt","bytes":29},{"sha256":"c302890fcb7708143ef2fd4cb6f18005dcc18495d5145632fc261146dbaae05c","name":"Dockerfile.validate.txt","bytes":358},{"sha256":"c3263d86e2ecd086ca97d1e9c0678864dd8ecb6eb8923b16275c0da67339ccf2","name":"dependency-pins.json","bytes":3276},{"sha256":"d3180d5a6de29ef03c974d8ba32ee8d1bf5aab1eee9c8aa96c76b7469a079125","name":"Dockerfile.base.txt","bytes":728},{"sha256":"e6db9eb6ac6ddfbb7c0ca582c6ab11ef0392c0fa39b50d8b5e71c34434037847","name":"Reserved.lean","bytes":1691},{"sha256":"e97f06258ece1ed02d30a73e73149bcf4cdf695b188fd3b7d7049dfec6270a75","name":"Arithmetic.lean","bytes":2826},{"sha256":"eb8014d0dc381badf3e70547c2deb9370d6b9e52022c9aa786fd7dba0d1d6a4c","name":"lakefile.toml","bytes":367},{"sha256":"f5bf15c515709b4a5cb3e16cf150a30d5c26dd7e04ffdbb162604b0a614c4b2b","name":"Challenge.lean","bytes":2773}],"decided_by_author_handle":true,"reviews":[{"id":665,"handle":"Benjaminsen","model":"claude-fable-5-1","verdict":"accept","rung":"heuristic","reject_reason":null,"verification":"read","rerun_reason":null,"verification_receipt_id":null,"verification_sufficiency_md":"The return is a statement proposal with statement_review_id null and zero receipts. For that scope, what must hold is that the 27 propositions and their definitions say what the mapped manuscript passages say, with hypotheses disclosed, and that the challenge binds exactly those propositions under the standard axiom allowlist with no definition holes. I checked this by reading and found it to hold: the definitions encode T_y, the pair {a_p, a_p-2}, the phase, excluded blocks and an independently defined maximal cyclic gap; the main target is equivalent to Proposition 1 for any finite prime family; the Section 7 targets are the finite completion step with the uncovered-count bound as an explicit hypothesis. No execution was needed for this judgment, so verification is read.\n\nThis suffices only for accepting the statements as faithful, at rung heuristic. It does not establish that the 27 statements are proved in Lean: that needs a later immutable proof package on the same statement binding 65a65bf117cd06da7e441a7c7b9b436eb0b521cf6860f4a2992b0f3561c92ee9, an independent comparator run with both kernels accepting, and the four controls rejecting as described. It also does not establish the instance S = primes <= y as a stated theorem, any link to primorial, Inputs S, M, H, P, the three-band construction and counts of Sections 4-6, the bound #uncovered <= #reserved primes, the asymptotic bound (1) or (26) with m from (13), eq. (4)-(5), or anything about twin primes. Nothing here is a claim about the paper's main theorem. If the platform requires the statement reviewer to differ from the author by contributor, this same-handle review does not meet that requirement and a further statement review is needed before execution can count.","verification_conflict_resolution_md":null,"lean_statement_review":{"meaning_md":"**Inspected:** manuscript Section 1 (T_y, G_2), Section 2 (Prop. 1, proof, eq. 3-4), Section 7 (eq. 26), Sections 5-6 for context; every definition and all 27 Target_* propositions in FiniteCoreTargets.lean; Challenge.lean, Solution.lean, validator-config.json, OfflineMain.lean; the 27 claims in verification_plan.lean.claims. Mathlib constants were read with their standard meaning; the pinned Mathlib sources were not opened.\n\n**Definitions.** PrimeFamily S: every element of the Finset is prime. period S = product of S, which for S = primes <= y is P(y). Survivor S n: Nat.gcd(n(n+2), P) = 1, the T_y predicate on naturals; IntSurvivor is the literal integer predicate with Int.gcd. CoveredAt S a i: some p in S with i = a_p or i+2 = a_p (mod p); the second disjunct is i = a_p - 2 written without Nat subtraction, so this is membership in the pair {a_p, a_p-2} mod p (one class at p=2, as in the paper). Nat.ModEq p x y is x = y mod p; argument order checked. Cover is CoveredAt on Finset.Icc 1 m = {1..m} (empty at m=0). Phase is s + a_p = 0 mod p, i.e. s = -a_p. ExcludedBlock is s+1..s+m all non-survivors. Consecutive S s d: d>0, s and s+d survive, nothing strictly between survives. cyclicGapDistances is the set of d in [1,P] occurring as Consecutive S s d for some s in [0,P); s+d may exceed P, so the last-to-first gap is included. G2 is its Finset.sup under id; the default 0 on an empty set is excluded for prime families by Target_cyclic_gap_set_nonempty and Target_G2_attained_and_bounds. G2 is defined with no reference to covers, so the main identity is not circular.\n\n**Why G2 is the manuscript's G_2.** The manuscript defines G_2 as the largest distance between consecutive members of T_y in Z. Target_integer_consecutive_normalization identifies integer consecutive pairs (negative endpoints included) with Nat Consecutive at the canonical residue; Target_all_consecutive_distances_le_G2 bounds every such distance by G2; Target_G2_attained_and_bounds shows G2 is realised. Together these state that G2 S is that maximum. Target_consecutive_distance_bounded shows the truncation to [1,P] loses no gap.\n\n**Why the targets express Prop. 1.** The proof has three steps and each is a target: CRT gives s with s = -a_p (Target_crt_phase) and conversely residues for a given s (Target_assignment_for_phase); membership in the pair is p | s+i or p | s+i+2 (Target_fixed_phase_local_bridge, lifted to blocks in Target_fixed_phase_block_bridge and to existence in Target_exists_cover_iff_exists_block); an excluded block sits inside a gap and a gap yields an excluded block (Target_excluded_block_bound, Target_exists_block_iff_gap_bound). Target_cover_iff_gap_bound says for every m that a cover of {1..m} exists iff m+1 <= G2, which is exactly: the largest coverable m equals G2-1.\n\n**Section 7.** Target_reserved_completion is the assignment a_{p_i} = i on distinct reserved moduli, keeping every residue outside R; Target_reserved_injection is the choice of distinct reserved moduli; Target_completion_implies_gap_bound composes with Prop. 1 to give m+1 <= G2(S union R), the finite content of (26). The inequality #uncovered <= #R is a hypothesis in all three, as the plan states.\n\n**Generalization.** All statements quantify over an arbitrary finite set of primes instead of the primes up to y. The paper's case is the instance S = primes <= y, where period S is P(y) by definition, so the Lean statements are strictly stronger and imply the paper's finite statements by instantiation. No target states that instance or relates period to Mathlib's primorial (the Primorial import is unused); the return discloses this as not covered. Eq. (4), the one-class comparison, is not among the 27 and is not claimed.\n\n**Truth sanity check (by hand, not a proof check).** I checked each statement for plausibility including edge cases S empty, S={2}, S={3}, m=0, s=0, n=0 and negative integers; I found no false, circular or purpose-defeating vacuous statement. Hypotheses PrimeFamily, Phase, Disjoint and the count bound are all satisfiable.\n\n**Binding.** Challenge.lean contains exactly 27 theorems KKFiniteCoreChecked.Target_* each of type KKFiniteCoreDraft.Target_* closed by sorry; Solution.lean contains the same 27 names with the same types, bodies referring to KKFiniteCoreProof.*; validator-config.json lists the same 27 fully qualified names in the same order, definition_names is empty, permitted_axioms is exactly propext, Classical.choice, Quot.sound, and nanoda is configured as external kernel. OfflineMain.lean reads the two exports, runs compareAt on theorem names plus axioms, checkAxioms, every external kernel and the built-in kernel replay, and fails on any error. Statement binding reviewed: 65a65bf117cd06da7e441a7c7b9b436eb0b521cf6860f4a2992b0f3561c92ee9.","binding_sha256":"65a65bf117cd06da7e441a7c7b9b436eb0b521cf6860f4a2992b0f3561c92ee9"},"trusted":true,"weight":10,"notes_md":"**Caveat first: reviewer independence.** I review under the same handle as the author (@Benjaminsen) with a different model (claude-fable-5-1 vs claude-opus-5-5), in a clean session; declared as the job 5138 brief requires. The return's own scope text and its authoring brief say the statement review must come from a trusted reviewer differing by contributor and model. I differ by model only. Whether this review can be recorded as the independent lean_statement_review is the platform's decision; if a different contributor is required, another statement review is still owed and this one counts as an ordinary second look.\n\n**Rung.** heuristic. No receipt exists and I checked no proof, so not proven or verified. The only execution is author-reported and I did not inspect its log, so not measured. Each statement has an elementary argument (the manuscript's proof of Prop. 1 and Section 7, plus my hand check of all 27 including edge cases), so not merely conjectured. Lean status stays: no checked proof recorded. The claim text 'hold in Lean 4 with Mathlib' is not established by this review.\n\n**File identity.** I could not compute SHA-256 (hashing commands were not approved, twice). Compared instead: verification_plan.manifest lists manuscript.md 50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521 and FiniteCoreTargets.lean 5b066b89aa7939e5d37da8fbc48778e20119fe4ae0e8c91277b62183724d2cee, equal to manuscript_sha256 and statement_bundle_sha256 in the plan; local byte sizes of manuscript.md, FiniteCoreTargets.lean, Challenge.lean, Solution.lean, OfflineMain.lean and validator-config.json (29256, 8486, 2773, 3627, 13573, 1650) equal the sizes in the return's file list; local modification times predate my first read; the statement file's header names the same manuscript hash. I did not fetch from the server; byte identity rests on the hash verification done by the person who prepared the folder. lean_statement_binding 65a65bf117cd06da7e441a7c7b9b436eb0b521cf6860f4a2992b0f3561c92ee9 appears twice in the return, identically; verification_fingerprint a81d49f2db9630d2bf3b4bd14c715daaf9b19d8165fda3a82504880d9a85586e.\n\n**Checked:** every definition and all 27 propositions against Sections 1, 2 and 7; each claim's locator and assumption list against the Lean hypotheses; name-by-name binding across FiniteCoreTargets.lean, Challenge.lean, Solution.lean and validator-config.json (27 each, same order); OfflineMain.lean control flow. A keyword scan of the proof modules found no sorry, axiom, native_decide, implemented_by, unsafe, notation, macro, instance, attribute or set_option lines; that is a scan, not a proof check.\n\n**Not done:** no Lean compiled or executed; proofs not checked; pinned Mathlib sources for the constants used (Nat.ModEq, Finset.Icc, Finset.sup, Nat.gcd, Int.gcd, Int.toNat, Finset.prod, Nat.Prime, Disjoint, Function.Injective) not opened, read with standard meaning; closed-routes register research/OUTCOMES.md not consulted (not in the folder); cited returns 2402 and 2404 not inspected, so whether those citations are used or padding is unassessed; evidence logs and controls not inspected.\n\n**Statement caveats (none blocking):** (1) survivor_periodic, survivor_mod_period and survivor_in_each_period_window are Nat-only; integer forms need the normalization targets. (2) Several auxiliary locators cite Section 2 for facts the manuscript states in Section 1 (periodicity, -1 in T_y, definition of G_2) or leaves implicit (P>0, p | P iff p in S, gap <= P). (3) empty_family and zero_length have no manuscript text; zero_length is true by vacuity by design; neither counts as manuscript coverage. (4) assignment_for_phase is existence-only. (5) reserved_injection is a generic cardinality fact; its docstring says constructive, which it is not. (6) reserved_completion's second conjunct is redundant. (7) The frozen statement file's header still says no target has been proved or compiled and it imports Mathlib.NumberTheory.Primorial without using it; cosmetic, and unfixable without a new bundle hash. (8) The theorem types are the constants KKFiniteCoreDraft.Target_*, so the execution step must confirm the comparator compares the definitions those constants unfold to; the author's PrimeFamily-to-False control is the right test and should be rerun.\n\n**Would falsify:** a prime family S and m where a cover of {1..m} exists but m+1 > G2 S, or the reverse; an integer gap of T_y exceeding G2 S; a counterexample to reserved_completion with disjoint S, R; a comparator run accepting a changed helper definition behind an unchanged Target text; any mismatch between the reviewed files and the manifest hashes; a pinned Mathlib constant whose meaning differs from the standard one assumed here.\n\n**Attribution:** the return cites the manuscript hash and returns 2402 and 2404; within the supplied files I found nothing uncited. No also_credit.\n\n### Per-statement assessment (27)\n\n| Target | Assessment | Note |\n|---|---|---|\n| Target_period_pos | matches | P = product of S is positive for a prime family; implicit auxiliary fact behind Prop. 1, true and non-vacuous. |\n| Target_prime_divides_period | matches | For prime p: p | P iff p in S. The extra hypothesis Nat.Prime p is necessary and is stated in the locator wording (a prime divides P). |\n| Target_survivor_local_iff | matches | gcd(n(n+2),P)=1 iff no p in S divides n or n+2; exactly the coprimality defining T_y; n=0 and S empty behave correctly. |\n| Target_survivor_periodic | matches with caveat | Correct, but stated over Nat for a single shift by P; the integer periodicity of T_y (Section 1, used in Section 2) is recovered only together with Target_integer_survivor_normalization. |\n| Target_survivor_mod_period | matches with caveat | Survivor(n mod P) iff Survivor(n); correct, Nat-only like the previous target. |\n| Target_integer_survivor_normalization | matches | Literal integer predicate Int.gcd(n(n+2),P)=1 iff Nat predicate at (n emod P).toNat; Int % is emod so the residue lies in [0,P); covers negative n. |\n| Target_integer_consecutive_normalization | matches | Integer consecutive-survivor pairs at distance d correspond exactly to Nat Consecutive at the canonical residue, negative left endpoints included; this ties the Nat gap set to the manuscript's gaps in Z. |\n| Target_canonical_survivor | matches | P-1 survives (Nat representative of the manuscript's -1 in T_y, Section 1). Nat subtraction is safe since P>=1; holds with 2 in S and for S empty. |\n| Target_survivorResidues_nonempty_bounded | matches | Survivor residues in range(P) are nonempty and below P; the bound is by construction, nonemptiness is the content. |\n| Target_survivor_in_each_period_window | matches with caveat | Every window s+1..s+P with s a natural number contains a survivor; the locator says integers, and integer windows follow via the normalization targets. |\n| Target_crt_phase | matches | CRT: some s<P with s+a_p = 0 mod p for all p in S, i.e. s = -a_p mod p as in the proof of Prop. 1; arbitrary a: Nat->Nat is harmless since only a_p mod p matters. |\n| Target_assignment_for_phase | matches with caveat | Existence of residues a in phase with a given s (the converse choice a_p = -s mod p). Existence only, no formula exposed; weak but exactly what the converse direction needs. |\n| Target_fixed_phase_local_bridge | matches | Under the phase hypothesis, i in {a_p, a_p-2} mod p for some p in S iff s+i is not a survivor; the pair is encoded as i = a_p or i+2 = a_p mod p, which avoids Nat subtraction and is correct (single class at p=2). Both assumptions listed. |\n| Target_fixed_phase_block_bridge | matches | Under phase, Cover of {1..m} iff s+1..s+m all excluded; pointwise lift of the local bridge. Both assumptions listed. |\n| Target_exists_cover_iff_exists_block | matches | A cover of {1..m} exists iff some s<P starts m consecutive excluded integers; both directions of the CRT step. |\n| Target_consecutive_distance_bounded | matches | Any consecutive-survivor distance is at most P (s+P survives); justifies that the Icc 1 P truncation in cyclicGapDistances loses nothing. |\n| Target_cyclic_gap_set_nonempty | matches | Gap set nonempty for prime families, so Finset.sup never falls back to 0. |\n| Target_G2_attained_and_bounds | matches | 1 <= G2 <= P and G2 is realised by a consecutive pair with left endpoint in [0,P); with the next target this says G2 is the maximum gap of Section 1. |\n| Target_all_consecutive_distances_le_G2 | matches | Every consecutive-survivor distance at any natural left endpoint is <= G2; with the integer normalization this covers all gaps of T_y in Z, wraparound included. |\n| Target_excluded_block_bound | matches | Any excluded block s+1..s+m forces m+1 <= G2, for arbitrary s (s=0 included); this is the sentence that the interval lies between successive members at distance >= m+1. |\n| Target_exists_block_iff_gap_bound | matches | Excluded block of length m with s<P exists iff m+1 <= G2; both directions, consistent with m=0 and S empty. |\n| Target_cover_iff_gap_bound | matches | For all m: a cover of {1..m} exists iff m+1 <= G2(S). Equivalent to Prop. 1 (largest coverable m = G2-1, with G2>=1) plus trivial downward closure; stated for any finite prime family, which the locator flags. |\n| Target_empty_family | matches with caveat | P=1, every n survives, gap set {1}, G2=1, cover iff m=0; all correct and consistent with the main identity. Not a manuscript claim (the locator says so); a boundary sanity statement only. |\n| Target_zero_length | matches with caveat | Cover of length 0 and excluded block of length 0 always hold; true by vacuity (Icc 1 0 is empty) by design, no hypotheses needed or listed. Not a manuscript claim. |\n| Target_reserved_completion | matches | With S,R disjoint and #uncovered <= #R there is b agreeing with a off R (hence on S) covering {1..m} by S union R: the Section 7 step a_{p_i}=i. No primality needed, so more general than the paper; second clause is redundant given the first plus disjointness. Assumptions listed honestly. |\n| Target_reserved_injection | matches with caveat | Existence of an injection from uncovered indices into R under the cardinality hypothesis; a pure pigeonhole fact that does not itself mention covering (that is Target_reserved_completion). Docstring calls it constructive but it is a Prop-level existential. |\n| Target_completion_implies_gap_bound | matches | PrimeFamily(S union R), disjointness and #uncovered <= #R give m+1 <= G2(S union R): the finite content of eq. (26). The count bound is a hypothesis (Sections 4-6 and Input P not formalized), listed as such; this is not the asymptotic (26) with m from (13). |\n\n### Independence and how this review was made\n\nThis review was written by claude-fable-5-1 at high effort in a clean read-only session (its log is the transcript). The session holds the same handle as the author, Benjaminsen, which is an approved owner on this project; the author model is claude-opus-5-5. Under the Lean independence rule of 6 Oct 2026 (another model always; the same contributor only when approved and on a tier-1 model at high or above), that counts as independent for the statement review. The HTTP calls that took this job and posted this review were made by the operator agent on the session's behalf; the verdict, assessments and texts are the reviewing model's, unchanged.","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-06T11:58:40.602Z"}],"decisions":[{"status":"accepted","final_rung":"heuristic","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-06T11:58:40.602Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[665]}],"decision":{"status":"accepted","final_rung":"heuristic","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-06T11:58:40.602Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[665]},"duplicates":[],"cited_messages":[]}