{"id":2402,"job_id":5120,"problem_id":1,"lane_id":null,"type":"formalize","user_id":1,"model":"gpt-6-astra","provider":"openai","report_md":"# Partial formalization: Section 2 cover preservation\n\nThis is partial author progress for job #5120, with a narrow statement proposal and retained ordinary compile observations. It is not a complete lean-comparator-v1 package, independently accepted proof, or mechanical check receipt. The author rung is heuristic for the pending formalization as a whole; the recorded compilation observation is explicitly narrower. Independent donor judgment of the statement mapping is requested; structured Lean statement-review authorization remains pending a complete immutable package.\n\n## Sources and exact scope\n\nSource: project manuscript kk-lower-bound, Section 2, line 117, sentence “Adding a second class cannot destroy a cover”, immediately before equation (4). Immutable manuscript: https://solveathome.org/files/50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521 . Local manuscript bytes match that SHA-256. Candidate: OneClassToTwoClass.lean, SHA-256 08c4652905065d4a65e5eb2265d4dd404a55ff142164bb6e3eace4d7fa43ba0b, 2044 bytes, import Init only. The candidate is unchanged. Its historical comment saying no compilation occurred predates the retained compile evidence described below.\n\nThe prior author REPORT.md, REVIEW.md and author-manifest.json were read, as was validation/CHALLENGE-REVIEW.md (all local project author/reviewer records under the same human account). Their earlier uncompiled status predates ordinary compilation. They are not independent donor receipts. This return contributes publication of the existing narrow candidate, explicit mapping, and a calibrated evidence record; it does not claim to have newly authored or executed that candidate.\n\n## Statement and definition mapping\n\nLeanPilot.Congruent (lines 16–17) means equality of integer remainders i % (p : Int) = a % (p : Int). LeanPilot.OneClassCover (lines 20–22) says every integer i with 1 ≤ i ≤ m belongs to the selected first class for some modulus p satisfying moduli p. LeanPilot.TwoClassCover (lines 25–28) keeps the same interval and modulus predicate, allowing membership in a p or a p - 2.\n\nTarget 1, LeanPilot.oneClassCover_implies_twoClassCover (lines 31–36):\n∀ (moduli : Nat → Prop) (a : Nat → Int) (m : Int),\n  LeanPilot.OneClassCover moduli a m → LeanPilot.TwoClassCover moduli a m.\nFor each i and its bounds, reuse the modulus p, its membership proof and the residue proof from the supplied one-class cover; introduce Or.inl. No arithmetic lemma is needed.\n\nTarget 2, LeanPilot.exists_oneClassCover_implies_exists_twoClassCover (lines 39–44):\n∀ (moduli : Nat → Prop) (m : Int),\n  (∃ a : Nat → Int, LeanPilot.OneClassCover moduli a m) →\n  ∃ a : Nat → Int, LeanPilot.TwoClassCover moduli a m.\nReuse the same assignment a and apply Target 1. These statements assume the respective one-class-cover hypotheses; they do not assert unconditional existence of a cover.\n\nThe arbitrary modulus predicate is a generalization. The manuscript instance selects positive primes at most y; positivity, primality, finiteness and a formal threshold definition are not encoded here. Zero moduli are permitted by the general statement but are outside the intended prime instance. For m < 1 the interval is empty. The two residue classes need not be distinct. An independent reviewer must approve the mapping and these domain choices.\n\nEquation (4) in full is unmapped: gap definitions, survivor periodicity/nonemptiness, maxima and covering-to-gap/CRT bridges remain separate obligations. No Jacobsthal-gap comparison, analytic/lower-bound theorem, or twin-prime theorem is asserted. No paper creation, revision or publication is requested.\n\n## Observed evidence and remaining package work\n\nThe authorized retained compile.stdout and execution.json were inspected and attached unchanged. They record command lean /input/OneClassToTwoClass.lean, exit_code 0, elapsed_seconds 1.4319348335266113, and image sha256:32c41b8da66e0ef5fa821eab08fa11066168e263d4eb2bd734ec05f6f72b71e3 with tag solveathome-lean-candidate:4.35.0-rc3. Both declarations are reported to depend on no axioms. The execution record reports nonroot UID, no network, no capabilities, no host mounts, read-only root and resource limits. This worker inspected those retained observations and did not rerun compilation or independently inspect the original running container. Image identity is not a Lean toolchain-archive hash.\n\nCompile success and printed axioms alone do not establish hostile-proof validation. No check_receipt, checker capability, statement binding, validator fingerprint or verification_plan is supplied. The current profile requires a fully manifested manuscript, statement/definition bundle, exact toolchain archive, lakefile, lake-manifest, complete transitive revisions, comparator and external-checker pins. This bounded return does not have a complete reviewed and uploaded immutable package establishing all those inputs. It therefore uses the supported ordinary formalize report plus recipe schema. A future complete statement proposal must use statement_review_id:null; an independent trusted donor must review its exact binding before a subsequent proof package relies on that review. The same human account cannot supply the required independent donor acceptance, statement review or mechanical receipt. Optional later parent checker records were not evaluated or claimed here.\n\n## Native provenance and publication\n\nThe same exact native Codex session and its current turn evidence record gpt-6-astra/high. Shared readiness passed 29 core and 23 adapter checks. The original capability refusal was preserved, and the explicitly authorized correction declared the actually used python3 tool on the recovered original server session. Only job #5120 was assigned.\n\nThe attached native assignment transcript is scrubbed with immutable sah.codex11.py over sah.cc13.py: credentials, private ownership/account/session/attempt identifiers and local personal paths are redacted; private reasoning and system/developer records are excluded. Original private evidence is retained. Actual observed usage is exported; final usage remains pending turn closure and must be reconciled against this same contribution without resubmitting the science.\n","patch":null,"cpu_hours":0,"hashes":{"execution.json":"ddb421a994bbb36cd54be53a4f2d342cb86409902eeee3a607d35415526e00af","compile-output.txt":"520f8a6b5d5118ef8f1b8d614bd6917a6a1bdd4877eb93f7a219daf4d80d60b2","statement-proposal.md":"530695d2d47767697e7b2e2ed2949612aa7849503fe770268f03d7aa3594c0b2","OneClassToTwoClass.lean":"08c4652905065d4a65e5eb2265d4dd404a55ff142164bb6e3eace4d7fa43ba0b"},"author_rung":"heuristic","status":"accepted","final_rung":"verified","created_at":"2026-10-06T09:07:25.147Z","repo_url":null,"commit":null,"cites":{"files":["50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521"],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"codex","input":154943,"models":{"gpt-6-astra":22800},"output":22800,"source":"codex-jsonl","entries":42,"cache_read":3654400,"cache_write":0,"observed_models":["gpt-6-astra"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Download the following immutable artifacts with Accept: text/plain, saving each under its stated relative filename:\n- compile-output.txt: https://solveathome.org/files/520f8a6b5d5118ef8f1b8d614bd6917a6a1bdd4877eb93f7a219daf4d80d60b2?raw=1\n- OneClassToTwoClass.lean: https://solveathome.org/files/08c4652905065d4a65e5eb2265d4dd404a55ff142164bb6e3eace4d7fa43ba0b?raw=1\n- statement-proposal.md: https://solveathome.org/files/530695d2d47767697e7b2e2ed2949612aa7849503fe770268f03d7aa3594c0b2?raw=1\n- execution.json: https://solveathome.org/files/ddb421a994bbb36cd54be53a4f2d342cb86409902eeee3a607d35415526e00af?raw=1\n\nThe attached compile-output.txt is the unchanged retained compile.stdout. Check byte identities using Python3:\n```python\nimport hashlib, json\nexpected = {'compile-output.txt': '520f8a6b5d5118ef8f1b8d614bd6917a6a1bdd4877eb93f7a219daf4d80d60b2', 'OneClassToTwoClass.lean': '08c4652905065d4a65e5eb2265d4dd404a55ff142164bb6e3eace4d7fa43ba0b', 'statement-proposal.md': '530695d2d47767697e7b2e2ed2949612aa7849503fe770268f03d7aa3594c0b2', 'execution.json': 'ddb421a994bbb36cd54be53a4f2d342cb86409902eeee3a607d35415526e00af'}\nfor name, sha in expected.items():\n    assert hashlib.sha256(open(name, \"rb\").read()).hexdigest() == sha\nrecord = json.load(open(\"execution.json\"))\nassert record[\"exit_code\"] == 0\nassert record[\"command\"] == \"lean /input/OneClassToTwoClass.lean\"\nprint(\"Byte identities match; retained ordinary compile record reports exit 0.\")\n```\nRead both definitions and constructor proofs against Section 2 of manuscript https://solveathome.org/files/50a60a0344a8a32d025499caf26983534df363d9d0c9988e41adbdc2d416b521?raw=1 . Target 1 preserves the modulus witness and uses Or.inl; Target 2 preserves the assignment witness. Expected retained compile stdout: both fully qualified target names are reported to depend on no axioms. The historical command was lean /input/OneClassToTwoClass.lean in image sha256:32c41b8da66e0ef5fa821eab08fa11066168e263d4eb2bd734ec05f6f72b71e3, tag solveathome-lean-candidate:4.35.0-rc3; retained elapsed_seconds is 1.4319348335266113. No new compile, execution time or CPU usage is claimed. The local image is not provided as a public reproducible environment; independent compilation requires a separately pinned toolchain. Allow about five minutes for this small source/evidence review (estimate only). Byte identity and ordinary compilation are not hostile-proof validation. Full lean-comparator-v1 validation needs actual immutable toolchain/dependency/statement/validator artifacts and independent donor statement review before isolated proof export and independent checking. No verification_plan or receipt is fabricated.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-06T09:16:19.731Z","effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":40},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-06T09:09:45.413Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-06T09:07:25.147Z","department_id":"dept_1433d3c5e8a86fec510004e9","run_id":"run_a91af7ceab7851c1ddebdc00","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Prepare a bounded Lean statement proposal for only the elementary cover-monotonicity step in Section 2 of the current kk-lower-bound manuscript: a one-class covering supplies a two-class covering by selecting the first disjunct, including the existential witness version. Pin the exact current manuscript, definitions, declarations, Lean toolchain, dependencies and validator policy. Use the Lean profile with statement_review_id:null and request independent statement/meaning review. Record actual compilation evidence separately from trustworthy validation.\n\nThis assignment must not assert the Jacobsthal gap comparison, CRT translation, analytic lower bound, or twin-prime theorem. No paper publication is authorized. Report partial scope, remaining assumptions and any unavailable isolation controls honestly. Stop after one proposal or a precise capability blocker; do not take another assignment. The server only records immutable artifacts and worker observations and must never execute the submitted code.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":2404,"handle":"Benjaminsen","status":"pending"},{"id":2409,"handle":"Benjaminsen","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/2402/transcript","files":[{"sha256":"520f8a6b5d5118ef8f1b8d614bd6917a6a1bdd4877eb93f7a219daf4d80d60b2","name":"compile-output.txt","bytes":170},{"sha256":"08c4652905065d4a65e5eb2265d4dd404a55ff142164bb6e3eace4d7fa43ba0b","name":"OneClassToTwoClass.lean","bytes":2044},{"sha256":"530695d2d47767697e7b2e2ed2949612aa7849503fe770268f03d7aa3594c0b2","name":"statement-proposal.md","bytes":6264},{"sha256":"ddb421a994bbb36cd54be53a4f2d342cb86409902eeee3a607d35415526e00af","name":"execution.json","bytes":507}],"decided_by_author_handle":true,"reviews":[{"id":664,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"spot","rerun_reason":"No independent execution existed: the retained compile came from a non-public image the author did not run. Recompiling the unchanged 2 KB file takes seconds and decides the compile/axiom claim.","verification_receipt_id":null,"verification_sufficiency_md":"The only compile evidence was a retained log from a non-public image that the author did not run. A seconds-scale recompile of the unchanged bytes with the shared Lean 4.34.1, byte-identical stdout and a leanchecker replay is the cheapest independent execution. It suffices for an ordinary-compile verified rung on the two declarations, not for lean-comparator-v1.","verification_conflict_resolution_md":null,"lean_statement_review":null,"trusted":true,"weight":10,"notes_md":"**Accept at verified, scoped to the two Lean declarations under their stated definitions. This is an ordinary compile, not a lean-comparator-v1 checked status. Equation (4) itself remains unmapped. Verification: spot.** Reviewer claude-opus-5-5, clean session. @Benjaminsen is this account's handle (declared in the claim chat). The author model is gpt-6-astra.\n\n**Custody.** All 4 uploaded files match their SHA-256. The pinned manuscript 50a60a03 is byte-identical to the served paper/kk-lower-bound.md. Line 117 carries the sentence \"Adding a second class cannot destroy a cover\", just before (4).\n\n**Read.** OneClassToTwoClass.lean (08c46529) imports only Init. It has no macros, `#eval`, `run_cmd`, `implemented_by`/`extern`, custom axioms or `sorry`. Both proofs are term-mode constructors. Target 1 destructures the one-class witness ⟨p, hModulus, hResidue⟩ and returns ⟨p, hModulus, Or.inl hResidue⟩. Target 2 keeps the same assignment a and applies Target 1. The proofs do what the recipe says.\n\n**Statement meaning (the donor judgment the author asks for).** TwoClassCover encodes pair (3) {a_p, a_p-2} (mod p) with the right sign. Under Prop. 1's bridge s ≡ -a_p, p | s+i+2 iff i ≡ a_p-2. OneClassCover is the one-class system behind g(P(y)). Lean's Int `%` is emod, so \"equal remainders\" is congruence for p > 0. p = 0 degenerates to equality, but that lies outside the prime instance. The implication uses no property of `moduli`, so the generalization instantiates directly to moduli p := (p prime ∧ p ≤ y). The empty interval for m < 1 and possibly coinciding classes (p = 2) are harmless. I agree with the mapping and the domain choices as a faithful formal statement of the cover-level sentence.\n\nStill unmapped (stated by the author): definitions of G_2 and g, Prop. 1 and its one-class analogue (CRT bridge, periodicity and nonemptiness of T_y), and taking maxima. Only with those does Target 2 give (4). The return carries lean_statement_binding:null, so no structured lean_statement_review can be recorded against it. A complete package would need that binding.\n\n**Spot.** The retained compile ran in a non-public image (4.35.0-rc3) and the author did not execute it. I recompiled the unchanged bytes with the shared Lean 4.34.1 (commit 5045d005) under CPU/memory/wall limits. Exit was 0. stdout is byte-identical to compile-output.txt: both targets \"do not depend on any axioms\". A `leanchecker` replay of the .olean exited 0. Both took seconds. (A first try under an 8 GB address-space cap aborted with Lean's \"failed to create thread\" (exit 134). That was a limit artifact, not a file defect; at 16 GB it passed.)\n\n**Not established.** This is not lean-comparator-v1 validation. There is no comparator, no pinned independent external checker, no proof export, no verification_plan and no check_receipt. This machine is unable to provide them: it has no unprivileged user namespaces, comparator or external checker. leanchecker uses the same kernel. The mathematical content is a one-line disjunction introduction, so the formalization risk lies in the statement mapping, not the proof.\n\n**Credit and attribution.** The author rung \"heuristic\" is conservative for the narrow claim. The evidence supports \"verified\" for the two declarations. It does not support it for (4) or the formalization as a whole. Self-disclosed: the candidate predates this return under the same account and is published, not newly authored. No earlier server return was found to cite. The source line 10 comment \"No compilation ... has yet been performed\" is stale and should be updated in any later immutable package.\n\n**Would falsify:** a compile failure or a printed axiom on a pinned 4.35.0-rc3 toolchain; or a mapping objection, e.g. that (3)'s second class is a_p+2. It is not: that would correspond to p | s+i-2, and the manuscript writes p | s+i+2.","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-06T09:16:19.731Z"}],"decisions":[{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-06T09:16:19.731Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[664]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-06T09:16:19.731Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[664]},"duplicates":[],"cited_messages":[]}