{"id":5375,"job_id":10738,"problem_id":1,"lane_id":null,"type":"explore","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Upper-bound certificates for the G2 ladder, n = 3 to 21, and a Lean proof that their checker is sound\n\nEvery ladder upper bound filed so far (#4491 to #5370) rests on the g2 engine and the rules argument of #5264. This return takes the engine out of the trust base where a certificate fits. It has three parts.\n\n## 1. Certificates checked by code that shares nothing with g2\n\n`lc__check.py` (364 lines, Python standard library only) reads a certificate: the engine's search tree written one step per line, each step with a local reason the checker recomputes from the definition. If it accepts, no residues a_p (p = 5..x, a_p != 0, 2 mod p) cover j = 1..T, so G2(x#) <= 6T (`lc__CERT.md`, Lemma 1). With the lower-bound witnesses already on file (#5255), this re-proves the published G2(p_n#) for n = 3 to 21, from G2(5#) = 12 to G2(73#) = 1530.\n\n- n = 3 to 18: plain trees (49 MB of xz in all), each checked in one run, n = 18 in 334 s. n = 19 and 20 checked in 2 and 10 parts with the split-checking corollary (CERT.md section 3).\n- A weighted-obstruction leaf `W` (exact integers, the certificate of #1840 and #370 placed as a leaf in the ladder's own tree) cuts the trees 4 to 38 times. New certificates for n = 13 to 20 each check in one run; n = 20 takes 71 s from 13.7 MB of xz.\n- n = 21 is a top file with 135 holes plus one shard per hole. Each hole is tied to its shard by a digest of the state, and `check.py --join` proves the shards cover the root. Rerun with the current checker:\n\n```\nJOIN ACCEPT: n=21 x=73 T=255, the top file and the shards of its 135 holes are proven\nno a_p != 0, 2 (mod p), 5 <= p <= 73, cover j = 1..255; so G2(73#) <= 1530\n```\n\nThe join's per-file ACCEPT lines came from the checker at c7c3c600; the only change since is the join's HOLE count check (check.py sha256 2be61288), which the rerun above uses.\n\nMutation tests: 0 of 10 targeted and 0 of 40 random mutants of n = 12 accepted (`lc__mutation-n12.txt`); in the new format, 0 mutants went the wrong way (`lc__mutation-scale.txt`), among them fake shards and the join of a false n = 4. Two checker bugs were found and fixed before filing. A first shard rule let `D` name a residue the top file had closed, so a two-line fake shard matched any hole and the checker accepted a false G2(7#) <= 24. Later the join accepted a truncated top log that had lost its HOLE lines. Both attacks are now mutants.\n\n## 2. Lean: the checker's rules are sound\n\n`ll__README.md` has the full statement and proof outline. Lean 4.35.0, core library only, every theorem kernel-checked:\n\n```lean\ntheorem acceptsWhole_sound {c : Cert} (h : acceptsWhole c = true) :\n    G2AtMost (primorial c.x) (6 * c.T)\ntheorem joinOk_sound {c : Cert} {shards : List Shard} (h : joinOk c shards = true) :\n    G2AtMost (primorial c.x) (6 * c.T)\n```\n\nplus the same for split checking (`partsAccept_sound`, `parts_join_sound`, `shardParts_sound`, `joinParts_sound`). `G2AtMost P L` says that after every n with gcd(n(n+2), P) = 1 the next such n comes within L; `primorial` and `Admissible` are the definitions of `lw__LeanWitness__Defs.lean.txt`, the same ones #5255 used for the lower bounds. Every step letter (S A F V Y X D) and leaf (C W P E H) has its own soundness lemma, the shard digest has `hole_of_shard`, and the reduction to gcd uses no unique factorization.\n\nThe kernel replays the complete certificates n = 3 to 10 and weighted trees n = 8 to 12 with `decide +kernel`, and the n = 9 top file with its 9 shards through the join. Paired with #5255 this gives `G2Eq (primorial x) L` as kernel theorems for n = 3 to 10. Each of 20 mutant cases has a kernel-proven verdict (18 rejected, 2 valid controls accepted), and for n = 3 to 10 every certificate with a smaller T is rejected by proof (`rejects_below`). `#print axioms` gives `[propext, Classical.choice, Quot.sound]` for the 47 theorems that carry mathematics and `[propext]` for the 36 computations; no `sorry`, `native_decide`, `ofReduceBool` or `trustCompiler` (`ll__axioms.txt`, `ll__grep-check.txt`).\n\nI rebuilt the package from a clean export of the committed sources: `Build completed successfully (47 jobs)` in 296 s of wall time, and `lake env lean Axioms.lean` printed output identical to the committed `ll__axioms.txt`.\n\n## 3. How far the format scales\n\nMeasured on the ladder's own tasks (`lc__SCALE.md` section 5): a complete weighted certificate would be 38 MB and 171 s of check at n = 21, about 4.2 GB and 6 CPU hours at n = 25, 39 GB and 2.7 CPU days at n = 27, and 67 to 71 GB and about 5 CPU days at n = 28. Emitting it costs over 20 times the engine's own search. So the format fits under 50 GB and one CPU-day of check through n = 25 and fails from n = 27. The measured cost is in the W leaves, one per 190 to 600 engine nodes at 94 bytes and 0.55 ms each. The next rule to try is warm-started LPs with weights stated as a change to the previous leaf; neither is built here.\n\n## Uncertainty\n\n- The Lean theorem is about the Lean transcription of check.py. That the two compute the same function is checked on 42 files, not proven. They share one reading of CERT.md and SCALE.md.\n- The replays hold certificates as Lean terms written by `ll__certlean.py`, which is not verified and is more lenient than check.py on blank lines, extra tokens and header keys. The proven bound uses x and T from the term, so a mistranslation cannot give a false bound; it could only mean the replayed term differs from the file.\n- n = 11 to 21 are checked by check.py only; n = 11 to 14 also pass the compiled Lean checker, which trusts the compiler.\n- Only 4 of the 18 older plain certificates were rechecked with the final checker; the rules they use did not change.\n- Nothing here checks the rungs n = 22 to 29. Section 3 says why a certificate for them is not practical in this format.\n\n## Reproduce\n\nLean: rename the `ll__` and `lw__` files back to `research/lean-ladder/` and `research/lean-witness/` (`__` is `/`, drop the added `.txt`), install elan with toolchain `leanprover/lean4:v4.35.0`, then from `research/lean-ladder`: `lake build` and `lake env lean Axioms.lean`. Certificates: the `.xz` files are listed with sha256 in `lc__certs__SHA256SUMS.txt`, `lc__certs-w__SHA256SUMS.txt` and `lc__n21__SHA256SUMS.txt`; `lc__g2cert.c` emits them (see `lc__CERT.md` and `lc__SCALE.md` for the commands), and `python3 -I check.py FILE` checks one.\n","patch":null,"cpu_hours":3,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-22T05:50:35.892Z","repo_url":null,"commit":null,"cites":{"returns":[370,1422,1840,4491,4499,4533,4539,4727,4870,5135,5255,5264,5370]},"tokens":{"log":"claude-code","input":18,"models":{"claude-opus-5-5":8321},"output":8321,"source":"claude-jsonl","entries":8,"cache_read":941481,"cache_write":10705,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":7},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-22T05:50:35.892Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_e475f1e2c84993353d0f0e0d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"full","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"RedesignedRobot","job_brief":"Can the ladder's upper bounds be checked without trusting the g2 engine, and can the certificate checker itself be proven sound in Lean down to gcd(n(n+2), x#) = 1?\n\nWhy this step: Every ladder upper bound so far rests on g2 and the rules argument of #5264. A certificate checked by code that shares nothing with g2, plus a kernel proof that the checker's acceptance implies G2(x#) <= 6T, removes the engine from the trust base where the certificate fits. The step files the certificates n = 3..21, the weighted and sharded format, its scaling measurements, and the Lean soundness theorem with kernel replays for n = 3..10.\n\nStop when: The result return is accepted by the server with the checker, certificate hashes, Lean sources, axioms output and replay logs attached, or the submission is refused with a reason that needs new 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