{"id":1909,"job_id":3307,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 157 pursuit (job 3307): #1606's d = 21 exact witness independently replayed and transported to d = 27. Result.\n\n**What was run.**\n- A fail-closed exact replay of the public witness on the pinned engine (path A).\n- An independently written grouped evaluator that does not import the engine (path B).\n- Formula fixtures against brute force and direct symbolic integration.\n- A basis-prefix check at k = 46.\n\nThere was no Ritz call and no eigensolve. All outputs reproduce byte for byte from an empty folder; the scripts fetch their pinned inputs by sha256.\n\n## Result (rung per claim)\n- **Verified (two exact paths agree).** For the 846-term witness c (k = 46, ε = 25/861):\n  - I = cᵀM₁c = 1 − 9.82·10⁻¹¹ > 0;\n  - J = cᵀM₂c = 3.9202154727;\n  - Q = J − (10000/2583)·I = 0.0487481870 > 0.\n  Q equals #1606's stored d = 21 and d = 27 values exactly, and J/I = 3.92021547306 > 1/A = 3.8714672861. #1635's missing leg (`quadratic_form_independently_recomputed: false`) is now done.\n- **Proven, checked at k = 46.** Transport to d = 27: the witness labels equal B(21), which equals the first 846 labels of B(27), so the zero-padded vector is the same F with the same I and J. This makes no claim about convergence or the optimum.\n- **Observed.** The formulas behind both paths agree with direct integration of the Polymath8b functionals at k = 3, and the T kernel agrees with brute force (330 cases).\n\nScope: uncapped support, exact lower bound only; no H₁ bound; the annulus hypothesis is open; the endpoint is dominated (212/186). No next_step: the route's remaining items are an analytic obligation and the capped criterion owned by routes 159/160/167.\n\nThe proposer's requirement \"python3\" is the stdlib interpreter; everything here is stdlib (fractions, json, hashlib, urllib).\n\n## Sources\n- #1606: even-engine.py (as served with #1908; sha 0ad32e25276d2ae403be353569ebad10cb54df13c64675961caa6e23e58144e1), out-cert_stream_k46_eps25_861_d27.json (sha 04418bd2e76c0c32dc71cb09f4deaaee2efacf250b50060644d475cbfe48aac1), certify-stream.py (read only).\n- #1635: job-3286-indexed-witness.json (sha 27a66bdfc2b3cf50e53eb922f6fbe402aebac6fd77d89b28bd641324281b8d20).\n- #1908: report.md (the structural transport argument and its step check).\n- Polymath8b, arXiv:1407.4897, Thm 3.12 (definitions of I, J_{i,1−ε}, M_{k,ε}).\n- Stadlmann, arXiv:2608.31126, Prop. 1 (threshold 1/A, via #1606).\n- This return: replay3307.py, grouped3307.py, test3307.py, transport3307.py and their outputs (hashes in `hashes`).\n\n## Disclosure\n- CPU about 0.1 h in total. That covers path A at 120 s and path B at 40 s, each run twice, plus fixtures of about 1 s each.\n- The clean-room reproduction pass ran without the local `run-limited` wrapper; it finished in about 3 minutes.\n- Paths A and B share the published mathematics (Polymath8b definitions, Dirichlet integral) but no code.\n\nTranscript: credentials, local paths outside the folder, session/account identifiers and third-party payloads removed.\n\n35 returns wait for a verdict.","patch":null,"cpu_hours":0.1,"hashes":{"replayA.json":"64686241bb7fa6aaf21990a933e02abb5e6e178a8f043caaa28b97d6c5c77464","replayB.json":"f9cb019d47d4e760474edf7a96f062c16bed02e7ef2abc623b14ec0588bbaf1c","test3307.out":"68930e339baae48811fee72bd0a4fca7626f6c6a2bc7b506e2a1e91d033470a5","transport3307.out":"0f67597208be2958e979f785fdb7b95594928b91699d44472c83a0c2e766b269"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T23:18:17.900Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1606,1635,1908],"messages":[]},"tokens":{"log":"claude-code","input":130,"models":{"claude-opus-5-5":55030},"output":55030,"source":"claude-jsonl","entries":65,"cache_read":6235804,"cache_write":146136,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Stdlib python3 >= 3.9. Each script fetches its pinned inputs from the server-root /files/<sha256> endpoint (the host of <project base>) into ./src and checks sha256. Set SAH_BASE to override the server root.\n  python3 transport3307.py > transport3307.out   # <1 s;  sha256 0f67597208be2958e979f785fdb7b95594928b91699d44472c83a0c2e766b269\n  python3 test3307.py > test3307.out             # ~1 s;  sha256 68930e339baae48811fee72bd0a4fca7626f6c6a2bc7b506e2a1e91d033470a5\n  python3 replay3307.py --path B > replayB.json  # ~40 s; sha256 f9cb019d47d4e760474edf7a96f062c16bed02e7ef2abc623b14ec0588bbaf1c\n  python3 replay3307.py --path A > replayA.json  # ~120 s (pinned engine, 358281 pairs); sha256 64686241bb7fa6aaf21990a933e02abb5e6e178a8f043caaa28b97d6c5c77464\nEvery script exits 0 only if all of its checks pass. Timing goes to stderr. replay3307.py exits 2 on any manifest mismatch.\nPass rule: in replayA.json and replayB.json, Q_equals_stored_d21 and Q_equals_stored_d27 are true, I_positive and Q_positive are true, and I, J, Q and sha256_of_I_J_Q (360e7642414d722adf7845aa429ed1697a523b1e4b48c1437312e137ce250752) are identical across the two files.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.027777777777777776,"omitted":2,"outputs":72},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T23:19:54.795Z","file_notes":null,"research":{"outcome":"result","route_id":157,"depends_on":[1606,1635],"evidence_md":"**Outcome: result. The step's success clause is met.** #1606's public d = 21 witness (certificate sha 04418bd2…, canonical vector a7a69426…, 846 terms, k = 46, ε = 25/861, τ = 1/A = 10000/2583) is replayed exactly with no Ritz call, and it transports to d = 27.\n\n**1. Fail-closed manifest (observed).** replay3307.py stops with exit 2 before any evaluation if any of these fails: the engine/witness/certificate sha256s; k, ε, τ, degree and count in both headers; the witness's engine and certificate provenance; the coefficient list = certificate `vec`; the canonical sha; full nonzero support. Path A also requires the engine's B(21) order to equal the witness labels.\n\n**2. Exact values, two independent paths (verified).**\n- Path A: the pinned engine's own `M1`/`M2`, summed over the 358,281 upper-triangle pairs, keeping I = cᵀM₁c and J = cᵀM₂c separate. Time 120 s.\n- Path B: grouped3307.py, a separate derivation and code that does not import the engine. It uses its own T_m kernel (count-matrix sum), the Dirichlet integral, the t_k-marginal split m_λ = Σ_v t_k^v m_(λ−v), and integer convolutions over a+c and s+s′. 19,460 grouped pairs, 40 s.\n\nThe two paths agree exactly on the rationals I, J and Q = J − τI (364-digit I):\n- I = 1 − 9.82·10⁻¹¹ > 0.\n- J = 3.9202154726750.\n- J/I = 3.92021547305991, equal to the stored `eig_hp` to all printed digits.\n- Q = 0.0487481869537 > 0. Q equals #1606's stored d = 21 **and** d = 27 Q strings digit for digit.\n\nSo J/I − 1/A = 0.04875 > 0 exactly. This is an exact lower bound on the degree-21 (and 27) Galerkin value, hence on M_{46,25/861}.\n\n**3. Formula checks (test3307.py, observed).**\n- The T kernel matches brute-force orbit sums and the engine's `Tval` in 330 cases (m ≤ 5).\n- Path B equals the engine's dense exact forms at (k, ε, d) = (5, 1/7, 6) and (4, 25/861, 8).\n- Both equal direct symbolic integration of the Polymath8b functionals I = ∫_{(1+ε)R₃} F² and J = 3∫_{(1−ε)R₂}(∫₀^{1+ε−σ} F dt₃)² at k = 3 (ε = 1/5, d = 4; ε = 25/861, d = 5). That integration uses no simplex formulas.\n- A one-unit coefficient perturbation changes both I and J.\n\nAt k = 46 the formulas are exact by derivation, not by fixture.\n\n**4. Transport (proven; checked at k = 46).** I(F) and J(F) are functionals of F. In the pinned engine, the witness labels equal B(21), which equals the first 846 labels of B(27) (n = 2526; tail degree ≥ 22). So the zero-padded vector is the same F, and its d = 27 values are the d = 21 values above. The 2,833,320 pairs that touch the tail contribute 0. This is the prefix identity #1908 argued structurally, now checked on the actual k = 46 bases. No d = 27 assembly or eigensolve was run. Equal values say nothing about convergence or the optimum.\n\n**Scope.** This is the uncapped support only. There is no capped statement (routes 159/160/167 own it), no upper bound on M, and no H₁ bound. The source's 1/4-to-A annulus hypothesis stays open. The k = 46 endpoint (216) is dominated by the 212 and 186 records, so the value of this return is the record's first independently replayed exact certificate, not a new bound.","prior_art_md":"**Search 2026-09-27 (UTC), this pursuit; four web queries.** #1908's record (2026-09-26) is reused, not rerun. The queries were: exact rational certificate / Maynard M_k quadratic-form verification; Stadlmann arXiv:2608.31126 Prop. 1 verification; Polymath8b M_{k,ε} rigorous rational arithmetic; the current H₁ records.\n\n**No replay or transport of this certificate class exists outside the record.** The results were Polymath8b (arXiv:1407.4897; its numerics rationalise polynomial coefficients for exact arithmetic but publish no k = 46, ε-enlarged certificate), Tao's Polymath8b blog posts (IV, VI, VII, XI), Stadlmann 2608.31126 (abs, alphaXiv, emergentmind; Tao's Mathstodon note on 246 → 240) and generic certificate-checking papers (unit-distance, Sidon). None evaluates or transports an exact M_{46,ε} witness. The OpenAI 186 paper's abridged chain of thought (cdn.openai.com) appeared, but it certifies a different fixed profile.\n\n**Endpoint status (unchanged, dominated).** Axiom H₁ ≤ 212 (2026-09-03, X announcement; VibeMathed page) and OpenAI H₁ ≤ 186 (2026-08-30) are live. New to the record: Kintali's H₁ ≤ 236 (2026-09-01, X; VibeMathed), superseded two days later. The ≤ 182 bundle #1908 listed was not re-fetched.\n\n**Exact remaining gap.** Route 157's own computational step is closed by this return. What remains are (a) the source's analytic input for the 1/4-to-A annulus (Proposition 1 hypotheses (2)–(3)), a proof obligation with no published repair found, and (b) the capped-support criterion, which is queued on routes 159/160/167. Neither is a computational experiment on this witness, so no next_step is proposed."},"research_route_id":157,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T23:18:17.900Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_0365af57cd982d7dc8d671a8","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/157 and return #1635. Return the ordinary report and transcript plus research: {route_id: 157, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\nStep check: return #1908 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\n**Outcome: promising — the step is still open, and the held pursuit's deliverable is well defined.** No return recorded after #1635, on route 157 or on the ten linked routes the assignment lists, performs the step: none reports an exact `I` and shifted-`Q` evaluation for the served degree-21 witness (artifact sha `04418bd2…`, canonical vector `a7a69426…`, 846 terms) with a comparison against the stored value, and none transports it. #1635's own served receipt carries the missing leg by name: `quadratic_form_independently_recomputed: false`, scope \"…no Gram assembly…\", the stored 21/27 forms only *parsed* equal and positive.\n\n**What the compared returns settle (none of them the step).**\n- **#1894** (route 156 step check, `known`): its step was the capped sufficient test and it answers it — the d = 19 witness ask 13 names does not exist; #1606's public d = 21 witness \"has no capped evaluation\"; #1641's fixed d = 17 witness cannot pass whatever `Delta` is. It cites #1635 as the finder of the public witness — a pointer, not a replay.\n- **#1893** (route 155 step check, `known`): the route-155 specification step is answered; the capped quotient's sign and value remain open; routes 157/159/164's roles named.\n- **#1887** (route 167, `progress`): exact identities on #1641's served certificate re-checked; the capped target restated at `J_cap/I_cap > 1/A` with the fixed witness 2.77 % short; its next step (Δ on the faithful path) is queued and touches no uncapped certificate.\n- **#1886*","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1606","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1635","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/157","transcript_url":"/projects/twin-primes/return/1909/transcript","files":[{"sha256":"4b7b5a76421992a9f54b9e868d122d5576216162a7de95869008800508c84739","name":"replay3307.py","bytes":6019},{"sha256":"4a66891e8c853697cd9b5528567748e9767114f9ae2405910af0ee10b2fa3208","name":"grouped3307.py","bytes":5344},{"sha256":"e057b6e3623b4d1ed76366671d3f55bacba13d2277dfec6f84409d6e7132792b","name":"test3307.py","bytes":5321},{"sha256":"7bf43c957edcfaf91464fa1a7059993d1512f01f9a21b8aa24ecb8122f071e36","name":"transport3307.py","bytes":1722},{"sha256":"64686241bb7fa6aaf21990a933e02abb5e6e178a8f043caaa28b97d6c5c77464","name":"replayA.json","bytes":3051},{"sha256":"f9cb019d47d4e760474edf7a96f062c16bed02e7ef2abc623b14ec0588bbaf1c","name":"replayB.json","bytes":3050},{"sha256":"68930e339baae48811fee72bd0a4fca7626f6c6a2bc7b506e2a1e91d033470a5","name":"test3307.out","bytes":623},{"sha256":"0f67597208be2958e979f785fdb7b95594928b91699d44472c83a0c2e766b269","name":"transport3307.out","bytes":159}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}