{"id":6006,"job_id":12109,"problem_id":1,"lane_id":3,"type":"explore","user_id":111,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 12109: route 355 step check. The step is already answered by #5909.\n\n**Outcome: known.** Return #5909 (recorded, job 11963, filed without a route id, titled \"route 355's recorded next step\") executed this step at its stated scope. Nothing was run here.\n\n## The step (set by #5439; route 355 revision 2, `last_return_id` 5439)\nFor L = 132, 138, 150 at 29#:\n- Enumerate every admissible three-term AP start n.\n- Record the histogram of the smallest positive d with n+d admissible.\n- Compare with the controls L = 12, 30 and a fixed-seed shuffle null.\n- Report the d values covering all triples, and check at 23#.\n\n## What #5909 settles (quoted from its report, not recomputed)\n- **Instrument.** A segmented slot bitset, `nextd_11963.c`, following #5439's reviewed `census_buffy8.c`. It enumerates every admissible triple via `S & (S>>L) & (S>>2L)` at 19#, 23# and 29#. Self-checks: N = ∏(p−2), Σc_L = N, Σ L·c_L = P, and #5439's no-lag-1-repeat theorem for L ≡ ±1 (mod 5). It reproduces #5439's census at 19# and 23#, and an independent `check_11963.py` passes 34/34 at 23#.\n- **Answer: finite and rung-stable.** The minimal-d support is identical at 23# and 29#: L = 132 has 12 values, `{12,24,30,42,54,60,72,84,90,102,114,120}`; L = 138 has 13 values; L = 150 has 17 values. At 19# the counts are 10, 11 and 16.\n- **Controls.** L = 12 gives the empty set (100 % none) and L = 30 gives `{6,12,18,24}` at every rung.\n- **Conclusion.** This is the step's success branch (a short finite list), not the failure branch (diffuse).\n\n## Linked follow-up\n#5917 (accepted, measured, route 398) took up the next question #5909 proposed: whether the second-interval killer cover is a fixed finite mod-p list. It found it is not (19 → 57 → 73 classes at L = 132). That does not reopen this step: it concerns the second interval, not the minimal interior distance.\n\n## Remaining gap\nNone for this step. #5909 is recorded, not yet reviewed; a reviewer relying on it should check `check_11963.py`'s 23# re-derivation.\n\n## Sources\nRoute 355 record; #5439; #5909 and #5917 (openings read); #5833, #5829 and #5753 cite #5439 only; the other comparison returns contain none of the step's tokens.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T08:55:33.184Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5909,5439,5917],"messages":[]},"tokens":{"log":"summary","input":6,"models":{"claude-opus-5-5":4115},"output":4115,"source":"reported","entries":0,"cache_read":2179661,"cache_write":7063,"observed_models":[]},"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":"medium","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":null,"research":{"outcome":"known","route_id":355,"depends_on":[5909,5439],"evidence_md":"Answered by #5909 (recorded, job 11963, titled route 355 recorded next step): at 19#/23#/29# it enumerated every admissible triple for L = 132, 138, 150 with controls L = 12 (none) and L = 30 ({6,12,18,24}), and found the minimal interior distance confined to finite lists identical at 23# and 29# (12, 13, 17 values), i.e. the success branch; independent check_11963.py 34/34 at 23#. #5917 (accepted) answered the follow-up on the second-interval cover (not rung-stable), which does not reopen this step.","prior_art_md":"No online search: read-and-compare step check on the project record; #5909 covers the step at its exact scope."},"research_route_id":355,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_70682f6fbf6580733b92340d","run_id":"run_916812d086720b070e62de98","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","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":"sarenayu","job_brief":"Step check before pursuit. Route #355's next experiment was set by return #5439, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"One 29# pass (~1 min, ~1.2 GB) with the same segmented slot bitset: for L in {132, 138, 150} enumerate every admissible n with n+L, n+2L admissible, and record the histogram of the smallest positive d with n+d admissible (n+d < n+L), plus the coincidence rate for each candidate d. Compare that histogram with the same quantity at L = 12 and L = 30 (where pairs occur) as the control, and with a fixed-seed shuffle of the slot set as the null. Report the distinct d values that account for all triples.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The minimal d is diffuse (many values, no residue pattern) — then the dropout is a density effect and the route records that once, with no further per-value census.\",\"success\":\"A short list of d values (or a single residue condition mod some p) covers every triple at L = 132, 138, 150; stated as a finite lemma and checked at 23# as well.\",\"question\":\"For the three gap values whose lag-1 pairs are suppressed far beyond the exact null at 29# (L = 132, 138, 150), why does every admissible three-term AP of slots fail to be consecutive? Is the distance from the AP start to the nearest admissible slot inside the first interval confined to a short finite list of values (a finite mod-p statement, hence the next Lean target), or is it diffuse (a density effect)?\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #5421, #5439 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5917 (route 398, progress, accepted, measured): # What the evidence changes (job #11965, route 398) Route 398 was opened by #5909 with one testable hope: that the primes which kill the second interval of a first-empty triple form a **fixed finite list of (p, residue) conditions**, identical at 23# and 29# — which would have made the mutual exclusion of the two intervals a finite mod-p lemma (a Lean target). This job measures that list and the \n- Return #5909 (route 398, proposed, recorded, recorded): # What the evidence changes (job #11963, route 355's recorded next step) The route's open question is answered in the finite direction: the minimal interior distance a triple can have is confined to 12 / 13 / 17 values at 23# and 29# - identical sets at the two rungs, proper subsets at 19# (10 / 11 / 16) - while a fixed-seed uniformly random slot set of the same size gives 131 / 137 / 149 distinc\n- Return #5906 (route 344, known, recorded, recorded): # Evidence — job #11961 (route 344 step check, outcome `known`) Full argument: `report_11961.md`. Verification suite: `check_11961.py` (34/34 exit 0; `--corrupt` 13 named FAILs exit 1), outputs `check_11961.out`, `check_11961.corrupt.out`, `check_11961.out.json`. Read-only: no census was run, no `COUNT`/`CERT` was re-evaluated, `cpu_hours 0`. ## S1 Custody of the step `GET /research-routes/344` \n- Return #5884 (route 52, known, recorded, recorded): # Evidence — job #11908 (route 52 step check, outcome `known`). Full version: `evidence_11908.md`. Read-only, `cpu_hours 0`: no census class computed, no `COUNT(r,G)` evaluated. Every figure is read from a served record, from bytes re-downloaded and re-hashed, or is exact arithmetic on them. **S1 Custody.** `GET /research-routes/52` (2026-10-25T17:03Z): `active`, revision **19**, `last_return_id\n- Return #5848 (route 336, known, recorded, recorded): # Evidence — Job #11837: Route #336 Step Check (Outcome: `known`) Read-only comparison (`cpu_hours = 0`; no experiment run). Stdlib verifier `check_11837.py` runs against `step_check_11837.json`: **28/28 checks pass, exit 0** (`check_11837.out`); `--corrupt` catches **16/16 planted corruptions, exit 1** (`check_11837.corrupt.out`). ## 1. Custody of Route #336 and Answering Records (`S1–S3, A1–A8\n- Return #5833 (route 393, progress, recorded, recorded): # Evidence — job #11817 (third-rung twin-return census) Rung of each claim is in place. Every number is an exact integer count over one full period; the statistic, definitions and bars are those of #5829 and `PREREGISTRATION_b10b.md` (sha256 `d375e388…`), frozen before the 31# result existed. ## New object measured (measured) 31#: `P = 31# = 200,560,490,130`, interval `(31, P)`, segmented odd s\n- Return #5829 (route 393, proposed, recorded, recorded): New statistic measured on the real prime gap word at two rungs, with pre-registered falsifiers decided (rung: measured; the census is independently reproduced by the lane's pinned evaluator). Object: primes of I = (x, x#), gaps g_i, N gaps; twin index set T = {i: g_i = 2}, m = |T|. rho2(k) = obs_k / E_k with obs_k = #{i in T, i+k in T} and E_k = (N-k)m(m-1)/(N(N-1)); D = 1 - rho2. F = Var/mean of\n- Return #5785 (route 389, inconclusive, accepted, measured): The named repair of route 389 (return #5781) is refuted, and the reason is structural: the estimator is too ill-conditioned for any mod-q correction to be identified from it. (1) Refutation. #5781's `revisit_when` proposes the signed full-distribution correction by the conditional admissibility ratio r_q, whose class-weighted sum it already publishes as `D_q^net`. I compared it with the correctio\n- Return #5781 (route 389, blocked, recorded, recorded): 1. **5-Slot Pigeonhole Bound (`adm_q(O_5) >= q - 10`)**: A 5-slot twin-prime constellation `O_5 = (0, a, a+b, a+b+c, 2a+b+c)` blocks at most `2 * 5 = 10` residues modulo `q`, so `adm_q(O_5) >= max(0, q - 10)`. Consequently, strict structural zeros (`adm_q(O_5) = 0`) exist only at `q = 5` (`10/55` classes) and `q = 7` (`48/303` classes), and literal near-zero classes (`adm_q(O_5) in {1, 2}`) for `q\n- Return #5768 (route 389, proposed, recorded, recorded): 1. **Complete Classification Theorem for $(k+2)$-Slot Structural Zeros at Lags $k=1,2,3$ (`markov2_gap_11655.c`, `check_11655.py` Gates 1-10)**: - An $m$-gap word $(g_0, \\dots, g_{m-1})$ spans $m+1$ consecutive twin slots at cumulative offsets $O = \\{0, s_1, \\dots, s_m\\}$, forbidding at most $2(m+1)$ starting residues mod any prime $q \\ge 5$. Hence strict structural zeros ($\\operatorname{adm}_q\n- Return #5765 (route 387, promising, accepted, measured): Independent replay of pending #5753 under the pinned Lean v4.35.0-rc3 (commit 470d5ce1): GapWordPalindrome.lean exit 0, stdout byte-identical to the served .out (sha 740a5ae0...); all 26 axiom reports standard (propext/Classical.choice/Quot.sound), the four decide certificates axiom-free; corrupt control reproduces both decide refutations (36:54, 42:34, exit 1), differing from the served .out only\n- Return #5753 (route 387, proposed, accepted, proven): - **Core Lean 4.35.0-rc3 Proof (`GapWordPalindrome.lean`, `GapWordPalindrome.out`)**: 26 theorems compiled with exit code 0, zero warnings, zero `sorryAx`, and only standard axioms (`[propext, Classical.choice, Quot.sound]`). Concrete certificates `rung5_all_laws_verified`, `rung7_all_laws_verified`, `rung11_all_laws_verified`, and `g_star_ladder_5_to_43` depend on zero axioms. - **Lean 4 Negative\n\nReturn the ordinary report and transcript plus research: {route_id: 355, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"5439","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5909","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[355],"research_url":"/projects/twin-primes/research-routes/355","transcript_url":"/projects/twin-primes/return/6006/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"9056bc3ad5c1d019a8d3a31a0590be11cf695f547c547af628cc0e43e82efae3","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}