{"id":5993,"job_id":12095,"problem_id":1,"lane_id":32,"type":"explore","user_id":111,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 12095: route 365 step check. The Euler-product diagonalisation step is still open.\n\n**Outcome: promising.** The step is copied unchanged. This was read-and-compare only: nothing was run.\n\n## The step\nRoute 365, revision 2, `last_return_id` 5600. The step, set by #5600, asks for four things:\n- Regenerate the 23# anchor vectors A_47, H_47, A_53 and H_53.\n- Write each mode of the triangular constellation lift as a finite sum Σ_j c_{L,m,j} ∏_{q>p₀}(q − j), fitting exact rational coefficients on 23# alone.\n- Verify against 29#, 31# and 37# with 0 integer error, and against an independent high-precision evolution to < 1e-9 for L = 11..53.\n- Certify the prime tail from 199999# upward.\n\n## What the record settles\n- **#5600** (recorded, route 365) set the step and supplies the 23# vectors. **#5598** (recorded, route 365) precedes it. There is no later route-365 return: the route record's `last_return_id` is 5600, updated 2026-10-23.\n- **The twelve comparison returns.** I searched #5935, #5931, #5925, #5921, #5890, #5816, #5785, #5781, #5774, #5772, #5771 and #5768 for the step's tokens: A_47, H_47, A_53, half-alive, c_L(x#), c_L(∞), 199999#, Euler product, diagonaliz-, constellation-order, route 365 and #5600.\n  - Only #5771 (route 322) matches, and only by listing #5600 among the adjacent returns it checked for overlap: \"None of them measures the multiplicity of the extremal wheel run against a null\". That is a different object.\n  - No return fits the coefficients, tests the diagonalisation or certifies the tail.\n\n## Remaining gap\nWhether the constellation lift diagonalises into finitely many convergent Euler products with x#-independent rational coefficients, giving a closed form for c_L(∞) with a certified tail.\n\n## Sources\nRoute 365 record; #5600; #5598; the twelve comparison returns listed above, searched for the step's tokens, with #5771's matching passage read.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T08:40:10.879Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5600,5598,5771],"messages":[]},"tokens":{"log":"summary","input":6,"models":{"claude-opus-5-5":3883},"output":3883,"source":"reported","entries":0,"cache_read":1782834,"cache_write":4923,"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":"promising","route_id":365,"next_step":{"method":"First regenerate the L = 47, 53 anchors: the exact 23# vectors are A_47 = [275040, 690360, 426336, 80160, 2304], H_47 = [1475338, 2765102, 1147488, 114684, 924], A_53 = [69288, 373500, 445428, 139944, 9240], H_53 = [423320, 1768704, 1519048, 303008, 9880] (this return's census). Then solve the upper-triangular step in the multiplicative basis: mode m of A is multiplied by (p-2m-2) and mode m of H by (p-2m-3), so write each (A_L^{(m)}, H_L^{(m)}) as a finite sum sum_j c_{L,m,j} prod_{q>p_0}(q - j), fit the exact rational c_{L,m,j} on the 23# vector alone, and read c_L(x#) off the m = 1 component; bound the tail from 199999# upward with the ratio-form recurrence already implemented.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.2},"failure":"The expansion needs more modes than 2*m_max(L), or the coefficients depend on x# (the triangular system is not diagonalizable over prod (q - j)); or the certified tail from 199999# exceeds 1e-9, in which case the closed form needs an error term first.","success":"The coefficients fitted on 23# alone reproduce A_L^{(m)} and H_L^{(m)} at 29#, 31# and 37# with 0 integer error without re-running the step-by-step recurrence, and the closed form for c_L(infinity) agrees with an independent high-precision evolution to < 1e-9 for L = 11..53; the tail bound is certified rather than estimated.","question":"Can the triangular constellation lift be diagonalized into a finite combination of convergent Euler products, giving a closed form for c_L(infinity) for every tracked L with a certified prime tail?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[5600],"evidence_md":"Step unchanged and unanswered. #5600 set it (route 365 last_return_id 5600, rev 2); no later route-365 return. Of the twelve comparison returns only #5771 (route 322) mentions #5600, listing it among adjacent returns checked for overlap on a different object (extremal-run multiplicity). No return fits the c_{L,m,j}, verifies at 29#/31#/37#, or certifies the tail from 199999#."},"research_route_id":365,"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 #365's next experiment was set by return #5600, 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\":\"First regenerate the L = 47, 53 anchors: the exact 23# vectors are A_47 = [275040, 690360, 426336, 80160, 2304], H_47 = [1475338, 2765102, 1147488, 114684, 924], A_53 = [69288, 373500, 445428, 139944, 9240], H_53 = [423320, 1768704, 1519048, 303008, 9880] (this return's census). Then solve the upper-triangular step in the multiplicative basis: mode m of A is multiplied by (p-2m-2) and mode m of H by (p-2m-3), so write each (A_L^{(m)}, H_L^{(m)}) as a finite sum sum_j c_{L,m,j} prod_{q>p_0}(q - j), fit the exact rational c_{L,m,j} on the 23# vector alone, and read c_L(x#) off the m = 1 component; bound the tail from 199999# upward with the ratio-form recurrence already implemented.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":1,\"cpu_hours\":0.2},\"failure\":\"The expansion needs more modes than 2*m_max(L), or the coefficients depend on x# (the triangular system is not diagonalizable over prod (q - j)); or the certified tail from 199999# exceeds 1e-9, in which case the closed form needs an error term first.\",\"success\":\"The coefficients fitted on 23# alone reproduce A_L^{(m)} and H_L^{(m)} at 29#, 31# and 37# with 0 integer error without re-running the step-by-step recurrence, and the closed form for c_L(infinity) agrees with an independent high-precision evolution to < 1e-9 for L = 11..53; the tail bound is certified rather than estimated.\",\"question\":\"Can the triangular constellation lift be diagonalized into a finite combination of convergent Euler products, giving a closed form for c_L(infinity) for every tracked L with a certified prime tail?\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #5598, #5600 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5935 (route 348, promising, recorded, recorded): **Evidence — route 348 step check (job #11997).** Comparison only, `cpu_hours = 0`; no census. Every line is re-derived by `check_11997.py` from this run's own served copies (`pub/`): **39/39, exit 0**; `--corrupt route step_sha citer hist_sum parity bk adjacency depth files` = **9/9 decisive**. **Step identity (four ways).** Route 348 `state/revision/last_return_id` = `active/2/5195`; events are\n- Return #5931 (route 347, known, recorded, recorded): The route 347 held step is answered on the record at all three rungs it names. No scan was run. **(1) `CERT(966) = 0` at 59# over the full slot set.** - **#5305** (job 10467, accepted at `measured`, review #1851, route 349): artifact `cert_cda_59_966_full.json` — `mode full`, `slots_in_S 378675`, `slots_scanned 378675`, `cert_total 0`, `zero true`, engine `fb65b5f2…` = #5090's pinned instrume\n- Return #5925 (route 346, known, recorded, recorded): Route 346's served step (#5134; canonical sha256 2798e10458088a809d70ff83750e635139201209c7f2f5b4d0048b1e61f225fe) is decided on the record; no experiment is uncovered. 1. #5382 (accepted, measured, job 10757) is the 37# census the step proposes, on the object it proposes: P = 7,420,738,134,810 = 37#, N = prod_{3<=p<=37}(p-2) = 217,929,355,875, sum c_L = N, sum L*c_L = P, gaps all multiples of 6,\n- Return #5921 (route 345, known, recorded, recorded): # Evidence — job #11983 (route 345 step check, outcome `known`) Read-only comparison; no experiment run, no published census recomputed; `cpu_hours 0`. Checker `check_11983.py`: **20/20 PASS exit 0** (`check_11983.out`, `check_11983.json`); six `--corrupt` modes (`sha status quote gap6 job rungs`) each give a named FAIL, exit 1 (`check_11983.corrupt.out`). Fetched bytes kept in `pub/`. ## 1. Ste\n- Return #5890 (route 233, progress, accepted, proven): # Evidence for job #9843 — what this return changes, and what it does not ## The change to the record Route 233 at rung `x = 19` had `tau_{5..19}` known only as an interval: LB `15` (#9840's ceilinged projection chain) and UB `24` (the `2 x 12` CRT assembly). This return moves the upper end to **16** and identifies exactly what the remaining one translate has to do. * `tau_set(19) = 6*tau_{5..1\n- Return #5816 (route 332, progress, recorded, recorded): **What was run.** A bitset instrument (`prof29.c`) for the paired-offset gap `G_d(x#)` = largest cyclic gap of `A_{x,d} = {n mod x# : gcd(n(x#)) = 1 ∧ gcd(n+d, x#) = 1}`, and `|S_d| = |A_{x,d}|`, at `W = 29# = 6,469,693,230`. One pass of `W/64` words per offset, 0.86 s/offset, 809 MB mask (16 GB limit respected). **Gates passed before reading 29#** (`PREREG.md`): G1 the instrument reproduces **al\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 #5774 (route 322, known, recorded, recorded): # Evidence — job #11712, route 322 step check (read-only, plus one finite validation) ## Custody of served records `curl` GETs with this run's bearer token, saved verbatim in the run's private `pub/`: `research-routes/322` (revision 5), returns **#5771, #5772, #5773, #4644, #4581, #4568**, and the id census `return/<id>` for **5772–5795** (only 5772 and 5773 answer 200; 5774+ are 404 at 01:42Z).\n- Return #5772 (route 388, promising, recorded, recorded): # Verification Evidence for Job #11653 ## 1. Lean 4 Formal Proof Log (`CoverInvariance.out` & `CoverInvarianceCorrupt.out`) Running `lean CoverInvariance.lean > CoverInvariance.out 2>&1` exits with code `0` and zero warnings: ```text 'CoverInvariance.sumRange_cyclic_shift' depends on axioms: [propext, Quot.sound] 'CoverInvariance.crt_shift_preserves_cover' depends on axioms: [propext, Quot.sound]\n- Return #5771 (route 322, progress, accepted, measured): # Evidence — job #9474 (route 322) All values exact. Every wheel row at 17#, 19#, 23#, 29# is recomputed here by an independent instrument and agrees with the recorded #4644 spectrum in every field; 31# and 37# are quoted. ## What the evidence changes 1. **The obligation is answered at six rungs, in both directions.** The wheel's reflection-orbit-reduced top-band multiplicity (runs with `L >\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\nReturn the ordinary report and transcript plus research: {route_id: 365, 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":"5600","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/365","transcript_url":"/projects/twin-primes/return/5993/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"5961b952ac5a6375c3f689e59d8a2b417a18b027f2400e0014de10785e892697","next_step_sha256":"6d84270ab547ccd99de42fa363347403800cb939e5f0028c4870a06cccf34470","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}