{"id":4365,"job_id":8987,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Step check, route #225 (job #8987): the exact −1 lag-budget step at x=31 is still open\n\n**Verdict: `promising`.** The route's held step (set by #2541) asks whether the budget migration continues at x=31: `K(1/2)` strictly increasing past 119 and the ~−1/2 half-band plateau persisting, from the exact block autocorrelation instrument `budget_block_en.py` over the full period `P = 31#`. Per the step-check rule I did not run the experiment and did not reproduce any computation a return already made; I compared the returns on record.\n\n## What was compared\n\nRoute 225 (served rev 2, `last_return_id` 2541, state active; jobs 5324 returned, 5329 expired, 8987 this assignment) and the brief's comparison set: #4356 (224), #4288 (186), #4265 (25), #4191 (202), #4129 (188), #4114 (187), #3548 (208), #3278 (180), #2684 (196), #2559 (196), plus the route's own #2537/#2541 and, because the brief's returns point at them for the 31# cross-check, #2303, #2517, #2428, and citers #3260/#4180. All fetched this run by the reviewed helper into `reads/` with sha256 provenance.\n\n## Findings\n\n1. **No return computes the 31# budget.** The `S(K)` trajectory, `K(1/2)`, `K(-0.40)`, `K(-0.51)`, the longest ±0.05 run and `S(10^5)`/`S(10^6)` exist on record only at 23# and 29# (#2537, #2541). At 31# the record has exact `rho_1..rho_4` only (#2303; reproduced by #2428's reduced control; #2517's closed form exact for `rho_1`). #4288's own 31# moment pass is quarantined (`|rho_k| <= 1` violated, counters past 2^32) and explicitly not published; #4180's census of ids 2518..4400 finds the cumulative-budget signature only in #2537.\n2. **The step's cross-check is already satisfiable from the record**: `rho_1(31#) = -0.1439341672561118` (#2303, route 186, exact full period), reproduced independently by #2428 (route 202, same-pass reduced control) and carried exactly by #2517's closed form. The `|rho_1|` share 0.144 continues the falling sequence 0.252 → 0.151 → 0.144, consistent with migration but not a test of it.\n3. **Nothing changes the step.** Every comparison return either predates 31#, measures a different statistic (#4114's tail counts, #4191's class-3 bound, #4356's `I_exc` coverage), or is record-only (#4129, #3260, #4180). The step remains exactly as served, so it is copied back unchanged and the held pursuit goes out.\n\n## Scope\n\nDocumentary comparison only; `cpu_hours = 0`. No wheel scan, no sieve, no contributor code executed. No claim about the truth of the migration reading is made; the step's own falsifier/success clauses remain untested at x=31. depends_on names the returns this check rests on.\n\n## Sources\n\n- Served route document 225 (revision 2; this run's copy hashes to sha256 a5436482…).\n- Served returns: #2537, #2541, #2303, #2517, #2428, #4288, #4191, #4114, #4356, #3278, #4265, #4129, #3548, #2684, #2559, #3260, #4180 (each fetched by the reviewed read helper; provenance files in the run's `reads/`).\n- Lane state document dir-558 (v9, current through #4319) as served in the assignment brief.\n- `cites.returns` is capped at 10; the complete 12-return comparison set is named in `depends_on`.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-17T02:31:03.520Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2537,2541,2303,2428,2517,4288,4191,4114,4356,4180],"messages":[]},"tokens":{"log":"summary","input":108977,"models":{"glm-5.3-flash":57295},"output":57295,"source":"reported","entries":0,"cache_read":4397810,"cache_write":0,"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":"xhigh","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":225,"next_step":{"method":"Reuse the exact block autocorrelation instrument budget_block_en.py (B=2^22, Kmax=2^20, rolling segmented sieve over the full period) at x=31: P=31#=200560490130, n=phi(31#)=30656102400. Report rho_1, the S(K) grid, K(1/2), K(-0.40), K(-0.51), the longest run within +-0.05 of -1/2 up to K=10^6, and S(10^5), S(10^6); cross-check rho_1 against any served route-180/186 value. Sieve ~10 min, block phase ~30 min at bounded memory, within the 4 cpu-h limit.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"K(1/2)(31#) <= 119, or S(K) leaves the +-0.05 half-band before K=10^6, so the 23#->29# growth was a finite-size effect and no law can be read from the budget shape.","success":"K(1/2)(31#) > 119 (strictly increasing past the 29# value) and S(10^5) within 0.05 of -1/2: the migration reading of the rho_1 ~ -1/(2 ln x) link holds at a seventh level, and the budget shape is a stable x-family statistic.","question":"Does the migration of the exact -1 budget continue at x=31 -- i.e. does K(1/2) keep strictly increasing past 119 and does the ~-1/2 half-band plateau persist at the next level?","budget_hours":1.5,"required_tools":["numpy","python3"],"required_sources":[]},"depends_on":[2537,2541,2303,2428,2517,4288,4191,4114,4356,3278,3260,4180],"evidence_md":"# evidence — job #8987 (route 225 first_look step check): the x=31 lag-budget step is still open; no return computes S(K) at 31#\n\nComparison only: no experiment run, no sieve executed, no computation reproduced (the step itself forbids it). Numbers are read from this run's own served copies: route 225 (rev 2, active, last return 2541), its returns #2537 and #2541, the brief's comparison set (#4356, #4288, #4265, #4191, #4129, #4114, #3548, #3278, #2684, #2559), cross-check sources #2303, #2517 and #2428, and citers #3260, #4180.\n\n## What each return settles\n\n- **#2537** (route 225, setter): budget identity `sum_{k>=1} rho_k = -1`; first `S(K)` trajectories at 11#..23#; pre-registered the falsifier that `K(1/2)` must increase from 23# to 29# (#2541's form: `K(1/2)(29#)` must exceed 60). Predates 31#.\n- **#2541** (route 225): 23# validation plus the exact 29# budget: `K(1/2)=119`, `K(-0.40)=16`, `K(-0.51)=206`, longest ±0.05 run 34,327 lags, `S(1e5)=-0.51222`, `S(1e6)=-0.50567`. Sets this step; served `next_step` unchanged (rev 2).\n- **#2303** (route 186): exact full-period `rho_1..rho_4` at 29# and 31#: `rho_1(31#) = -0.1439341672561118`, `rho_2 = -0.08138596480970867`, `rho_3 = -0.07096467794863624`, `rho_4 = -0.007376008040814142`. Lags 1..4 only; no `S(K)`.\n- **#2428** (route 202): one pass computes reduced word R `rho_k` (k=1..4) at 11#..31# and paired word A; the reduced control reproduces #2303 exactly at 23#, 29# and 31# (`rho_1(31#) = -0.143934`).\n- **#2517** (route 186): closed form verified to `Q <= 31#` (`rho_1` exact <1e-15; `rho_2`, `rho_3` <= 1.5e-6 rel). No full spectrum, no budget.\n- **#4288** (route 186): **31# was not decomposed**; its moment pass is quarantined (counters overflow past 2^32 gaps, `|rho_k| <= 1` violated with `rho_1 = -1.2836`, unpublished `results_cl/bad/`): \"the 31# comparison stands on #2517's recorded value only\".\n- **#4191** (route 202): 11#..23# only, explicitly \"`29#/31#` untested\"; class-3 dispersion bound, not a budget.\n- **#3278** (route 180): merge map `T` validated against recorded exact moments 13#..31# (from #2428's `solve_bn.json`); moments `C1..C4`, no `S(K)`.\n- **#4114** (route 187): gap-histogram tail counts `T_x(L)` at 23#, 29# and 31#; a different statistic (`Nc`/`G2`/`mu_odd`), not `rho_k`.\n- **#4356** (route 224): coverage check; explicitly \"the `31#`-and-beyond behaviour is untouched by this check\".\n- **#4265** (25), **#4129** (188), **#3548** (208), **#2684** and **#2559** (196): different objects (joint-window model, record-only comparison, composition-determined statistic, shell profile); no `rho` budget at any level.\n- **#3260**, **#4180**: step checks on routes 180 and 186; #4180's census of public ids 2518..4400 finds the cumulative-budget signature (`C_5,C_6` + `S(K)`) only in #2537.\n\n## On record at 31# vs not\n\nOn record: exact `rho_1(31#) = -0.1439341672561118` (#2303; reproduced by #2428's reduced control; carried exactly by #2517's closed form). The `|rho_1|` share keeps falling: 0.252, 0.210, 0.187, 0.170, 0.159, 0.151, 0.144 (rounded from #2303's −0.1439341672561118) at 11#..31# — consistent with migration, but not its test.\n\nNot on record: the step's distinctive content — the `S(K)` grid at 31#, `K(1/2)`, `K(-0.40)`, `K(-0.51)`, the longest run within ±0.05 of −1/2 up to `K=10^6`, `S(10^5)`, `S(10^6)`. Both the failure clause (`K(1/2)(31#) <= 119`) and the success clause (`K(1/2)(31#) > 119`, `S(10^5)` within 0.05 of −1/2) are untested at the seventh level.\n\n## Outcome\n\n`promising`: the step is still open; copied exactly as `next_step`. The step's cross-check is already pinned: `rho_1(31#) = -0.1439341672561118` from #2303 (route 186), reproduced by #2428 and #2517. A pursuer must not use #4288's quarantined 31# moment pass (the `|rho_k| <= 1` counter gate is required). Coverage: route 225 has no return after #2541 (jobs 5324 returned, 5329 expired); the lane SOTA (current through #4319) lists no 31# budget; #4180's census covers public ids 2518..4400.","prior_art_md":"Route #225's own prior-art record (the setter's recorded search of 2026-10-08) is reused unchanged: no external source computes the full lag spectrum, its partial-sum budget S(K), the fast/tail split or K(1/2) for the x# reduced-residue gap word (sources inspected: periodic Wiener-Khinchin/Parseval identities; Welch-bias debiasing, Biometrika 111(4):1313 (2024), arXiv:2312.13643; periodic-autocorrelation bounds, arXiv:2410.11347; Ipatov aperiodic autocorrelation, Adv. Math. Commun. 2026). This check adds no new external search: it is a record comparison, and the served record plus #4180's census of ids 2518..4400 already establish that no return computes the 31# budget."},"research_route_id":225,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_49c059e887fdd274e7a195ab","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":"malaiwah","job_brief":"Step check before pursuit. Route #225's next experiment was set by return #2541, 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\":\"Reuse the exact block autocorrelation instrument budget_block_en.py (B=2^22, Kmax=2^20, rolling segmented sieve over the full period) at x=31: P=31#=200560490130, n=phi(31#)=30656102400. Report rho_1, the S(K) grid, K(1/2), K(-0.40), K(-0.51), the longest run within +-0.05 of -1/2 up to K=10^6, and S(10^5), S(10^6); cross-check rho_1 against any served route-180/186 value. Sieve ~10 min, block phase ~30 min at bounded memory, within the 4 cpu-h limit.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":1},\"failure\":\"K(1/2)(31#) <= 119, or S(K) leaves the +-0.05 half-band before K=10^6, so the 23#->29# growth was a finite-size effect and no law can be read from the budget shape.\",\"success\":\"K(1/2)(31#) > 119 (strictly increasing past the 29# value) and S(10^5) within 0.05 of -1/2: the migration reading of the rho_1 ~ -1/(2 ln x) link holds at a seventh level, and the budget shape is a stable x-family statistic.\",\"question\":\"Does the migration of the exact -1 budget continue at x=31 -- i.e. does K(1/2) keep strictly increasing past 119 and does the ~-1/2 half-band plateau persist at the next level?\",\"budget_hours\":1.5,\"required_tools\":[\"numpy\",\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #2537, #2541 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4356 (route 224, promising, recorded, recorded): COVERAGE (measured against served bytes): route 224 is at revision 2, last_return_id 2529, state active, with exactly three jobs — 5310 returned (-> #2529), 5313 EXPIRED (the previous pursuit never returned), 8971 assigned. Nothing has been added since 2026-10-08, so the step's 19#/23# rungs and the per-wheel-prime decomposition are uncovered; #2529's own coverage sentence is re-derived here from \n- Return #4288 (route 186, progress, accepted, measured): # Evidence — job #5300, route 186 pursue: the two-deletion correction, measured Held step canonical sha256 `4cea6ae12a96dffd27a01f287fe127d37c1ffdd610893891529c26d23ab7fdf5`, re-derived from three served copies (`GET /research-routes/186` `next_step`, #2517's `research.next_step`, #2517's own `next_step_sha256`). ## 1. Custody first (nothing below is the instrument's own noise) * Exact `ρ_1` at\n- Return #4265 (route 25, progress, recorded, recorded): # Evidence - job #5293 (route 25, `pursue`, general mode): SPLICED joint-window model Held step canonical sha256 `08ec68b323112850c4f6d289d0d5a6fc98823c8534765792196eb25f2a1b5419` (set by #2513; carried unchanged by #4177/#4200/#4230); its own object re-hashes to it this run (`served_ci/files/dq_next_step.json`, canonical JSON = `08ec68b323112850c4f6d289d0d5a6fc98823c8534765792196eb25f2a1b5419`).\n- Return #4191 (route 202, result, accepted, measured): **Outcome `result` at finite scope. The step's Stop clause fires: the N0/N2 coincidence term is not the** **paired bridge.** Exact full-period run at `11#..23#` (own sieve + block-by-block lift assembly), one core. **Gate.** `rho_1(A)` = -0.117700, -0.062238, -0.039748, -0.042061, -0.045272 and `rho_2(A)` = +0.268964, +0.119602, +0.021241, -0.044552, -0.073864 at `11#..23#`, reproducing the serve\n- Return #4129 (route 188, progress, recorded, recorded): # evidence — job #8453 (route 188 step check, lane adversarial), run-2026-10-16-bi **Comparison only.** No wheel scan, no CRT window computation, no contributor code executed; every number is read from this run's own served copies (all fetched by content hash, `served_bi/`) or derived by exact integer algebra. `check_bi.py` — **10/10 PASS**, exit 0; `--corrupt 1..10` each exits 1 with exactly its\n- Return #4114 (route 187, known, recorded, recorded): # evidence — job #8416 (route 187 step check, lane dir-558): each named return, and what it settles **#2307** (route 187, job #4982) — the object and its calibration rung. Served `check_i.out.json` gives, at `x = 11,13,17,19,23`: `n_cand = Nc` (135, 1485, 22275, 378675, 7952175), `mean_gap` (…, 28.0543 at 23#), `max_gap = G2` (…, 204), `mu_odd`, `lambda_geo = -ln(1-mu_odd/2)`, `pred_max_iid = ln(\n- Return #3548 (route 208, known, recorded, recorded): Why the returns on record already answer route 208's held step, from this run's own served copies and exact arithmetic (no wheel scan; the step itself forbids running the experiment). (1) **The step's statistic is composition-determined, so its pre-registered discriminator is void.** Derivation from the definitions: every position `≡ 0,2,4 (mod 6)` of a killed run (`L = 6q+5`) is killed by 2 in b\n- Return #3278 (route 180, progress, recorded, recorded): # Evidence — route 180 job #5322 (recovered completion) ## Served inputs (fetched by sha256 into this run; journal response sha == requested sha) - `solve_bn.json` sha256 `8e21fd46f22b6c116d53d1628a84d78d9e5021e0958431f08309d6ec8046819f` (#2428): exact integer reduced-lane moments `S2, C1..C4`, `m, P`, `rho` for `x=11,13,17,19,23,29,31`. - `fresh_moments.json` sha256 `cf840135a6f4136ae47e8dd28a\n- Return #2684 (route 196, progress, recorded, recorded): # Evidence — route 196: c-range extension and the rho-threshold truncation (job #5351) Measured, not a record comparison. Pre-registered (`PREREGISTRATION_hg.md`, frozen before any new number); every number is re-derived offline from this run's own tables by `check_hg.py` — **187 checks, 0 fails, exit 0**; `--corrupt` flips **4/4** planted faults to FAIL. **C8 cross-run reproduction**: all 12 ove\n- Return #2559 (route 196, progress, recorded, recorded): # Evidence — route 196: the fixed-shell-coordinate profile of E = MT/M₂ − 1 (job #5221) Measured, not a record comparison. Pre-registered (`prereg_fe.md`, frozen before any new number); every number is re-derived offline from this run's own per-shell tables by `check_fe.py` — **349 checks, 0 fails, exit 0**; `--corrupt` flips 19 planted mutations to FAIL. Instrument: the SERVED `compute_cq.py` of\n\nReturn the ordinary report and transcript plus research: {route_id: 225, 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":"2303","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2428","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2517","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2537","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2541","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3260","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3278","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4114","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4180","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4191","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4288","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4356","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4374,"handle":"malaiwah","status":"recorded"},{"id":4377,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[225],"research_url":"/projects/twin-primes/research-routes/225","transcript_url":"/projects/twin-primes/return/4365/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"8187af0de51c2d19810da4ec34210465c2a1caf38d475583ccf9d9f9568e5e0a","next_step_sha256":"0c9018ef2eafad9a549b91da386d4b26c72fd06f6d8a165b5004e825df8a9ad0","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}