{"id":6105,"job_id":12251,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Step check — route 397 (job 12251, read-only)\n\nVerdict: `promising`. Route 397's served step (revision 2, set by return #5894) is still open: no\nreturn on record answers either part — (a) the k = 6, 7 extension of the half-period loss\nmeasurement, and (b) the closed form for the class-weight average `beta_k`. This was\nread-and-compare only: no experiment was run, no computation a return already made was reproduced.\n\n## 1. What was compared\n\n- The served route record `research-routes/397`: revision 2, state `active`, `obstacle` null,\n  `last_return_id` 5894, created 2026-10-25T17:28:26.066Z, updated 2026-10-25T18:45:33.094Z\n  (fetch sha256 `f99983d4e8b08ddc031600d1d3ce05565cfe37dded424f5942d3f29b599ce35a`). Its\n  `next_step.method` is #5894's k = 6, 7 step verbatim (method sha256\n  `6cef45bc3d0668a654cc1c7e916acc78fe9b8c73aea94c5a39b07b3256866c9b`, 814 characters); it is\n  reproduced field-for-field as this return's `next_step`.\n- Route 397's own returns: #5886 (recorded, job 11886) and #5894 (accepted, measured, job 11913,\n  review 2040) — none after the setter.\n- The brief's comparison list: #5947 (route 399, accepted, measured, job 12006, review 2048,\n  decided 2026-10-26T06:38:37Z) and #5943 (route 399, accepted, proven, job 12004).\n- Citation closure beyond the list: #5924 (applicability screening of #5894, recorded, job 11973)\n  and #5955 (dir-558 lane upkeep v37, recorded, job 12028) — the only returns citing #5894;\n  #5886 is cited only by #5894.\n- Open advisory findings #158871 and #158872 (review 2040, on `report_11913.md`).\n\nAll were fetched through the authenticated read helper with provenance sidecars; per-return\nmetadata, the token-scan hit table and false-positive contexts are machine-readable in\n`step_check_12251.json`.\n\n## 2. The linked returns work on different objects\n\n#5943 and #5947 study `R_m(H) = Phi_odd(H;m) − A_m(H)` — the #5616 decomposition — over all odd\nsquarefree conductors `m <= 5e4` and windows `H <= 3e5`: pointwise `R_m >= 0` refuted with exact\nwitnesses (worst `R_11445(5670) = -8.943314`), the aggregate `S(H) = Σ_m R_m(H)` positive at every\ntested `H`, with an H-independent negative cap `Σ_m min_ρ R_m = -72.1123`. None of the step's\nobjects appears in either report. Token scan over both full reports: **zero** occurrences of\n`beta_k`, `B(N)`, `B(1)`, `r_6`, `r_7`, `k = 6`, `k = 7`, CRT family, `qS`, any of the five\nmeasured `r_k` values, or the k = 5 normalised constant `0.3668`. Their only `half-period`\noccurrences (#5943: the negative R-window is \"centred exactly on the half-period `rho = m/2`,\nwhere `A_m` peaks\"; #5947: `rho = m/2` as the cheap detector for failing conductors, plus artifact\nnames) are A_m-window statements on general conductors, not the loss anchor `H = (m+1) / 2` for\n`B(N) / Phi_odd` on #5793's CRT family.\n\nThe distinction is substantive, not cosmetic: a negative `R_m` window says `A_m > Phi_odd` there;\nthe route-397 step compares #5792 §3's paired bound `B(N)` against `Phi_odd` — a different pair —\non the special family `m = Pq` with `qS ≡ 1 (mod P)`, k up to 7, `m ~ 1e25` and `1e29`, entirely\noutside #5947's `m <= 5e4` conductor range. Neither return evaluates `B(N)`, `beta_k`, or any\nmember of the family; #5947's cap-growth constant (two-point exponent 2.14 on route 399's\naggregate) is a different quantity on different objects and neither measures nor bounds the\nCRT-family loss.\n\n## 3. The step's two parts on the record\n\n- **(a) r_6, r_7: unmeasured.** The loss is measured only at k = 1..5 (#5894: 2.810955, 2.855922,\n  2.863046, 2.851536, 2.875735; #5886 has k = 1, 2 only and computes no k ≥ 3 value — its own\n  scope statement is \"no claim for other m, other H, or k >= 3\"). No\n  k = 6 or k = 7 loss value exists anywhere in the fetched record. #5924's screening recommends\n  the extension (\"the parameterized loss and check scripts reach k = 6, 7 cheaply\") and states it\n  ran no experiment and reproduced no computation; nothing was measured by it.\n- **(b) beta_k: defined, never evaluated.** The class-weight average\n  `beta_k = (1/m) Σ_h (1−2|h| / m) 2^{−z(h)}` over the `3^{omega(m)}` classes appears only inside\n  the step text and the route's own returns; no return evaluates it and no closed form exists on\n  record. The two later record-maintenance returns agree it is open: #5924 (\"Boundedness of `r_k`\n  in k is not established. Five values and a plateau are evidence, not proof; the closed-form\n  class-weight count is open\") and #5955's folding row for #5894 (\"Next step: a closed form for\n  the class-weight count\").\n\n## 4. Corrections carried by the record (advisory, open)\n\nTrusted review 2040's two findings on #5894's served text shape how the pursuit must execute the\nstep but do not answer it:\n\n- **#158871**: the lemma's increment from `N−1` to `N` is `(2/m) w(N)(h² − m²/(4N²))` (factor\n  `2/m` missing in the served text), and `B(0) − B(1) = w(1)(2m−1)/(2m)`, not `(2/m)w(1)`;\n  \"seven consecutive measured steps\" is five values (k = 1..5); \"`r_k = 2.86 ± 0.03`\" excludes\n  `r_1 = 2.811` — restrict to k = 2..5 or widen. Signs and the conclusion `min_N B(N) = B(1)` are\n  unaffected; the five gate values `r_1..r_5` stand (review 2040's exact recomputation, per\n  #5924's screening record).\n- **#158872**: attribution — #5886 §5 already states `Phi_odd` is cheap via the `3^{k+1}` CRT\n  classes and §6 proposes the class sum; the new instruments in #5894 are the `a_m` correction,\n  the `B(1)` lemma and the `w >= 0` truncation bound.\n\n## 5. Decision\n\n`promising`: the step is still open in both parts, so it is copied exactly as `next_step` (all\nserved fields verbatim: method, question, success, failure, compute, budget_hours,\nrequired_tools, required_sources, progress). No new step invented, no computation substituted, no\nstep text edited. The held pursuit goes out with this note; these returns never hold it again.\n\n## Not claimed\n\nNo new mathematics, no bound, no closure of (a) or (b). Nothing computed beyond the authenticated\nfetches, JSON parses and the token scan. No files were executed.\n\n## Sources\n\n- Route record: `research-routes/397` (revision 2; fetch sha256\n  `f99983d4e8b08ddc031600d1d3ce05565cfe37dded424f5942d3f29b599ce35a`).\n- Return #5886 (recorded, job 11886) and #5894 (accepted, measured, job 11913, review 2040) —\n  route 397's own returns.\n- Returns #5943 (accepted, proven, job 12004) and #5947 (accepted, measured, job 12006, review\n  2048) — route 399, the brief's comparison list.\n- Returns #5924 (recorded, job 11973) and #5955 (recorded, job 12028) — citation closure.\n- Findings #158871 and #158872 (open, advisory; review 2040, on `report_11913.md`).\n","patch":null,"cpu_hours":0,"hashes":{"step_check_12251.json":"4794d8e01a22a7f24ee486dedf70898296051986e6cd10f38c1cb4eea0ebc0f5"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T21:19:26.083Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5886,5894,5924,5943,5947,5955],"messages":[]},"tokens":{"log":"summary","input":190187,"models":{"glm-5.3-flash":96700},"output":96700,"source":"reported","entries":0,"cache_read":16066826,"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":397,"next_step":{"method":"Same objects and instrument (loss_11913.py, unchanged definitions): build the k=6,7 instances (the 6 and 7 smallest primes above 96 and 112, q = least prime > P with qS = 1 mod P, m = Pq ~ 1e25 and 1e29), Phi_odd by the 3^{omega(m)} class formula and B(1) = (m/2)[(1-1/m)^2 w(1) + Q(1)] with Q truncated at N0 = 200000 and closed by mu_m 2^{k+1}/N0. In the same run evaluate beta_k = (1/m) sum over the 3^{omega(m)} classes of (1-2|h|/m)2^{-z(h)}, i.e. the object that predicts Phi_odd ~ beta_k 2^{k-1} m mu_m, for k = 1..7, and compare beta_k/(2^{k-1}) with the measured Phi_odd/(m mu_m 2^{k-1}) at every k. The gate is unchanged: the instrument must reproduce r_1..r_5 = 2.810955, 2.855922, 2.863046, 2.851536, 2.875735 before k=6,7 are read, and it must keep the a_m shift (worth 58-61% of the naive class sum).","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.5},"failure":"r_6 or r_7 outside [2.70,3.00], or beta_k/(2^{k-1}) drifting by more than 5% per step: the k=1..5 plateau is a small-k effect, the loss is not bounded at the peak, and #5792 section 3 must be treated as peak-unusable; report the ratios and stop rather than fitting a constant.","success":"r_6, r_7 both inside [2.70, 3.00] and beta_k/(2^{k-1}) staying within 1% of its k=5 value 0.3668: the plateau is structural, so the ~2.86x loss at the peak is bounded over every measured k and route 397 may consume #5792 section 3 without a local-mass input, which #5793 forbids.","question":"Does the flat loss r_k = 2.86 +- 0.03 continue at k = 6, 7, and does the class-weight average beta_k = (1/m) sum_{h in classes}(1-2|h|/m)2^{-z(h)} follow a closed form, which would turn the measured plateau into a proof that the loss is bounded in k rather than a five-point observation?","budget_hours":1.5,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[5886,5894,5924,5943,5947,5955],"evidence_md":"# Evidence — job #12251, route 397 step check (read-only)\n\nQuestion: do the returns on record answer the step route 397 serves (set by #5894)? Method:\nauthenticated served-record fetch + full-text read of every compared return + distinctive-token\nscan. Nothing recomputed; no k = 6, 7 instance built, no `B(N)`/`beta_k` evaluated.\n\n## 1. The step is served unmoved\n\n`research-routes/397`: revision 2, active, obstacle null, `last_return_id` 5894, updated\n2026-10-25T18:45:33.094Z (fetch sha256 `f99983d4…`). `next_step.method` = #5894's k = 6, 7 step\nverbatim (sha256 `6cef45bc…`, 814 chars: the k = 6, 7 CRT-family instances, class-formula\n`Phi_odd`, the N0 = 200000-truncated `B(1)`, `beta_k` for k = 1..7, gate = reproduce `r_1..r_5`\nfirst, keep the `a_m` shift). Copied\nfield-for-field as this return's `next_step`. All sources fetched via the authenticated read\nhelper into `RUN/reads/` with provenance sidecars.\n\n## 2. Comparison rows\n\n| return | route | status/rung | job | objects | establishes | (a)? | (b)? |\n|---|---|---|---|---|---|---|---|\n| #5886 | 397 | recorded | 11886 | B(N) / Phi_odd, k = 1,2 | r_1 = 2.8110, r_2 = 2.8559; grid min N = 1; no k ≥ 3 value computed (its O(m) tail obstacle) | no | no |\n| #5894 | 397 | accepted/measured | 11913 | same, k = 1..5 | r_1..r_5 measured; lemma min_N B = B(1); class formula; truncation bound; sets this step | no | no |\n| #5924 | — | recorded | 11973 | screening of #5894 | runs nothing; \"Boundedness of `r_k` in k is not established... the closed-form class-weight count is open\" | no | no |\n| #5943 | 399 | accepted/proven | 12004 | R_m = Phi_odd − A_m, m ≤ 5e3 | pointwise R_m ≥ 0 refuted (R_1085(527) = −0.0592); aggregate S(H) positive | no | no |\n| #5947 | 399 | accepted/measured | 12006 | R_m, m ≤ 5e4, H ≤ 3e5 | 86 failing conductors, worst −8.943314; H-indep cap −72.1123; S(H) > 0 on [101, 5e4] | no | no |\n| #5955 | — | recorded | 12028 | lane upkeep v37 | folds #5894/#5943; \"Next step: a closed form for the class-weight count\" | no | no |\n\ndepends_on = all six: the route's own returns (5886, 5894), the brief's comparison list (5943,\n5947), the citation-closure maintenance returns (5924, 5955).\n\n## 3. Token scan\n\nOver each full `report_md`: #5943 and #5947 contain **zero** of `beta_k`, `B(N)`, `B(1)`, `r_6`,\n`r_7`, `k = 6/7`, CRT family, `qS`, any of the five `r_k` values, `0.3668`. Their only\n`half-period` hits are A_m statements: #5943 \"the negative set ... centred exactly on the\nhalf-period `rho = m/2`, where `A_m` peaks\"; #5947 \"`rho = m/2` is the cheap detector\" + artifact\nnames. #5924's two `k = 6` hits are its recommendation (\"scripts reach k = 6, 7 cheaply\") in a\nreport that \"runs no experiment\"; #5955's `B(N)`/`B(1)`/`class-weight` hits are its #5894 folding\nrow.\n\n## 4. Why the linked returns cannot answer\n\n#5943/#5947 compare `Phi_odd` against `A_m` (#5616's aligned class) over squarefree conductors\n≤ 5e4 at windows ≤ 3e5. The step compares #5792 §3's bound `B(N)` against `Phi_odd` at the single\nanchor `H = (m+1) / 2` on the family `m = Pq` (`qS ≡ 1 mod P`), whose k = 6, 7 instances (~1e25,\n~1e29) lie 5 and 9 orders beyond that range — a different inequality pair. #5947's cap constant\n(−72.1123) concerns route 399's aggregate `S(H)`; `beta_k` occurs in neither report.\n\n## 5. Unanswered part of the obligation\n\n(a) No `r_6`/`r_7` measurement exists anywhere on record; the furthest extension recommendation\n(#5924) executed nothing. (b) `beta_k` is defined only in the step text and the route's own\nreturns; no evaluation, no closed form — named open by #5924 and #5955. Context that stands:\n`r_1..r_5` flat (max/min 1.0230) and `min_N B(N) = B(1)`, with review 2040's advisory corrections\n(#158871: increment factor `2/m`, `B(0) − B(1) = w(1)(2m−1)/(2m)`, five values, `r_k = 2.86 ± 0.03`\nexcludes `r_1`) carried beside the served text; #158872 fixes attribution (#5886 §5/§6 class-sum\nidea; new instruments: `a_m` correction, `B(1)` lemma, `w ≥ 0` truncation bound).","prior_art_md":"# Prior art (coverage comparison, no new method)\n\nThis return claims no new method or bound; it is a coverage decision on route 397's served step,\nso the prior-art check IS the record comparison. Inspected (all via the authenticated read\nhelper, local sha256 recorded in provenance sidecars): `research-routes/397` revision 2 (fetch\nsha256 `f99983d4e8b08ddc031600d1d3ce05565cfe37dded424f5942d3f29b599ce35a`, updated\n2026-10-25T18:45:33.094Z); route 397's own returns #5886 (recorded, job 11886) and #5894\n(accepted, measured, review 2040); the brief's comparison returns #5943 (route 399, accepted,\nproven, job 12004) and #5947 (route 399, accepted, measured, job 12006, review 2048); the\ncitation-closure returns #5924 (applicability screening of #5894, job 11973) and #5955 (dir-558\nlane upkeep v37, job 12028); open advisory findings #158871/#158872 on `report_11913.md`\n(review 2040).\n\nThe step's own academic prior art is the route's served record: #5894's `prior_art_11913.md`\n(2026-10-25 ~19:40Z) ran two search-level queries — (1) truncated `sum_n e(na/m) over n^2`\n(root-of-unity power sums) and polylogarithm kernels; (2) the CRT-product Fourier weight\n`prod (1+cos(4 pi n xi_p/p))` with class support and local-mass bounds — with no hit on the\nstep's objects, and recorded limits (no arXiv API; the folder's recorded 403/0-entry access gap;\nno paywalled PDFs; no MathSciNet/zbMATH). That search record is reused unchanged; this return\nadds no new literature search because it proposes no method. The lane's served state of the art\n(dir-558 v39, current through #5999) contains no `beta_k` evaluation, no k = 6, 7 loss\nmeasurement, and no closed form for the class-weight count."},"research_route_id":397,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_4c10cd22eab51fd05dc239d2","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 #397's next experiment was set by return #5894, 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\":\"Same objects and instrument (loss_11913.py, unchanged definitions): build the k=6,7 instances (the 6 and 7 smallest primes above 96 and 112, q = least prime > P with qS = 1 mod P, m = Pq ~ 1e25 and 1e29), Phi_odd by the 3^{omega(m)} class formula and B(1) = (m/2)[(1-1/m)^2 w(1) + Q(1)] with Q truncated at N0 = 200000 and closed by mu_m 2^{k+1}/N0. In the same run evaluate beta_k = (1/m) sum over the 3^{omega(m)} classes of (1-2|h|/m)2^{-z(h)}, i.e. the object that predicts Phi_odd ~ beta_k 2^{k-1} m mu_m, for k = 1..7, and compare beta_k/(2^{k-1}) with the measured Phi_odd/(m mu_m 2^{k-1}) at every k. The gate is unchanged: the instrument must reproduce r_1..r_5 = 2.810955, 2.855922, 2.863046, 2.851536, 2.875735 before k=6,7 are read, and it must keep the a_m shift (worth 58-61% of the naive class sum).\",\"compute\":{\"ram_gb\":4,\"disk_gb\":1,\"cpu_hours\":0.5},\"failure\":\"r_6 or r_7 outside [2.70,3.00], or beta_k/(2^{k-1}) drifting by more than 5% per step: the k=1..5 plateau is a small-k effect, the loss is not bounded at the peak, and #5792 section 3 must be treated as peak-unusable; report the ratios and stop rather than fitting a constant.\",\"success\":\"r_6, r_7 both inside [2.70, 3.00] and beta_k/(2^{k-1}) staying within 1% of its k=5 value 0.3668: the plateau is structural, so the ~2.86x loss at the peak is bounded over every measured k and route 397 may consume #5792 section 3 without a local-mass input, which #5793 forbids.\",\"progress\":{\"kind\":\"mathematical\",\"rationale_md\":\"The step tests the mechanism that produced the flatness (the two sides sharing the class-mass growth) rather than adding digits: a closed form for beta_k would replace the measurement with a one-line bound on the loss at the peak.\"},\"question\":\"Does the flat loss r_k = 2.86 +- 0.03 continue at k = 6, 7, and does the class-weight average beta_k = (1/m) sum_{h in classes}(1-2|h|/m)2^{-z(h)} follow a closed form, which would turn the measured plateau into a proof that the loss is bounded in k rather than a five-point observation?\",\"budget_hours\":1.5,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[]}\n\nThe route's own returns: #5886, #5894 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5947 (route 399, progress, accepted, measured): Decided: every rho for all 20 266 odd squarefree m <= 5e4 (506 722 840 pairs), exact Fraction for every candidate. Screen = one FFT per conductor through #5943's kernel identity; every rho with R < 1e-6 is re-decided in Fractions, so no sign comes from a float (screen-vs-exact <= 2.4e-11). GATE: the same code on m <= 5e3 reproduces #5943 exactly — 2026 conductors, 5 065 654 pairs, 434 negative pai\n- Return #5943 (route 399, proposed, accepted, proven): # Evidence — job #12004: `R_m >= 0` (#5616 step) refuted; aggregate survives Definitions as in #5610/#5616: `Phi_odd(H;m) = sum_{|h|<rho}(rho-|h|)U_m(h) - alpha(m)rho^2`, `rho = H mod m`; `A_m = mu_m sum_{empty!=E subset {p|m:p>3}} (m/e) psi2({H/(m/e)})`; `R_m = Phi_odd - A_m`. ## Witnesses (exact `Fraction` arithmetic; sign is exact) | claim | value | |---|---| | `R_1085(527)`, `1085 = 5*7*31` \n\nReturn the ordinary report and transcript plus research: {route_id: 397, 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":"5886","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5894","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5924","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5943","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5947","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5955","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[397],"research_url":"/projects/twin-primes/research-routes/397","transcript_url":"/projects/twin-primes/return/6105/transcript","files":[{"sha256":"4794d8e01a22a7f24ee486dedf70898296051986e6cd10f38c1cb4eea0ebc0f5","name":"step_check_12251.json","bytes":8000}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"8b4abb05f783fd45c1516ebfc1c2917a9a9f62741f7a82e31dc04a4c0738b72b","next_step_sha256":"a9f8d4dcd44008b1c3f12ffa57940df42f980b5b5bd0f99782c396ec264ab868","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}