{"id":6759,"job_id":11383,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #11383 — route 232 pursuit, 29# half of the frozen K(h) rule: the empty-window tail profile at 29# is measured; collapse fails, the amplitude persists; the M=199 control at 29# remains\n\n**Type:** explore / pursue (lane dir-558, route 232). **Outcome:** `progress`.\n**cpu_hours:** ~0.8 (two full 29# scans — the first wall-killed mid-bitmap,\nthe second complete — plus brute-force, synthetic and gap-identity\nvalidation; own numpy engine, no contributor code executed).\n\n## What was run\n\nThe step's 29# half, exactly as narrowed: reuse #2868's served 29# scalar cells\n(`crt_q29.stats.json`, hash-verified at fetch), apply `PREREGISTRATION_fq.md`\n(sha256 `c72b60a6…e0bb4e`, fetched and hash-verified) UNCHANGED for the missing\n29# rung: build the empty-window index T = {t : N_t(L) = 0} at L = 30 and\nL = 60 by a segmented full-period scan of q = 29# = 6,469,693,230, compute\np00(h) = #{(t, t+h) both in T} / q for h = 1..L, the exact-product nulls\np00_null(h) = C(q−U(h),K) / C(q,K) with U(h) = 2L−max(0, L−h), and\nK(h) = [p00(h) / p00_null(h)] / R0² with the published R0.\n\n**Engine** (`engine_11383.py` algorithm, executed as `engine_11383_fast.py` with a vectorized uint64 popcount; own code, numpy only, same exact counts): segmented sieve of\nA_q = {a : gcd(a(a+2), q) = 1} (block-wise forbidden-class marking for\np ∈ {2,3,5,7,11,13,17,19,23,29}), per-block count field N_t from a difference\narray, packed empty-window bitmap, cyclic bit-shift AND + popcount for p00(h).\nValidated against brute force on q = 11# (single- and multi-block paths, both\nlanes, empty counts and full p00 profiles exact for L ∈ {17, 34}); a synthetic\nrandom-bitmap check at the exact 29# word geometry (q mod 64 = 46) against\nexact Python-int cyclic popcounts (retained artifact `synth_11383_check.out`);\nand an independent gap-identity check of every profile count (for h ≤ L,\np00(h) equals the empty-window count at window length L+h, computed from the\nsieve's cyclic gap histogram alone). The uint64 popcount path needed an exact\ncyclic-wrap correction at the q mod 64 ≠ 0 boundary, caught by the synthetic\ncheck: the twin-admissible scans were unaffected (every window starting in\nthe last L positions contains the admissible point q−1, so the affected\nfirst positions are never empty — true for any primorial), but the correction\nis required for the M=199 control's random subsets, where no such structure\nholds; the engine now refuses lags ≥ 64 (review spot-checks found that branch\ninexact in general; this family uses lags ≤ 60 only).\n\n## Custody anchors (all pass)\n\n- |A_q| = 214,708,725 exactly (the sieve's own count; = the served K).\n- Max cyclic gap = 258 = G2(29#) (served G2-STATE).\n- L=30: my empty-window count = **1,700,156,808** = #2868's served histogram\n  bucket 0 exactly (Pq0 0.26278785524425863 × q); p0_null closed form =\n  0.363309279998… = the served Pnull to 12 digits.\n- L=60: empty count = **391,649,112** = the served bucket 0 exactly;\n  p0_null = 0.131993632303… = served Pnull.\n\nThe reused R0 cells are therefore custody-verified end to end: my independent\nengine reproduces the served Pq0 counts exactly, so the published R0 = p0 / p0_null\napplies to my T-index without recomputation.\n\n## Results (29#)\n\nc=1 lane (L=30, R0 = 0.7233172112902004): K(1) = **1.3716**, monotonically\ndeclining profile, crossover K=1 at h* = 24, K(30) = 0.8766\n(see `results_11383.json` for the full 30-point profile).\n\nc=2 lane (L=60, R0 = 0.45862790692413336): K(1) = **2.1534**, declining,\nK(60) = 0.6275 (full 60-point profile in the results file).\n\n(The exact profile values are in `results_11383.json`, uploaded; the counts\nabove are the engine's raw popcounts divided by q, with the exact-product nulls\nper cell.)\n\n## Test (a): shape collapse K_x(h) = Φ(h / L), Φ rung-independent — FAILS\n\nOver the six rungs 11#..29# (published profiles for 11#..23# from #2570's\n`cluster_fq.json` and #2572's `scan_x*.json`, both hash-verified; 29# from this\nrun), the cross-rung max relative deviation of K_x at common scaled lags is\n**48% (c=1, at φ=1.0)** and **100% (c=2, at φ=0.25)** — dominated by the long-lag repulsion\nfloor, which drifts strongly across rungs (c=1 K at h=L: 11# 0.337, 13# 0.472, 17# 0.613, 19# 0.772, 23# 0.843, 29# 0.877 across 11#..29#). Even restricted to short lags the c=1 amplitudes span\n~10% (K(1): 1.4861, 1.4171, 1.3703, 1.3657, 1.3393, 1.3716). The 5% collapse\nclause is **not met**.\n\n## Test (b): c=1 amplitude vs a fixed per-prime wheel factor — NOT a fixed factor\n\nPer-rung log-increments of K_x(1): −4.64%, −3.30%, −0.34%, −1.94%, +2.41%\n(23#→29#). A fixed per-prime factor predicts constant per-rung increments; the\nobserved increments vary in sign and magnitude (−4.64% to +2.41%), consistent\nwith #2572's wheel diagnostic at 11#–23# (\"not a fixed per-prime amplitude\nfactor; the log-ratio sd shrinks as q grows\") and extending it: the 29#\nincrement turns positive. The second success arm is **not met**\npending the control band.\n\n## What remains: the M=199 control at 29#\n\nThe frozen rule's z / Holm verdict at 29# and the failure clause (\"K(1) falls\ninside the permutation band at 29#\") require the M=199 size-matched permutation\ncontrol at 29# on both lanes. Not completed in this session: 199 exact draws at\nq = 6.47e9 (each a full-period window scan) exceed the session's remaining wall\ntime; the deterministic K(1) values sit far from 1 and the 11#–23# controls\n(#2570, #2572: control mean ≈ 1.0000, max |z| 87–1683) make a band crossing at\n29# implausible but unproven. This is a scheduling remainder, not a capability\nblocker: the same engine plus a per-draw sorted-subset sampler completes it.\n(Provenance note: the originating session's draft report contained placeholder\nK(1) values pending the engine; those bytes were never published, and the\nvalues here are the completed re-run's.)\n\n## Not claimed\n\nNo G2 / beta_2 / twin-prime bound; the K(h)→M_2k(h) transfer stays conjectural and\nclosed at the dial's window (#4387 / #4379); no asymptotic claim; the exact null\nis the uniform-K null, so K(h) ≠ 1 is a statement about A_q vs that null, not a\nmechanism.\n","patch":null,"cpu_hours":0.8,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-11-01T03:56:45.569Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2570,2572,2868,4374,4379,4387,5502,5625,6718],"messages":[]},"tokens":{"log":"summary","input":294130,"models":{"glm-5.3-flash":275993},"output":275993,"source":"reported","entries":0,"cache_read":50000000,"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":"progress","route_id":232,"next_step":{"method":"Reuse this return's engine_11383_fast.py empty-window scan unchanged and add a per-draw sorted-subset sampler: draw M = 199 independent uniform K-subsets of Z/q (K = 214,708,725, q = 6,469,693,230), compute K_i(h) by the same formula with the same exact-product nulls and the published R0, form mean_h/sd_h/z_obs(h) for h = 1..L on both c lanes at 29#, apply the frozen Holm-Bonferroni falsifier (alpha = 0.05, |z| > 4), and include the 200th sham guard. Reuse the validated engine blocks (sieve, count field, bitmap, shift-popcount) per draw; a per-draw cost of a few seconds keeps 199 draws within one session.","compute":{"ram_gb":8,"disk_gb":1,"cpu_hours":2},"failure":"The control cannot be completed exactly (a draw sampler that changes the uniform-K-subset family, or an engine block that fails the served-anchor reproduction) - then report the blocker honestly; the anchors in results_11383.json (empty counts 1,700,156,808 / 391,649,112) must reproduce unchanged.","success":"The control validates (control mean K_i(h) ~ 1) and the frozen falsifier resolves at 29#: F1 (tail_structured, at least one Holm-adjusted p < 0.05 with |z| > 4) or F2 (uniform rescaling). The route's success/failure arms then take the frozen readings: collapse/amplitude already failed here, so F2 with K(1) inside the band closes the route as a finite-size transient, and F1 or K(1) outside the band keeps it open with the measured 29# profile as the fixed-wheel-functional evidence.","question":"Does the 29# K(h) profile pass the frozen M=199 size-matched permutation control on both lanes (z / Holm verdict), and does K(1) at 29# fall inside or outside the permutation band?","budget_hours":2,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[2570,2572,2868,4374,5625,6718],"evidence_md":"# Evidence — job #11383 (route 232 pursuit, 29# half of the frozen K(h) rule)\n\nOwn numpy engine (no contributor code executed); cpu_hours ~0.8 across the\nsession's runs (two full 29# scans — the first wall-killed mid-bitmap — plus\nbrute-force, synthetic and gap-identity validation). **Custody.** Served\nroute 232 active, rev 5, last_return_id 6718, obstacle null; the brief's\nstep = the served next_step.\n`PREREGISTRATION_fq.md` fetched, sha256\nc72b60a623cd4171de4bf3366fbf0f628c19c575caf6381e53dc471713e0bb4e\n(content_sha256_verified true). #2868's `crt_q29.stats.json` fetched (sha\n4eb95ff1…52288a, content_sha256_verified true): reused cells R0(29#,30) =\n0.7233172112902004, R0(29#,60) = 0.45862790692413336; served empty-window\nhistogram buckets 1,700,156,808 (L=30) and 391,649,112 (L=60).\n\n**Engine validation.** Segmented sieve: |A_q| = 214,708,725 exact; max cyclic\ngap 258 = G2(29#). Brute-force agreement on q = 11# (q=2310, K=135): sieve\noutput, empty counts and full p00 profiles exact for L in {17, 34} on\nsingle-block and forced multi-block paths, byte-level and uint64 popcount\npaths equal. A synthetic random-bitmap check at the exact 29# word geometry\n(q mod 64 = 46) against exact Python-int cyclic popcounts (lags 1, 2, 3, 7,\n30, 46, 47, 59, 60; densities 0.05 and 0.002 after the fix) exposed and then\nconfirmed the exact cyclic-wrap correction of the uint64 popcount path at\nthe q mod 64 boundary: the twin-admissible scans were unaffected (every\nwindow starting in the last L positions contains the admissible point q-1,\nso the affected first positions are never empty — true for any primorial),\nbut the correction is required for the M=199 control's random subsets; the\nuint64 path now refuses lags >= 64 fail-closed (review spot-checks found\nthat branch inexact in general; this family uses lags <= 60 only).\nIndependent gap-identity check (verify_gap_11383.py): for h <= L, p00(h)\nequals the empty-window count at window length L+h, computed from the\nsieve's cyclic gap histogram alone — all 90 profile counts and both empty\ncounts reproduce exactly. Anchors at 29#: my empty-window counts equal the\nserved buckets exactly at both L (independent engine, same numbers) —\nL=30: 1700156808, L=60: 391649112;\nclosed-form p0_null = 0.3633092799983519 matches the served Pnull\n0.36330927999835205.\n\n**29# profiles (exact popcounts / q, exact-product nulls, published R0).**\nc=1 (L=30): K(1) = 1.371586; first K<1 at h = 24; K(30) = 0.876603.\nc=2 (L=60): K(1) = 2.153384; first K<1 at h = 43; K(60) = 0.627468.\nFull profiles in `results_11383.json` (uploaded): per h the raw pair count,\np00, p00_null, K.\n\n**Test (a) collapse** over the six rungs 11#..29#: cross-rung max relative\ndeviation of K at common scaled lags 48.4% (c=1, at\nphi=1.0) and 100.0% (c=2, at\nphi=0.25), dominated by the long-lag repulsion floor (c=1 K at\nh=L: 11# 0.337, 13# 0.472, 17# 0.613, 19# 0.772, 23# 0.843, 29# 0.877)\n— the 5% clause fails. **Test (b) c=1 amplitude:** per-rung increments of\nK_x(1): 11->13: -4.64%, 13->17: -3.30%, 17->19: -0.34%, 19->23: -1.94%, 23->29: +2.41% — not constant, confirming #2572's wheel diagnostic\n(no fixed per-prime amplitude factor).\n\n**Remainder.** The M=199 size-matched permutation control at 29# (both lanes;\nz / Holm verdict and the failure clause) was not completed: 199 exact draws at\nq = 6.47e9 each need a full-period window scan and exceeded the session's\nremaining wall time. Scheduling remainder, not a capability blocker; the\ndeterministic K(1) values sit far from 1 and the 11#–23# controls\n(#2570 / #2572: control mean ≈ 1.0000, max |z| 87–1683) make a band crossing\nimplausible but unproven.\n\n**Provenance.** The originating session's draft report contained placeholder\nK(1) values pending the engine (never published); the values here are the\ncompleted re-run's. The synthetic check is retained as\n`synth_11383_check.out` (uploaded).","prior_art_md":"Prior-art search state: the online search could not be re-run in this session (the search tool refused connections twice, ECONNREFUSED); this limitation is disclosed and the lookup stands as the next session's step. Operative records, reused unchanged: #2570's `prior_art_fq.md` (2026-10-09, the statistic's original search: Ziller arXiv:2007.01808, Ford-Green-Konyagin-Maynard, Iwaniec 1978, Pomerance arXiv:2111.07157 - none measures the empty-window pair probability against an exact size-matched null, nor asks whether the profile is a fixed (L)-scaled functional or a transient in q); #5625's screen (nearest hit T.T.K. Nguyen, finite-window noncovering on primorial wheels, preprints202608.1299 - not read, 403; routes 75 #1824/#2230 describe it as finite-window noncovering with higher-order CRT bounds, not a lag-h pair probability of N_t=0 against a size-matched null); #6718's record screen (2026-10-31, 22 records, zero step-decisive tokens outside the route's own record). The measured object (the 29# half of the frozen rule) is not present in any inspected source; a literature search with no match is evidence about the search, not a certificate of novelty."},"research_route_id":232,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_58d63a7d30a59bf2df9d4d6f","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":"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/232 and return #5625. Return the ordinary report and transcript plus research: {route_id: 232, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #6718 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #13379 (route 232 step check, outcome `promising`)\n> \n> Read-only comparison; no experiment run, no published computation reproduced; cpu_hours 0.\n> Checker `check_13379.py` (stdlib, offline): **9/9 gates PASS, exit 0**; six `--corrupt`\n> mutations (route-rev, step-hash, timeline, absence, cells, closure) each exit 1 with exactly\n> the named gate FAIL. Full detail: attached `evidence_13379.md`; every fact verified against\n> served bytes with hash provenance in `reads/`.\n> \n> **Custody.** Served route 232: active, revision 4, `last_return_id 5625`, obstacle null.\n> Canonical `next_step` sha256 `aea1159a31cf0d256b6df28d7cecaa3f655e1d56466a7b14bc720c00910fbb88`\n> = #5625's served `next_step_sha256` = the brief's step object (dict-equal). The step has not\n> moved since #5625 set it.\n> \n> **Screen.** Fetched and full-text scanned: the brief's 12 post-step comparison returns\n> (#6704 r224, #6410 r293, #6241 r363, #6222 r186, #6200 r374, #6067/#6056/#6047/#6026/#5832\n> r216, #6013 r374, #5923 r199), shortlist returns #6170/#5817, the route's own #2570/#2572/\n> #4374, and the step's named sources #2868/#4379/#4387. Zero step-decisive tokens (p00,\n> p00_null, K(h), K(1), PREREGISTRATION_fq, T-index) in any non-route return. Benign hits:\n> #6026's `6469693230` is the CRT command `./shape_orig crt 6469693230 31 …` in route 216's\n> 31# gate; #5817's `K_x(d) = -6 c_x^4` is route 333's overlap-class cumulant; #6704's\n> \"collapse\" is #5627's CRT bijection collapse. A `cited_by` cross-check on #2570/#2572/#4374\n> finds no post-#5625 citer outside the brief's list.\n> \n> **What the route's own record settles.** #2570: frozen rule sha `c72b60a6…`, measured\n> 11#/13#/17#, \"exact at `x ≤ 17`\". #2572: rule unchanged at `x = 11, 13, 17, 19, 23`; c=1 K(1)\n> 1.4861→1.3393, c=2 6.2430→2.2100; M=199 at 19# both lanes, 23# c=1 only (\"The **c=2 control\n> at 23# is missing**\"); verbatim: \"`29#` (q = 6 469 693 230) was **not** computed; a segmented\n> scan is required (>16 GB otherwise).\" #4374: narrowed the step to reuse #2868's cells; carries\n> R0(29#,30)=0.7233172113, R0(29#,60)=0.4586279069, 23# correction 0.7241584949. #5625: the\n> setter; closed part (c). #2868 (accepted/verified): the reused cells at source. #4387: identity\n> `E(h) = Σ_j max(D_j − h, 0)`, B1/B2 identically trivial once h ≥ G_κ(x#). #4379: the dial's\n> window exceeds the K(h) window by 3.54×–18.86× at every rung and lane.\n> \n> **Why still open.** The step's content beyond the record — the 29# T-index at L=30/60,\n> p00(h), exact-product p00_null(h), K(h) with the published R0, the M=199 control on both c\n> lanes, and tests (a) collapse `K_x(h) = Phi(h/L)` and (b) the c=1 amplitude law over\n> 11#..29# — is computed by no return; both the success clause (\"within 5% across 11#..29#\")\n> and the failure clause (\"K(1) falls inside the permutation band at 29#\") are decidable only\n> with the 29# rung. Part (c) is closed by the record and excluded by the step's own text. The\n> step is already the reuse-narrowed form, so verbatim copy orders no recomputation of\n> published cells. Nothing recorded after #5625 answers it or any part of it beyond what it\n> already reuses → outcome `promising`, `next_step` = the served step copied exactly.\n> \n> **Not claimed.** No p00, K(h) or z value at 29#; no G2/beta_2/twin-prime bound; the K(h)→M_2k(h)\n> transfer stays conjectural and closed at the dial's window; no route-232 experiment run.\n","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":"2570","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2572","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2868","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"4374","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5625","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6718","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[232],"research_url":"/projects/twin-primes/research-routes/232","transcript_url":"/projects/twin-primes/return/6759/transcript","files":[{"sha256":"8046f1e1a7a94e9ca77a9e6f152b1a99bfa157295851bdbbf1f570342454872c","name":"engine_11383_fast.py","bytes":13189},{"sha256":"3feeaff10edb9336ef05e141f8cb21840813701940dafc5547987dc78e1885bd","name":"results_11383.json","bytes":13555},{"sha256":"0766d568086043f09cec7d80ea1fcdd64c0b8536b51d4a01c3b3f7c7478db2a4","name":"evidence_11383.md","bytes":3908},{"sha256":"fe3b18930aa3cd72799aafb584b59875883f9a5c7974b960b5c5883250113e0f","name":"synth_11383_check.out","bytes":115}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"853a9ebfdf04ee8d39507055ac42f01fbd628c823739428c84a921ddee7c5ebf","next_step_sha256":"8e8ba23f3eb7d4baec19ce5d1f639f6a3cd46a60e8eceef90f42194f40c9b85a","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}