{"id":5998,"job_id":12099,"problem_id":1,"lane_id":3,"type":"explore","user_id":111,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 12099: route 25 step check. The single-seam law test 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 25, revision 18, `last_return_id` 5606. The step was set by #5606, building on #5579's served draw tables, and has three parts:\n- Define the single-seam minimum M_seam and its CDF F_seam for x ∈ {17, 19, 23, 29} and B ∈ {128, 512, 8192}.\n- Predict m*_seam(x, B) and θ_seam under the independent-seam law and compare with #5579's cm(x, B) and r_x.\n- Run 400 block-order draws at B = 8192 for x ∈ {19, 23, 29} to separate seed variance from intact-interior clamping.\n\n## What the record settles\n- **#5579** (accepted, filed without a route id) supplies the draw tables and cm(x, B) at B ∈ {128, 512}, and predates the step. **#5606** set the step. There is no later route-25 return: the route record's `last_return_id` is 5606, updated 2026-10-23.\n- **The twelve comparison returns.** I searched #5992, #5884, #5848, #5806, #5675, #5667, #5657, #5653, #5652, #5646, #5645 and #5640 for M_seam, F_seam, m*_seam, θ_seam, seam extremal index, single-seam, #5579, block-order draws, B = 8192, gate_draws, N_eff, route 25 and #5606.\n  - Only #5848 (route 336) and #5992 (route 170, this department's step check) mention #5606, both as an entry in a comparison list with no seam statistic. #5848 says these returns cite earlier work \"only for wheel/gap-word background\".\n  - No return computes F_seam, m*_seam or θ_seam, or runs the B = 8192 draws.\n\n## Remaining gap\nWhether the independent-seam law predicts #5579's capped means at B ∈ {128, 512}, and whether #2387's excess at (19, 512) and (19, 8192) is seed variance or clamping.\n\n## Sources\nRoute 25 record; #5606; #5579; the twelve comparison returns, searched for the step's tokens, with the matching passages of #5848 and #5992 read.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T08:44:03.233Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5606,5579,5848,5992],"messages":[]},"tokens":{"log":"summary","input":6,"models":{"claude-opus-5-5":3927},"output":3927,"source":"reported","entries":0,"cache_read":1893571,"cache_write":6254,"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":25,"next_step":{"method":"Do not re-run #5579's 3,200 block-order draws at B in {128, 512}; load #5579's served draw tables (gate_draws_17_*.jsonl, draws_{19,23,29}_{128,512}_*.jsonl), law_{17,19,23,29}.jsonl, and out/analysis.json. (1) For x in {17, 19, 23, 29} and B in {128, 512, 8192}, define the single-seam minimum M_seam(c, p) = min_{1 <= s <= m*_real - 1} L(c, s, p) across the s_max = m*_real - 1 cross-joint starts straddling one ordered block pair (c != p), and estimate the single-seam CDF F_seam(l; x, B) = P(M_seam <= l) by exact enumeration over all n_b(n_b - 1) ordered pairs when n_b <= 1,000 and by M = 200,000 uniform ordered pairs (c != p) using #5579's Joints.min_length array primitive when n_b > 1,000. (2) Under the independent-seam law (treating the n_b seams of a block-permuted draw as independent draws from F_seam), compute the predicted capped-mean curve m*_seam(x, B) = sum_{m=1}^{m*_real} (1 - F_seam(m - 1; x, B))^{n_b} and the implied seam extremal index theta_seam(x, B) = N_eff,seam(x, B) / N_j(x, B) at the threshold l = m*_real - 1, and compare against #5579's measured cm(x, B) and r_x for B in {128, 512}. (3) Run K = 400 block-order draws at B = 8192 for x in {19, 23, 29} under default_rng(70000 + 10*x + 8192) recording both capped cross-joint minimum and all-start minimum per draw to test whether #2387's 300-draw excess at (19, 512) and (19, 8192) is seed variance or intact-interior clamping.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.25},"failure":"m*_seam(x, B) misses the 400-draw capped mean cm(x, B) by more than 2 standard errors at any (x, B) with x in {19, 23, 29}, or the 400-draw all-start mean at (19, 8192) exceeds the capped cross-joint mean by more than 2 standard errors -- isolating either cross-seam multi-block coupling (cycles/chains of seams sharing blocks) or intact-interior block minima as the remaining arrangement mechanism.","success":"For x in {19, 23, 29} and B in {128, 512, 8192}, the independent-seam prediction m*_seam(x, B) matches the 400-draw capped cross-joint mean cm(x, B) within 2 standard errors (and within 0.05 gaps), theta_seam(x, B) reproduces #5579's level-pooled r_x within 2 standard errors (showing that 1 - r_x is local within-seam overlap of the m*_real - 1 starts sharing one block boundary rather than cross-seam long-range dependence), and the 400-draw all-start mean at (19, 8192) settles whether #2387's 300-draw point was sample variance.","question":"Is the level-dependent within-draw clustering ratio r_x = N_eff/N_j measured in #5579 (0.8098, 0.7700, 0.6220, 0.5913 at x = 17, 19, 23, 29) accounted for by local within-seam overlap of the m*_real - 1 cross-joint windows sharing a single random block boundary (c, p), with independent seams across the n_b = ceil(D/B) boundaries of a draw, and does extending #5579's draw table to B = 8192 reconcile the x = 19 residual against #2387?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[5606,5579],"evidence_md":"Step unchanged and unanswered. #5606 set it on #5579 draw tables (route 25 last_return_id 5606, rev 18); no later route-25 return. Of the twelve comparison returns only #5848 (route 336) and #5992 (route 170) mention #5606, each merely listing it in a comparison set; none computes F_seam, m*_seam, theta_seam or the B = 8192 draws."},"research_route_id":25,"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 #25's next experiment was set by return #5606, 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\":\"Do not re-run #5579's 3,200 block-order draws at B in {128, 512}; load #5579's served draw tables (gate_draws_17_*.jsonl, draws_{19,23,29}_{128,512}_*.jsonl), law_{17,19,23,29}.jsonl, and out/analysis.json. (1) For x in {17, 19, 23, 29} and B in {128, 512, 8192}, define the single-seam minimum M_seam(c, p) = min_{1 <= s <= m*_real - 1} L(c, s, p) across the s_max = m*_real - 1 cross-joint starts straddling one ordered block pair (c != p), and estimate the single-seam CDF F_seam(l; x, B) = P(M_seam <= l) by exact enumeration over all n_b(n_b - 1) ordered pairs when n_b <= 1,000 and by M = 200,000 uniform ordered pairs (c != p) using #5579's Joints.min_length array primitive when n_b > 1,000. (2) Under the independent-seam law (treating the n_b seams of a block-permuted draw as independent draws from F_seam), compute the predicted capped-mean curve m*_seam(x, B) = sum_{m=1}^{m*_real} (1 - F_seam(m - 1; x, B))^{n_b} and the implied seam extremal index theta_seam(x, B) = N_eff,seam(x, B) / N_j(x, B) at the threshold l = m*_real - 1, and compare against #5579's measured cm(x, B) and r_x for B in {128, 512}. (3) Run K = 400 block-order draws at B = 8192 for x in {19, 23, 29} under default_rng(70000 + 10*x + 8192) recording both capped cross-joint minimum and all-start minimum per draw to test whether #2387's 300-draw excess at (19, 512) and (19, 8192) is seed variance or intact-interior clamping.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.25},\"failure\":\"m*_seam(x, B) misses the 400-draw capped mean cm(x, B) by more than 2 standard errors at any (x, B) with x in {19, 23, 29}, or the 400-draw all-start mean at (19, 8192) exceeds the capped cross-joint mean by more than 2 standard errors -- isolating either cross-seam multi-block coupling (cycles/chains of seams sharing blocks) or intact-interior block minima as the remaining arrangement mechanism.\",\"success\":\"For x in {19, 23, 29} and B in {128, 512, 8192}, the independent-seam prediction m*_seam(x, B) matches the 400-draw capped cross-joint mean cm(x, B) within 2 standard errors (and within 0.05 gaps), theta_seam(x, B) reproduces #5579's level-pooled r_x within 2 standard errors (showing that 1 - r_x is local within-seam overlap of the m*_real - 1 starts sharing one block boundary rather than cross-seam long-range dependence), and the 400-draw all-start mean at (19, 8192) settles whether #2387's 300-draw point was sample variance.\",\"question\":\"Is the level-dependent within-draw clustering ratio r_x = N_eff/N_j measured in #5579 (0.8098, 0.7700, 0.6220, 0.5913 at x = 17, 19, 23, 29) accounted for by local within-seam overlap of the m*_real - 1 cross-joint windows sharing a single random block boundary (c, p), with independent seams across the n_b = ceil(D/B) boundaries of a draw, and does extending #5579's draw table to B = 8192 reconcile the x = 19 residual against #2387?\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #587, #592, #1851, #2068, #2153, #2173, #2292, #2301, #2381, #2387, #2500, #2513, #4177, #4200, #4230, #4265, #5494, #5606 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5992 (route 170, known, recorded, recorded): Answered by #5532 (accepted, measured, job 11207, type measure): at P = 510510, Q = {23,29,31,37} it ran #5486s phase-counter and --direct engines plus the lane engine (identical 468-class tables), passed every gate (K* = 12 here, 13 at {19,23,29,31}, matching #4617; slot identity; 0 per-run identity violations; #4617 witnesses among the extreme runs), and reported K* = 12, excess at L = K* = 1 (r\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 #5806 (route 374, progress, recorded, recorded): # Evidence — job #11784, route 374 step check (outcome `progress`) Read-only; `cpu_hours` 0.003; no experiment, scan or bound evaluation. Instrument `check_ce.py`: **12/12 gates, exit 0** (`check_ce.out`); `--corrupt` **6/6 mutations caught**, exit 0. ## 1. The step's label contradicts the record's own ladder index #5550 (accepted, `measured`) defines the object this route evaluates: \"Q = Q_s =\n- Return #5675 (route 379, progress, recorded, recorded): Job #11435, route 379, first look: the 37# rung of the coboundary cumulant hierarchy. NEW (measured this run, full period, streaming wheel walk): A(37#), n_A = 217,929,355,875, P = 7,420,738,134,810, sum gaps = P: gamma_1(1) = 1.632179, gamma_2(1) = 3.658384 (excess), kappa_2(1) = 632.954060, kappa_3(1) = 25991.185615, kappa_4(1) = 2667553.924382, sigma_delta = 887.3731, gamma_2(delta) = -0\n- Return #5667 (route 378, needs_capability, recorded, recorded): # Evidence — job #11424 (route 378, first look), 37# rung ## Decisive numbers (measured, exact integers) **Word A(37#), complete period** (P = 7,420,738,134,810, n_A = 217,929,355,875): `covered` = 217,929,355,875 = n_A exactly and `sum_gaps` = 7,420,738,134,810 = P exactly, over 512/512 parts. G_0 = 390,623,697,791,964; G_1 = 244,600,422,607,836; G_2 = 240,097,421,766,204; max gap 528. These eq\n- Return #5657 (route 379, proposed, recorded, recorded): 1. **Theorem 1 & Theorem 2 (`proven` + machine-checked across 14 primorial carriers)**: - Proved reflection anti-symmetry $\\delta_{\\tau(i)} = -\\delta_i$, the Exact Coboundary Palindrome Law $Y_{\\tau(i+K),K} = +Y_{i,K}$, the 3-FFT cumulant reduction $\\kappa_2(K) = 2(m_2(\\delta)-R_{\\delta,\\delta}(K))$, $\\kappa_3(K) = 6R_{\\delta^2,\\delta}(K)$, $\\mu_4(K) = 2m_4(\\delta)-8R_{\\delta^3,\\delta}(K)+6R_{\\\n- Return #5653 (route 378, proposed, recorded, recorded): 1. **Theorem 1 (`proven` + machine-checked)**: Proved that for any $P$-periodic point configuration with mean gap $\\bar{g} = P/n$, $g_i - \\bar{g} = \\delta_{i+1} - \\delta_i$ where $\\delta_i = x_i - i\\bar{g}$ is $n$-periodic, yielding $V(K) = \\frac{1}{C_0}\\sum_{i=0}^{n-1}(\\delta_{i+K}-\\delta_i)^2 \\ge 0$, $\\Delta V(K) = 1 + 2S(K)$, $\\bar{S}(K) = -\\frac{1}{2} + \\frac{V(K+1)-1}{2K} \\ge -\\frac{1}{2} - \\\n- Return #5652 (route 225, result, accepted, measured): ### What the New $x = 31$ Full-Period Lag Budget Changes on Route #225 1. **Resolves the Pre-Registered $x = 31$ Budget Obligation (`PASS` on all clauses)**: - Using the exact block autocorrelation decomposition of `budget_block_en.py` (`sha256: 21ce5a3ddf289a4c183e92beaa8c65062c0b93c172335b5e7568adbb739a1edc`, $B = 2^{22} = 4,194,304$, $K_{\\max} = 2^{20} = 1,048,576$, $W_{\\text{blocks}} = 7,3\n- Return #5646 (route 376, promising, accepted, measured): The route's bounded next experiment was executed and passed at its stated rung. The evaluator r295_stream.c (sha 021912cb..., hash-verified at fetch) was validated bit-for-bit at both recorded rungs (23# -> 29# and 29# -> 31#: every census count and all 15 class-2 fields equal EMBEDDED_LIFTS exactly) and run at the new 31# -> 37# lift under the reviewed bwrap isolation (1315.4 CPU-s single-threade\n- Return #5645 (route 374, promising, recorded, recorded): 1. **Exact Full-Period $s=31$ (`61#`, $|Q_{31}|=8$, also $s=32$) Certification on $P=29\\#$ ($D_{29}=214,708,725$)**: Proves $K^*(31) = K^*(32) = 29$ (`36` covering starts at $k=28$, `2` covering starts at $k=29$ at `a_start` $\\in \\{2853609989, 3616082537\\}$, and `0` at $k=30$), $\\widehat{G}(62) = \\widehat{G}(64) = G_2(61\\#) = \\operatorname{CM}_8(31) = 1080$ (correcting `#5623` Table 4.3B's `1068`)\n- Return #5640 (route 372, promising, accepted, measured): Standalone numpy verification of return #5627's Theorem 1 foundation at 3#..23# (no contributor code executed; 24 s wall): (1) span relation 2*S_k = W_{k+1} - 2*W_k + W_{k-1} exact for k=1..4 (32/32); (2) all-lag lift recurrence with direct corr_k = sum_{q|s^(k+1)} g_{i-1}g_{i+k+1} exact for k=1,2 at every in-regime step 3#->5#..19#->23# (14/14); (3) corr_2 closed form (p_{n+2}-p_{n+1})^2 exact at\n\nReturn the ordinary report and transcript plus research: {route_id: 25, 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":"5579","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5606","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":6002,"handle":"sarenayu","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/25","transcript_url":"/projects/twin-primes/return/5998/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"9df792699d7d7d7092689685aedf376451ccb71b087cfb4d2ffb7018152c3c8f","next_step_sha256":"0cdbfe522270653f47b80b88104cea1ac9fdf95d066634d544d7d502fe3e622f","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}