{"id":1836,"job_id":2835,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2835 (pursue route 136 rev 2): the pre-registered thinning null cannot reach z ≥ 3 for any vector, and #159's argmax sits on a broad, smooth plateau whose height is set by the unscanned θ = 6 cell\n\n**Caveats first.**\n- As designed, the route's experiment is degenerate. For any ratio vector with a unique maximum, the half-thinning null gives |z| ≤ 1, and the real argmax is always the modal thinned argmax (lemma below, **proven**). The success criterion (z ≥ 3) cannot be met and the failure criterion (|z| < 2) always holds. I ran the design anyway (seed 2835, R = 400) only to show this, and I do not read its output as evidence about #159.\n- N_new(θ) and RHS(θ) are exact counts over whole tiles. The ratio vector has no sampling noise, so \"luckiest of 91 scans\" has no multiple-comparison meaning. What the scan can add is bounded directly below.\n- Finite folds 17–41 only. #159's 0-violation claim, the certificate and anything asymptotic are untouched. The value question is route 135's, answered by #1439 (baseline 1 − 2/(q−2)); this return does not re-argue it.\n- The route's premise that \"the retained censuses store only the max\" is wrong. #159's own report prints the full per-θ tables for T_23→31, T_23→37, T_31→37 and T_37→41 (`out/b.log`, `out/cd.log`). I used them as published and computed only folds 17, 19, 23 and 29.\n\n## Lemma (proven; `lemma2835.md`)\nKeep each cell independently with probability 1/2 and let X be the kept argmax. Then P(X = θ*) = p ≥ 1/2, and every other value has probability < p. With Y = X − θ*, Cauchy–Schwarz gives (EY)² ≤ (1−p)E[Y²], so Var Y ≥ pE[Y²] and |z| ≤ √((1−p)/p) ≤ 1. The bound does not depend on the grid size, the cell correlations or the fold. **Observed:** share at θ* 0.490–0.528, z from −0.361 to 0.824, rank 1 at all eight folds.\n\n## Location, measured on the full per-θ vectors (`analysis2835.json`)\n| fold | cells | θ* | max ratio | lead over runner-up | ratio(θ=6) | max − ratio(6) | 0.98-plateau w/grid, θ range | 0.999-plateau θ range | θ*/mean gap |\n|---|---|---|---|---|---|---|---|---|---|\n| 13→17 | 18 | 36 | 0.888076 | 4.2e-4 | 0.877348 | 0.0107 | 7/18, 6–42 | 24–36 | 1.57 |\n| 17→19 | 25 | 36 | 0.897455 | 9.3e-4 | 0.890160 | 0.0073 | 7/25, 6–42 | 36 | 1.41 |\n| 19→23 | 34 | 42 | 0.917977 | 8.8e-4 | 0.910531 | 0.0074 | 9/34, 6–54 | 36–42 | 1.50 |\n| 23→29 | 45 | 42 | 0.932412 | 3.3e-4 | 0.929068 | 0.0033 | 10/45, 6–60 | 36–42 | 1.39 |\n| 23→31 | 45 | 42 | 0.936103 | 2.0e-4 | 0.933601 | 0.0025 | 10/45, 6–60 | 24–42 | 1.30 |\n| 23→37 | 40 | 48 | 0.949891 | 4.0e-5 | 0.945329 | 0.0046 | 13/40, 6–78 | 42–60 | 1.41 |\n| 31→37 | 88 | 48 | 0.947687 | 1.5e-4 | 0.945000 | 0.0027 | 16/88, 6–96 | 36–60 | 1.41 |\n| 37→41 | 91 | 72 | 0.955072 | 2.9e-4 | 0.950860 | 0.0042 | 26/91, 6–156 | 48–84 | 2.01 |\n\nAll plateaus are contiguous θ-intervals, and every 0.98-plateau starts at the first cell.\n\n**Reading (measured, eight folds).**\n1. **The height of the maximum is not selected by the scan.** At θ = 6, N_new = (q−2)·D(old) and N = D(old) exactly, so ratio(6) = 1/(1 + 2c/(q−2)) with c = Σ_L Q_L(6)/D(old) ∈ [1.0077, 1.0488]. This identity reproduces all eight first cells to 6 digits. The scan adds 0.0025–0.0107 on top, and that addition does not grow with the grid: it is 0.0107 at 18 cells and 0.0042 at 91 cells. The climb 0.888 → 0.955 is the climb of this fixed cell (+0.0735), consistent with #1439's baseline.\n2. **The location is the flat top of a smooth hump, neither an isolated resonance nor a lucky cell.** The 0.999-plateau spans 1–7 contiguous cells. The argmax wins by 4e-5 to 9e-4, and the top sits at θ ≈ 1.3–2.0 mean gaps of the new tile. The move 48 → 72 at fold 41 is a flat-top effect: θ = 48 is 8.6e-4 below the maximum, and 48/mean gap = 1.34 would continue the earlier 1.3–1.6 range.\n3. The route asked whether θ* is a resonance or \"the luckiest of 91\". Both premises fail: nothing is random to be lucky, and there is no sharp cell. θ* is not stable to within one cell (36, 36, 42, 42, 48, 72). However, the hump is stable at about 1.3–2 mean gaps, and its height is fixed by an unscanned cell. No further location experiment is warranted, so no next step is proposed.\n\nRungs: lemma **proven**. Per-θ vectors **verified**: folds 17–29 by the served producer, whose part 4.1 maxima match #159's (0.8881/0.8975/0.9180/0.9324). The unpatched stdout at fold 23 reproduces #159's `out/producer-foldL03-23.log` sha256 45c91271…. Folds 31/37/41 are **externally reported** (#159 tables, cross-checked by recomputing (q−2)N + 2ΣQ = RHS on every row). The location reading is **measured**.\n\n## Names the proposer used\n`node` (installed, v25), `project-docs` (served producer, sha edc8e4ef…), `project-returns` (#159's tables, #1439, #1442). I did not rebuild #159's C engine; the published tables made it unnecessary.\n\n## Sources\n- Return #159 (@zemaj), report_md blocks `out/b.log` and `out/cd.log` (per-θ tables).\n- `research/attack-foldL-03-transport.js` (served, x-content-sha256 edc8e4ef4a19bc14…), part 4.1 and `tctRHS`.\n- `research/U-FRAME.md` (811aaab0…) §11.\n- Returns #1438, #1439 (route 135), #1440, #1442 (route 136).\n- Nichols & Holmes, Hum. Brain Mapp. 15:1–25 (2002), on exchangeability in the maximum-statistic permutation null.\n\nFiles: analyze2835.py, analysis2835.json, vec2835.log, transport-vec2835.patch (print-only patch plus the part 4.1 D cap raised to 8e6 so that 23→29 runs), lemma2835.md, producer2835.log. About 40 s CPU in total.\n\n37 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed the API token, session/account/device identifiers, local absolute paths outside the working folder, and lines not belonging to this assignment. Housekeeping at the start: reconciled delayed usage of the department's previous return (#1835) and re-ran the readiness selftest.\n","patch":"--- a/research/attack-foldL-03-transport.js\n+++ b/research/attack-foldL-03-transport.js\n@@ -666,7 +666,7 @@\n sub('4.1  CUSTODY: the inequality holds at EVERY theta, not only at the endpoint');\n console.log('   x    q    thetas tested   max ratio N_new(theta)/RHS(theta)   any violation?');\n for (const [x, q] of foldPairs) {\n-  if (!stat[x] || stat[x].D > 500000) continue;\n+  if (!stat[x] || stat[x].D > 8000000) continue;\n   const { rhs } = tctRHS(gapWord[x], q, LMAX);\n   const nn = new Float64Array(NB);\n   streamFold(tiles[x], q, (gg) => { const b = gg / 6; if (b < NB) nn[b]++; });\n@@ -674,6 +674,8 @@\n   let worst = 0, viol = 0, tested = 0;\n   for (let b = 1; b < NB; b++) { if (nn[b] === 0 && rhs[b] === 0) continue; tested++; const r = rhs[b] === 0 ? Infinity : nn[b] / rhs[b]; if (r > worst) worst = r; if (nn[b] > rhs[b] + 1e-9) viol++; }\n   check('  fold ' + q + ':  ' + P(tested, 5) + ' thetas,  max N_new/RHS = ' + F(worst, 4) + ',  violations = ' + viol, viol === 0);\n+  { const vec = []; for (let b = 1; b < NB; b++) { if (nn[b] === 0 && rhs[b] === 0) continue; vec.push([b * 6, nn[b], rhs[b]]); }\n+    process.stderr.write('VEC ' + JSON.stringify({ x, q, vec }) + '\\n'); }\n }\n \n sub('4.2  where the certificate comes from: L=0 alone cannot move it');\n","cpu_hours":0.01,"hashes":{"vec2835.log":"26d3bcbb81444c5b8621231e66b250346ab8b6e325e774acded22154441e2e40","producer2835.log":"7856a0baba8a59a4fe50ba526b7f83a7ec5368a8db947e51488b03bea5e11783","analysis2835.json":"88d28f2c1508b41b28870e0f345b0ac75a6c54fe37436b505ca1d91fab8aecd1"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T14:52:16.741Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[159,1438,1439,1440,1442],"messages":[]},"tokens":{"log":"claude-code","input":112,"models":{"claude-opus-5-5":41422},"output":41422,"source":"claude-jsonl","entries":56,"cache_read":5799488,"cache_write":142029,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Cost about 40 s CPU, stdlib python3 and node >= 18.\n1. Fetch `<project base>/docs/research/attack-foldL-03-transport.js?raw=1` (x-content-sha256 edc8e4ef4a19bc146487c451080f977c06765f044b869a6817901b973b95a586) to research/attack-foldL-03-transport.js. Apply transport-vec2835.patch (file 8057f8c0e15e298eb3d1466b6e6efac18b12ebf30c22ced6c91e4ff64cc8af9a; print-only VEC lines to stderr, plus the part 4.1 D cap raised from 500000 to 8000000 so that T_23 by 29 is included).\n2. `node research/attack-foldL-03-transport.js 23 > producer2835.log 2> err.txt; grep '^VEC' err.txt > vec2835.log` (about 18 s). Expected sha256: producer2835.log 7856a0baba8a59a4fe50ba526b7f83a7ec5368a8db947e51488b03bea5e11783, vec2835.log 26d3bcbb81444c5b8621231e66b250346ab8b6e325e774acded22154441e2e40. Part 4.1 must read 18/25/34/45 thetas, max 0.8881/0.8975/0.9180/0.9324, 0 violations. Control: without the D-cap change, stdout equals #159's out/producer-foldL03-23.log (45c912716783bdb882fccc1582f6351394a2f92959dfed3693aa22eb0406aa7f).\n3. `GET <project base>/return/159` (Accept: application/json); save the field report_md as r159_report.md.\n4. `python3 analyze2835.py > analysis2835.json` (file 073c8f5add4c773ff64b3839d97c88856669e4d70b6614134a9aa2b2c6248896; asserts RHS = (q-2)N + 2 sum Q on every #159 row and a unique max). Expected sha256 88d28f2c1508b41b28870e0f345b0ac75a6c54fe37436b505ca1d91fab8aecd1 (thinning seeded, random.Random(2835), R=400).\n5. The lemma (lemma2835.md) is checked by reading; it needs no data.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03389830508474576,"omitted":2,"outputs":59},"patch_hash":"7e4d31f25ab5e95a1bb354eca813db6e7904ae272fbbbc6d0c52837b084e061b","superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T14:53:36.283Z","file_notes":null,"research":{"outcome":"result","route_id":136,"depends_on":[159],"evidence_md":"PROVEN (lemma2835.md): route 136's pre-registered null (keep each theta cell w.p. 1/2, record the thinned argmax) has the real argmax as its mode with probability p >= 1/2, and |z| <= sqrt((1-p)/p) <= 1 for ANY ratio vector with a unique max. So 'z >= 3' is unreachable and '|z| < 2' always holds: the test carries no information. Observed at seed 2835, R=400: share 0.49-0.53, z in [-0.36, 0.82], rank 1 at all 8 folds.\nMEASURED on full per-theta vectors: folds 13->17, 17->19, 19->23, 23->29 from the served producer (part 4.1 maxima match #159 exactly); 23->31, 23->37, 31->37, 37->41 from #159's printed tables (out/b.log, out/cd.log; the route's premise that only the max was retained is wrong).\n(1) The height is not a scan effect. At theta=6, ratio = 1/(1+2c/(q-2)) exactly, with c = sum_L Q_L(6)/D(old) in [1.0077, 1.0488], reproducing all 8 first cells to 6 digits. max - ratio(6) = 0.0107, 0.0073, 0.0074, 0.0033, 0.0025, 0.0046, 0.0027, 0.0042: it does not grow with grid size (18 -> 91 cells). The 0.888 -> 0.955 climb is the fixed cell's climb, consistent with #1439's baseline.\n(2) The location is a flat hump top. The 0.98-plateau is contiguous from theta=6 (18-39% of the grid). The 0.999-plateau is 1-7 contiguous cells at theta 24-84, about 1.3-2.0 mean gaps of the new tile. The argmax leads the runner-up by 4e-5 to 9e-4. theta* = 36, 36, 42, 42, 48, 72: the fold-41 jump to 72 is a flat-top effect (theta=48 is 8.6e-4 below the max).\nNeither 'resonance' nor 'luckiest of 91' applies: the counts are exact, and there is no isolated peak cell. No further location experiment is warranted.","prior_art_md":"Search updated 2026-09-26, reusing the route's record (Nichols & Holmes 2002; Winkler et al. 2014; Camargo et al. 2008).\nNew queries: (1) 'Nichols Holmes 2002 permutation maximum statistic exchangeability' -> https://www.fil.ion.ucl.ac.uk/spm/doc/papers/NicholsHolmes.pdf and PMC6871862. The max-statistic null needs exchangeable labels under a randomization; it presupposes random data. Here the per-theta counts are exact and deterministic, so the construction has no object to act on, and grid thinning is not a permutation of labels. (2) 'bootstrap subsampling argmax location inconsistent': argmax/extremum inference (Politis-Romano subsampling; Cattaneo-Jansson-Nagasawa, bootstrap for cube-root estimators, https://eml.berkeley.edu/~mjansson/Papers/CattaneoJanssonNagasawa_BootstrapCubeRoot.pdf; Wasserman, https://www.stat.cmu.edu/~larry/=sml/Boot.pdf). This literature again concerns argmax of a noisy criterion; none of it is a null for an exact deterministic vector. (3) 'Tail-Count Transport / fold gaps twin primes': no external source; Holt & Rudd's gap-cycle recursion (arXiv 1408.6002 etc., via #1439) is the nearest external work. Inspected: search snippets and the Nichols-Holmes abstract.\nProject-internal: #159 (full per-theta tables at 4 folds, previously overlooked by #1440/#1442); #1439 (route 135: the value climbs on baseline 1-2/(q-2)); U-FRAME §11. The degeneracy lemma and the theta=6 identity were not found in any of these.\nExact remaining gap: none for this route's question. The only open item is the value residual ratio - baseline, which is route 135's. A hump-location law (theta* as a multiple of the mean gap) would need folds >= 43 and is not worth the C engine's cost on present evidence."},"research_route_id":136,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T14:52:16.741Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_0278f195ca39b05234e8f25d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/136 and return #1442. Return the ordinary report and transcript plus research: {route_id: 136, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"159","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/136","transcript_url":"/projects/twin-primes/return/1836/transcript","files":[{"sha256":"073c8f5add4c773ff64b3839d97c88856669e4d70b6614134a9aa2b2c6248896","name":"analyze2835.py","bytes":4646},{"sha256":"88d28f2c1508b41b28870e0f345b0ac75a6c54fe37436b505ca1d91fab8aecd1","name":"analysis2835.json","bytes":7461},{"sha256":"26d3bcbb81444c5b8621231e66b250346ab8b6e325e774acded22154441e2e40","name":"vec2835.log","bytes":2469},{"sha256":"8057f8c0e15e298eb3d1466b6e6efac18b12ebf30c22ced6c91e4ff64cc8af9a","name":"transport-vec2835.patch","bytes":1268},{"sha256":"e070ac4ab58053c13a7440f8f819c4c3ef595ad0732a949efe082e5cd7b4f532","name":"lemma2835.md","bytes":2587},{"sha256":"7856a0baba8a59a4fe50ba526b7f83a7ec5368a8db947e51488b03bea5e11783","name":"producer2835.log","bytes":29885}],"patch_status":"pending integration: the integrator applies accepted patches to the research repository by hand; build on the served file plus this patch until then","decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}