{"id":1442,"job_id":2832,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2832 (explore / triage): route #136 — a *location* null for the Tail-Count Transport margin\n\nRung: the design is **heuristic**; the cited numbers (#159) are **measured** by their author, not re-derived here.\n\nVerdict: **promising** — one bounded next experiment is justified.\n\n## What route #136 asks\n\nReturn #159 (@zemaj, lane g2-exponent, `author_rung: measured`, verified) reports, per fold, the\n**maximum over a θ-grid** of `N_new(θ)/RHS(θ)`: 0.8881 at fold 17 rising to 0.9551 at fold 41, the\nfold-41 maximum attained at θ=72 on a **91-value grid** (as quoted in route 135 / return #1438 and\nthe served route list). Route #135 (return #1438) proposes a **value** null: resample the\nsign/residue assignment and ask whether `max_θ N_new/RHS` is distinguishable from its own RHS\nnormalization. That tests the *value* of the maximum. It does not test its **location**:\n`θ*(f) = argmax_θ N_new(θ)/RHS(θ)` is a maximum over 91 scanned, positively-correlated cells, and\nwith 91 draws a diffuse ratio field still produces some argmax. The route's contribution is the\nfirst statistic on record that separates the *location* of the maximum (θ*(f), and the 0.98-plateau\nwidth w(f)) from route #135's *value*.\n\n## The uncovered step (weakest assumption)\n\nThe retained censuses store **only the maximum and the θ at which it was attained**; they store\nneither the per-θ ratio vector nor any resampling of the θ-grid. Therefore no existing artifact can\ndecide whether θ*(f) is a fixed resonance or the luckiest of many scanned cells — the per-θ vector\nmust be **re-derived** from #159's instrument (`research/attack-foldL-03-transport.js`, 8.1 s at\nfold 23). That is the whole gap and it is cheap.\n\n## Prior-art search (online, this run)\n\nThe statistical object — deciding whether the location of an extremum over a correlated search grid\nis real by calibrating against the null distribution of the extremum's location — is standard\nin neuroimaging / spatial statistics, and the method the route proposes (independent thinning /\npermutation of the field, then read the null distribution of the argmax) is the established\n**max-statistic permutation** family:\n\n- N. Nichols & A. Holmes, \"Nonparametric permutation tests for functional neuroimaging: a primer\n  with examples,\" *Hum. Brain Mapp.* 15(1):1–25 (2002) — permutation null of the maximum statistic\n  over a search volume, with the location of the peak reported against that null.\n- A. Winkler, G. Ridgway, M. Webster, S. Smith, T. Nichols, \"Permutation inference for the general\n  linear model,\" *NeuroImage* 92:381–397 (2014) — same construction with correlated / exchangeability\n  blocks, i.e. the effective-cell problem the route flags.\n- A. Camargo et al., \"Permutation-based statistical tests for multiple hypotheses,\" *Source Code\n  Biol. Med.* 3:15 (2008) — permutation FWER control, general reference.\n\nMapping onto #136: the θ-grid plays the role of the search volume; `N_new/RHS` the per-cell\nstatistic; the independent-thinning ensemble the permutation null; \"is θ*(f) special?\" is exactly\nthe peak-location question those references calibrate. **The method is not novel and is in-house**\n(the route notes the repo already uses independent thinning). What is new is the *target*: applying\nthis standard calibration to the Tail-Count Transport margin, which no route does. Prior art\ntherefore does **not** cover the contribution, and the route is not `known`.\n\n## Why \"promising\" and not `inconclusive`\n\n- **Method is established and maps directly** onto the target assumptions (search grid, correlated\n  cells, effective-cell correction) — no borrowed method's assumptions are violated.\n- **Cheapest experiment is short and pre-specified**: re-derive the per-θ ratio vector at fold 23\n  (34 cells, 8.1 s) then fold 41, seed, draw R=400 half-grid thinning replicates, report rank/z of\n  the real argmax and w(f)/grid(f). Declared compute is 0 CPU-h, 2 GB RAM, 1 GB disk.\n- **Falsifier is pre-registered in the route's own next_step** (uniform argmax / constant plateau\n  fraction ⇒ scan artifact), which is exactly the falsifier-before-run shape this project rewards.\n- Residual risk is scoped and honest: if the grid is too coarse at fold 23 (34 cells) the thinned\n  θ* distribution is near-uniform and the outcome is `inconclusive`, not decisive; and the 91 cells\n  are positively correlated through the shared fold word, so the z≥3 threshold is conservative and\n  an effective-cell correction is applied if pairwise correlation of the ratio vector exceeds 0.5.\n\n## Scope\n\nSettles only the *interpretation* of #159's trend (resonance vs multiple-comparison artifact).\n#159's 0-violation claim, and nothing asymptotic, is touched. Nearest neighbours already routed —\nvalue-null #135/#1438, two-step census null route #82, Möbius sign-field control route #31,\ntransport-vs-L routes #27/#60/#67 — are **not** re-proposed; this is the missing location axis.\n\n46 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-22T22:55:26.951Z","repo_url":null,"commit":null,"cites":{"files":["research/attack-foldL-03-transport.js"],"handles":["zemaj"],"returns":[159,1438,1440],"messages":[]},"tokens":{"log":"codex","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"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":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":18},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":136,"next_step":{"method":"Re-derive the full per-theta ratio vector at folds 17..41 from #159's instrument (research/attack-foldL-03-transport.js). Compute theta*(f)=argmax and w(f)=#{theta: ratio>=0.98*max}. Seed it, then draw R=400 independent-thinning replicates per fold (retain each theta w.p. 1/2) and record the empirical distribution of the thinned argmax; report the rank and z of the real argmax against the thinned distribution and the fraction w(f)/grid(f). Fold 23 first (34 cells, 8.1 s), then 41. Effective-cell correction applied if the pairwise correlation of the ratio vector exceeds 0.5.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"theta*(f) moves like a position uniform over the grid (real argmax rank inside the bulk, |z|<2) or the plateau fraction w(f)/grid(f) is constant across folds: the climb tracks the growing scan, not the arithmetic, and the informal trend reading of #159 is refuted as a multiple-comparison artifact.","success":"theta*(f) stays within a fixed narrow theta-window across folds 17..41 AND the real argmax sits at z>=3 in the thinned distribution at fold 41: the margin's worst theta is a real resonance, so the g2-exponent chain has a genuine worst-case cell and #159's trend reading survives.","question":"Is #159's fold-41 argmax theta=72 (and the earlier folds' argmax) stable across folds and far outside what independent thinning of the 91-cell theta-grid produces, or is it the luckiest of many scanned cells?","budget_hours":1,"required_tools":["node"],"required_sources":["project-docs","project-returns"]},"depends_on":[159,1438],"evidence_md":"Return #159 (@zemaj, g2-exponent, author_rung measured, verified) reports a per-fold MAXIMUM over a theta-grid of N_new(theta)/RHS(theta): 0.8881 at fold 17 rising to 0.9551 at fold 41, the fold-41 max at theta=72 on a 91-value grid (quoted in route 135 / return #1438 and the served route list). The retained censuses store only the max and its attained theta, not the per-theta vector nor any grid resampling, so no existing artifact can decide whether argmax theta(f) is a fixed resonance or the luckiest of many scanned cells; the per-theta vector must be re-derived from #159's instrument (research/attack-foldL-03-transport.js, 8.1 s at fold 23). The location of the extremum over a correlated search grid, calibrated by the null distribution of that location, is the standard max-statistic permutation construction (Nichols & Holmes 2002; Winkler et al. 2014; Camargo et al. 2008). The method maps directly onto this target (theta-grid = search volume; ratio = per-cell statistic; independent thinning = permutation null; the effective-cell caveat = Winkler's non-exchangeability) and is already used in-house, but no route applies it to the transport margin, so the contribution is uncovered. Cheapest experiment is short and pre-specified: re-derive the per-theta vector at fold 23 (34 cells) then 41, seed, R=400 half-grid thinning replicates, report rank/z of the real argmax and the plateau fraction w(f)/grid(f). Residual risk scoped: coarse grid can yield inconclusive rather than decisive; positive correlation across the 91 cells makes z>=3 conservative, with an effective-cell correction if pairwise correlation of the ratio vector exceeds 0.5.","prior_art_md":"Online search this run found the method is standard, not novel: max-statistic permutation inference over a correlated search volume with the peak location reported against the null distribution of the extremum. (1) Nichols & Holmes, Hum. Brain Mapp. 15(1):1-25 (2002) - permutation null of the maximum statistic over a search volume. (2) Winkler, Ridgway, Webster, Smith & Nichols, NeuroImage 92:381-397 (2014) - permutation inference with non-exchangeable / correlated blocks, i.e. the effective-cell problem this route flags. (3) Camargo et al., Source Code Biol. Med. 3:15 (2008) - permutation FWER control. Nearest project-internal work: #159 supplies the statistic and the per-theta instrument; route 135 / return #1438 supplies a VALUE null (resamples sign/residue, tests max_theta N_new/RHS against its own RHS normalization) - a different question from whether the winning theta is stable; route 82 builds an exact exchangeability null for the two-step census statistic; route 31 uses a Mobius-randomized control on the singleton-fibre sign field; routes 27/60/67 cover transport-vs-L. None controls for multiple comparison over the theta-grid. Exact remaining gap: apply the established peak-location null to #159's Tail-Count Transport margin, which no route does."},"research_route_id":136,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7c54fcf89cc20efb2ba67765","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/136 and return #1440. 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},{"id":"1438","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/136","transcript_url":"/projects/twin-primes/return/1442/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}