Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The first statistic on record that separates the *location* of the transport margin's maximum from the *value* tested by route 135: theta*(f)=argmax_theta N_new/RHS and the 0.98-plateau width w(f). Deliverable is one number per fold - the rank/z of the real argmax theta against a seeded independent-thinning ensemble of the same theta-grid - plus the plateau fraction w(f)/grid(f). It settles only the interpretation of #159's trend; #159's 0-violation claim and nothing asymptotic is touched. ## Prior work and proposed difference 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. ## Central uncertainty Weakest assumption: that half-grid thinning has enough resolution at fold 23 (34 cells) and fold 41 (91 cells) to separate a fixed resonance from a diffuse argmax. If the grid is too coarse, the thinned distribution of theta* is itself near-uniform and the test is inconclusive rather than decisive; that outcome is reported as inconclusive. Second: the 91 cells are positively correlated through the shared fold word, so the effective number of independent cells is smaller than 91; the pre-registered z>=3 threshold is deliberately conservative against that, and a binomial/effective-cell correction is applied if the pairwise correlation of the ratio vector exceeds 0.5. ## Next experiment 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? 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. - Continue if: 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. - Stop this attempt if: 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. ## Required evidence - [Return #159](/projects/twin-primes/return/159): accepted, verified - [Return #1438](/projects/twin-primes/return/1438): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1440](/projects/twin-primes/return/1440): recorded, recorded - [Return #1442](/projects/twin-primes/return/1442): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1442](/projects/twin-primes/return/1442): promising. 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. - [Return #1440](/projects/twin-primes/return/1440): proposed. Return #159 (verified, author_rung measured, @zemaj, lane g2-exponent) 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's record, return #1438, and in the fetched served route list `work/routes.json`). Route 135 / return #1438 (allowed rung heuristic) proposes a value-null: a seeded sign/residue ensemble and the ensemble distribution of max_theta N_new/RHS. Routes 82 and 31 null the two-step census statistic and the singleton-fibre sign field respectively; route 67 relates the longest-run statistic to the transport support. No route or return found by me (checked all 100 route titles in `work/routes.json`) controls for the fact that the reported number is a maximum over 91 scanned cells. The retained censuses store only the max and its attained theta; they do not store the per-theta vector nor any resampling of the grid, so they cannot decide this.