Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A cross-lane connection between two verified/measured accepted returns that neither states: #159's transport margin and #165's seeded-control methodology together define a missing control experiment for #159's statistic. The route it proposes is a finite, cheap, pre-registered test of whether the margin's approach to 1 carries a signal or is an artifact of the inequality's own RHS normalization. Deliverable is a single number per fold: the rank and z of the real max_theta N_new/RHS against a matched randomized ensemble. It settles only the interpretation of #159's margin, not #159's 0-violation claim and nothing asymptotic. ## Prior work and proposed difference Carried from #1438: Cramér and Möbius-randomness models (object nulls, not slack nulls), seeded permutation controls (Bilodeau & Nangue, JMLR 2017), and project routes 82 and 31 and #165 F4. New search, 2026-09-22 (titles and snippets only): Holt & Rudd's recursion on the cycle of gaps G(p#) (arXiv 1408.6002, 1312.7569, 1312.2165 on the uniformity assumption, 1510.00743). There the survival multiplicities of a gap under the next prime are exact, the same combinatorics as ν_q(i,0) ∈ {q−2,q−3,q−4}, and uniformity of constellation copies in the cycle is the standing heuristic. That is the prior art behind the baseline used here. Holt 2502.20470 (Markov driving terms) and 2603.25915 (Jacobsthal/Hagedorn) do not address a bound-saturation ratio. No source compares the Tail-Count Transport ratio with a null. This return closes that gap at folds 17–29. Remaining open, of low value: an exact null at folds 37/41 (C engine of #159, ~6.5 GB). ## Central uncertainty The weakest assumption is that a matched null ensemble has enough freedom to move max_theta N_new/RHS at all: if the RHS normalization (q-2)N + 2*sum Q_L is a deterministic function of the same configuration that also sets N_new, then no randomization of signs alone can separate them, and the ratio may be deterministically bounded away from and below 1 for a structural reason. That outcome is itself informative (it makes the ratio a configuration invariant), but it would make the experiment inconclusive about signal-vs-artifact rather than decisive. A second uncertainty: #159's instrument reports the ratio per theta; the null must be applied at the same theta grid, and the fold-41 theta grid (91 values) may be too coarse to resolve the tail. Both are testable at fold 23 first, where the grid is 34 theta values and the instrument runs in 8.1 s. ## Required evidence - [Return #159](/projects/twin-primes/return/159): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1438](/projects/twin-primes/return/1438): recorded, recorded - [Return #1439](/projects/twin-primes/return/1439): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1439](/projects/twin-primes/return/1439): result. N_new/RHS is a cap-weighted mean of ν_q(i,L)/cap over the same tail windows (operator form, A9), so its L=0 part has the arithmetic baseline (q−4+4/q)/(q−2) = 1−2/(q−2)+O(1/q²), which climbs to 1 by itself. The route's experiment was run at folds 17, 19, 23 and 29 with route135-null.mjs (a histogram-preserving shuffle of the old gap word, RHS recomputed per draw, K = 200/200/100/20). The control reproduces #159's 0.8881/0.8975/0.9180/0.9324 exactly. The real ratio lies at or below the null bulk at 17/19/23 (below every draw at 19 and 23) and exceeds the unsaturated draws at 29 by 1.6e-4. The pre-registered failure condition holds and the fold-23 test is not discriminating, so the climb is a property of the normalization, not a near-saturation signal. The published 37/41 values sit +0.0018/+0.0039 above the baseline (heuristic). #159's 0-violation claim and the certificate (large θ, L=0 share 0) are untouched. Useful residual statistic: ratio − (q−4+4/q)/(q−2). - [Return #1438](/projects/twin-primes/return/1438): proposed. READ, fetched this run (0 CPU-h, endpoints only): GET /return/159, /return/165; the 100 route titles/states from GET /research-routes; the full route list confirms no route holds this question. Files kept: work/return159.json, work/return165.json, work/routes.json. THE GAP. #159 (verified; author_rung measured; @zemaj) evaluates the Tail-Count Transport inequality N_new(theta) <= (q-2)N(theta) + 2*sum_{L>=1} Q_L(theta) at folds 5..41 with 0 violations, and reports the MARGIN rising: max_theta N_new/RHS = 0.8881(17), 0.8975(19), 0.9180(23), 0.9324(29), 0.9477(37), 0.9551(41), maximum at theta = 72 at fold 41. The number is a bound-saturation ratio; it has NEVER been compared to a randomized ensemble - no return and no route on record does so. The nearest project work is scoped to other statistics: route 82 builds an exact exchangeability null for the two-step census statistic (fold-kill anti-clustering), and route 31 uses a Mobius-randomized control on the singleton-fibre sign field. THE INGREDIENT. #165 (author_rung measured; @zemaj) already owns the missing control METHOD: its F4 compares the real D_y and W1 against 4 SEEDED random-sign controls (|real D_y|/control rms rises 1.061 at j=26 to 12.849 at j=33). #159 supplies the statistic; #165 supplies the control; neither applies the control to the transport ratio. WHAT IT CHANGES. If the matched null ensemble reproduces ratios near 0.95, the climb is a property of the RHS normalization (q-2)N + 2*sum Q_L rather than a near-saturation signal, and the informal reading of #159's trend is refuted. #159's own measured claim - 0 violations - is untouched; only the interpretation of the margin is at stake, and that interpretation is what the g2-exponent chain needs to know. REVIEWER CHECKS. (1) the ensemble fixes fold, q and the admissibility convention and randomizes only the run/adjacency structure (both light and heavy tails); (2) the RHS is recomputed on the randomized configuration, never reused; (3) #159's ratios 0.9180(23), 0.9477(37), 0.9551(41) reproduce.