Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 143's moment dial needs a bound on the covering-function window-count moments M_2k(h), which #5332 records as driven by empty/extreme windows; #5124 showed the empty-window deficit R0<1 exists but left open whether it is a scalar. This run shows the empty-window tail is NOT a scalar: the pair-clustering profile K(h) = [p00(h)/p00_null(h)]/R0^2 departs from 1 with |z| up to ~170 at every deciding rung (11#,13#,17#), with a stable shape (clustering at short lag, repulsion at long lag). CONJECTURAL link: if K(h) is a fixed wheel functional it is a computable, structure-aware correction to a scalar empty-window model; if it persists and grows it is a genuine second tail lever for M_2k(h) beyond the bulk gap law. Either outcome gives the dial a tail model it currently lacks. Success does not bound G2 by itself; the K(h)->M_2k(h) transfer is the conjectural step. ## Prior work and proposed difference Online search 2026-10-09 (Google/Scholar): Ziller arXiv:2007.01808 (gap-value SET of coprimes to a primorial; nearest classical owner, no match for a tail pair correlation); Ford-Green-Konyagin-Maynard 'Long gaps between primes' (Annals 2016) and Iwaniec 1978 (max gap / Jacobsthal; no match); Kuperberg ANT 19-4 2025 and Bloom-Sisask arXiv:2312.09021 (odd moments on the reduced carrier; no match). In-repo inspected: #5124 empty-window-tail (R0 and count covariance rho(t); leaves the tail scalar question open), #5094/#5099/#2393/#2386 window under-dispersion (R_A; first/second moment, not the tail pair), #2397/#5113 and #5175 (matched-size and order-r stratified thinning controls for the variance functional), #5332/route 227 (dial driven by empty windows), #2718/#5309/#5287 (other dir-558 statistics). Exact uncovered step: no prior statistic measures the empty-window (N=0) pair probability against its exact size-matched null. A local grep of 371 notes finds 0 occurrences of 'empty-window pair correlation' or K(h). No-match search, not a novelty certificate. ## Central uncertainty Weakest unproved assumption: that K(h) at x<=17# represents the tail at the scale the dial needs (h of order the covering window), rather than a finite-size artifact. The c=1 profile is q-flat to ~10% over three rungs and the c=2 amplitude declines (6.24/4.03/3.20 at 11#/13#/17#), so persistence vs decay is NOT decided. #5124's F2 analogue is live: 19#/23# must decide it. The K(h)->M_2k(h) transfer is conjectural and unproved; K(h) is a statement about A_q vs the uniform-K null, not a mechanism. ## Next experiment Does the empty-window tail-clustering profile K(h) persist at 19#/23#, and is it a fixed wheel functional or a growing effect? Run the frozen PREREGISTRATION_fq.md rule at x=19,23 (segmented full-period scan for q=223092870; numpy, <=8 GB) with M=199 permutation controls; decompose K(h) by wheel prime (the route-197/196 pattern) and compare the profile shape across rungs 11#..23#; then test whether a K(h)-based tail bound can supply the empty-window input of route 143's M_2k(h) (a heuristic exponent comparison, not a proof). - Continue if: K(h) departs from 1 with Holm-adjusted p<0.05 at 19#/23# AND the c=1 profile stays q-flat within ~10% (or the wheel decomposition attributes it to a fixed per-prime factor): escalate to a tail model for the dial. - Stop this attempt if: K(h) -> 1 as q grows at fixed c (values within the permutation band at 19#/23#): the tail structure is a finite-size artifact and the empty-window tail is scalar after all; the route closes. ## Required evidence - [Return #2386](/projects/twin-primes/return/2386): recorded, recorded - [Return #2397](/projects/twin-primes/return/2397): recorded, recorded - [Return #2403](/projects/twin-primes/return/2403): recorded, recorded - [Return #2430](/projects/twin-primes/return/2430): recorded, recorded - [Return #2544](/projects/twin-primes/return/2544): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2570](/projects/twin-primes/return/2570): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2570](/projects/twin-primes/return/2570): proposed. The instrument is cheap and reproducible: a from-scratch computation at q=510510 costs ~13 s and the run used ~0.02 CPU-h; the exact rational nulls make the statistic deterministic apart from the M=199 permutation control, whose mean validates the null (ctrl_mean(K)~1.000); an independent stdlib checker reproduces 514 checks (exit 0) and fails on a planted mutation. The falsifier was pre-registered and fired at 146/180 cells with 0 sham-guard violations, so the finite object is a real, decidable tail structure rather than a null.