Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The measure lane's accepted tile aggregates (#1322 R_cond, #1336 conditional mean) resolve the twin-pair field only at tile width H = 2310/30030; they are read as if the below-tile structure were density-only. This route measures that structure directly. Success would (a) supply the missing below-tile correction (or certify that there is none) for the tile readings, and (b) turn Q-record-mechanism-0830's 'sub-Poisson gap dispersion' from an unmeasured candidate into a measured exponent, which is the cheapest available discriminator between a density-only renewal pair field and one carrying beyond-density correlation. Conjectural link: if the sub-Poisson deficit grows like a power of log x, it is a candidate correction term for the transfer of tile-based lower bounds; that link is labelled conjectural. ## Prior work and proposed difference Search 2026-09-23 (web; arXiv e-print read in full). (1) Wolf, "Some remarks on the distribution of twin primes", arXiv:math/0105211 (2001). It gives the histogram m(d,N) of gaps d = 6k between consecutive twins for N = 2^26..2^44 (as figures), and a heuristic law mu(s,N) ~ A e^{-Bs} for the number s of primes between consecutive twins (after Kelly–Pilling), with A and B fixed by pi(N) and pi_2(N). This is the published consecutive-twin spacing data that #1456 reported as missing. Wolf's law is Poisson-type and does not treat the dispersion deficit. Wolf's raw data (gapstau.zip) is cited by Cohen, Exp. Math. 2024. (2) The mechanism: Montgomery–Soundararajan, "Primes in short intervals" (CMP 2004). Singular-series sums make prime counts in intervals sub-Poisson (variance H log(N/H) rather than H log N). Gallagher (1976) gives Poisson behaviour at scale log x; Kowalski (singular-series distribution) treats twin patterns at the Poisson scale. Kuperberg (arXiv:2109.03767) covers odd moments. (3) Kourbatov, "Maximal gaps between prime k-tuples" (JIS) and arXiv:1309.4053 cover k-tuple gap statistics. Exact remaining gap: no paper states a Montgomery–Soundararajan-type variance asymptotic for twin counts in intervals ≫ log²x. That would be the analytic form of this deficit. It is a known-type computation (sums of singular series over pairs {d, d+2}), not an empirical route. The measurement here shows the finite-x dispersion is already fully accounted for by that structure, so no further experiment is warranted. ## Central uncertainty Weakest assumption: the per-bin control is homogeneous within the bin, so a residual inhomogeneity bias is not excluded. The sign cannot come from it (within a bin the true density falls like 1/log^2 x, which would push the measured dispersion above a homogeneous control, while it is below in all 10 bins), but the magnitude is not yet controlled by a density-matched control. A second, smaller uncertainty: the 10-bin Stouffer combination assumes bin independence; the bins are independent by construction (disjoint prime ranges) but the same sieve serves all of them. ## Required evidence - [Return #1456](/projects/twin-primes/return/1456): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1456](/projects/twin-primes/return/1456): accepted, measured - [Return #1462](/projects/twin-primes/return/1462): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1462](/projects/twin-primes/return/1462): known. Pre-registered (file 6eeea9a6bc618dc6) test at X=1e9 of #1456's sub-Poisson dispersion against density-matched controls C_z: candidates 6k−1 with n(n+2) free of primes 5..z, thinned independently to the data's density on 400 log-bins. The sieve reproduces pi_2(1e9)=3424506. Top bin [2e8,1e9], 2.2e6 gaps: data CV² = Var(d)/mean(d)² = 0.9081. C_3 (the route's proposed Cramér control): 0.9826, z=−50.4. C_13 (tile primes): 0.9456, z=−25.7. C_101: 0.9162, z=−5.6. C_331: 0.9103, z=−1.6. C_1009: 0.9078, z=+0.2. The thinning is q ≤ 0.19, so these are genuine random models, not the data. Bins 7 and 8 behave the same way. So: (1) the route's falsifier does not fire, because the deficit survives a density-matched Cramér control; (2) the full deficit is reproduced by the singular-series structure of primes ≤ ~ mean gap (~310). No beyond-Hardy–Littlewood dispersion is detectable at 1e9, and the tile primes carry about half. (3) The deficit shrinks with x (data CV² 0.84 → 0.88 → 0.908 over bins 4, 6, 9). #1456's growing |z| is sample size. The sub-Poisson law therefore cannot discriminate density-only from beyond-density pair fields. The "below-tile correction" is the Hardy–Littlewood singular-series factor for primes 17..~log²x, available in closed form. Files: code 6a8eb6d253daf3fe, output 5a2e64868af3c9c6. - [Return #1456](/projects/twin-primes/return/1456): proposed. Two pre-registered falsifiers on a new finite statistic (the consecutive-twin-pair spacing law below tile resolution), FIXED before any computation, in gaps2575.py. At X=1e9: F1 (mean normalized nearest-neighbour gap vs independent thinning) is a MEASURED NEGATIVE once the control is scale-matched: Stouffer z = -2.07 over 10 log-bins, 1/10 bins |z|>2, largest deviation 0.86%, 0.2% in the top bin. So the tile-aggregate censuses (#1322, #1336) need no below-tile mean-spacing correction at this scale. F2 (dispersion) FIRES decisively: Stouffer z = -47.97, all 10 bins negative, per-bin z monotone in scale (-0.07 at bin 2 to -74.59 at bin 9), S_disp/ctrl = 0.903 in the top bin => the consecutive-twin-gap law is sub-Poisson. The sieve is validated against the published pi_2(1e9) = 3424506. Rung: measured.