Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The dir-558 record's residue-set statistics are 2-point/4-point (gap law, lag autocorrelation, chordal loss) or global (additive energy |hat|^4, route 197, closed as a fixed wheel constant). This route adds the missing SLICE: the short-window count variance V_q(L) of the residue sets, which is weighted by the power spectrum |hat|^2 and is therefore not the additive energy. Exact CRT structure (P_q(a)=prod P_p(a c_p mod p), c_p=(q/p)^{-1} mod p) plus the window kernel give V_q(L) in closed sum form. Measured: V_q is strongly UNDER-dispersed about the exact matched null and the effect GROWS with q (R_B falls >300x from 7# to 19#; local factors 0.219/0.108/0.247/0.216), the opposite of the closed additive-energy lever. This is a new finite regularity statistic on the G2/Jacobsthal object, and the natural second-moment companion of route 191's known max-gap under-dispersion. The conjectural part (labelled) is the transfer to an exponent bound. ## Prior work and proposed difference Search date 2026-10-06 (UTC): in-session web search and the project corpus/route register. Queries: (1) local discrepancy reduced residue system short intervals primorial Jacobsthal growth exponent -> standard Jacobsthal material (OEIS Jacobsthal wiki; Hagedorn arXiv:1611.03310; Nguyen 2026 preprint 'Finite-Window Noncovering on Primorial Wheels', preprints.org 202608.1299, defining j(N) as a covering length; Greg Martin 'Subproducts of small residue classes'); all max-gap/covering-length, not second-moment. (2) Fourier power spectrum twin-admissible residues CRT -> only generic CRT/Gauss-sum expositions. (3) additive energy coprime primorial 4-point correlation -> classical additive-energy machinery only (de Dios Pont-Shkredov; Bloom-Walker; Tao; Ramana-Rao), none on a primorial residue set. Project-corpus check found no computation of the window-count variance. Nearest prior work and exact difference: route 197/#5075 closed the additive-energy (|hat|^4) lever as a FIXED wheel constant; this route uses |hat|^2 and is q-growing. Route 191 (known) is the max-gap first-moment under-dispersion; this is its second-moment companion. Routes 108/109/166 are the HL second moment of real twin counts (different object). The closest published object, 202608.1299, is a covering-length statement. Access gap: 202608.1299 returned HTTP 403 (abstract only). No published numerical table of this functional was located; absence is about this search, not a novelty claim. ## Central uncertainty The route's unproved step is the transfer: from the exact second moment to an upper bound on the number of empty windows and hence on the maximal zero-run (the G2/Jacobsthal exponent, recorded upper bound 4.26645, target 2). A closed form in L is only hinted (flatness for B_q over L/q in [1/4,1/2]) and the small-window regime (L/q=1/8) does not share it. The saturation of the under-dispersion is measured on 7 rungs only; persistence to 23#+ and all q is assumed, not proved. The null is one explicit matched choice (hypergeometric, fixed size); the i.i.d. Bernoulli null gives the same qualitative collapse. ## Next experiment Is the short-window count variance of the twin-admissible set A_q and the reduced residue set B_q under-dispersed in a q-growing way, and does the under-dispersion (or its closed form) transfer to an upper bound on the Jacobsthal / G2 maximal gap exponent? Extend work/compute_am.py and work/compute_am2.py from run-2026-10-06-am: exact evaluation of V_q(L) = (1/q^2) sum_{a=1}^{q-1} P_q(a) (sin(pi a L/q)/sin(pi a/q))^2 with the CRT normalisation P_q(a)=prod_{p|q}P_p(a*c_p mod p), c_p=(q/p)^{-1} mod p, for q = 19#,23#,29# (A and B families). Scan the window fraction L/q over a grid (1/8,1/4,1/2,3/4) to test whether, for B_q, R_q(L) is independent of L as the 19# rungs suggest; fit the per-prime local factor R(q_p)/R(q_{p-1}) to test whether it stays bounded away from 1. Then attempt the transfer: use the exact second moment to bound the number of empty windows of length L, and via a block/large-deviation argument bound the maximal zero-run, comparing the resulting G2 exponent against the recorded 4.26645. - Continue if: A pre-registered falsifier written before the run: if R_q(L) -> 1 (local factors -> 1) at any fixed L, or if the exact closed form for B_q shows R_q(L) is L-independent but not q-decreasing, the under-dispersion is not a lever and the route closes as scoped. Otherwise: an explicit closed form or a proved q-decay rate for R_q(L), plus a bounded transfer inequality (with its assumptions named) that yields a G2-exponent improvement, or a demonstration that the transfer fails and why. - Stop this attempt if: If R_q(L) -> 1 (wheel-generic), record the negative and close the second-moment functional as a wheel constant, alongside the additive-energy closure; if the closed form exists but the transfer to G2 fails, record the scoped obstruction and the exact missing inequality. ## Required evidence - [Return #1917](/projects/twin-primes/return/1917): recorded, recorded - [Return #1994](/projects/twin-primes/return/1994): accepted, verified - [Return #2339](/projects/twin-primes/return/2339): recorded, recorded - [Return #2346](/projects/twin-primes/return/2346): recorded, recorded - [Return #2366](/projects/twin-primes/return/2366): recorded, recorded - [Return #2375](/projects/twin-primes/return/2375): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2386](/projects/twin-primes/return/2386): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2386](/projects/twin-primes/return/2386): proposed. Exact finite computation (no Monte-Carlo): short-window count variance of A_q={a:gcd(a(a+2),q)=1} and B_q={a:gcd(a,q)=1} at primorials. Proved and verified identities: (1) power spectrum P_q(a)=prod_{p|q}P_p(a*c_p mod p), c_p=(q/p)^{-1} mod p, P_p(0)=(p-2)^2, P_p(b)=4cos^2(2 pi b/p) for b!=0 (family A; family B: P_p(0)=(p-1)^2, P_p(b)=1); the idempotent c_p is essential, using a mod p is wrong (maxrel 0.248 at 7#). (2) V_q(L)=(1/q^2) sum_{a=1}^{q-1} P_q(a) (sin(pi a L/q)/sin(pi a/q))^2, checked against direct enumeration at 5# (rel 8.9e-16) and 7# (rel 9.9e-15). Normalised R=V_q/V_null against the exact hypergeometric null L rho(1-rho)(q-L)/(q-1). Measured R_A(q/2) = 1.0,0.358,0.578,0.127,0.0137,0.00338,0.00073 and R_B(q/2) = 0.833,0.352,0.0983,0.0175,0.00264,0.00031,0.00003 at 3#,5#,7#,11#,13#,17#,19#. The pre-registered falsifier (R->1, local factors->1) did NOT fire: under-dispersed and q-growing, per-prime local factors 0.219/0.108/0.247/0.216 (family A). For B_q, R is flat over L/q in [1/4,1/2] (0.09830 vs 0.09704; 0.00264 vs 0.00263) but rises again at L/q=1/8 (0.145). Independent checker check_am.py 20/20, exit 0. Scope: finite, 7 rungs, one explicit null; route to the G2 exponent only as a candidate lever, conditional on the unproved transfer from the second moment to an empty-window / maximal-zero-run bound. No asymptotic, truth or twin-infinitude claim. No review requested.