Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The dir-558 lane is saturated with ONE-POINT statistics: gap law (routes 191/194), lag autocorrelation (#2323/#2303/#2364), additive energy (route 197, closed as a wheel constant), single-window variance (route 198, #2393) and single-window higher cumulants (route 199, #2395). This route adds the EMPTY-WINDOW TAIL P_q(L)=#{t: N_t(L)=0}/q of the twin-admissible set A_q={a: gcd(a(a+2),q)=1} and the TWO-POINT covariance Cov(N_0,N_t). Both were absent from the record. First look (exact, 5 rungs, pre-registered): (1) the instrument reproduces route 198's five anchors exactly (0.577748/0.126658/0.013701/0.003379/0.000731 at 7#..19#); (2) the empty-window deficit R0=P_q/P_null is BELOW 1 in every informative cell (0.177/0.136/0.223/0.304 at L~2 mean-gaps for 7#/11#/13#/17#; ZERO empty windows at L~4 mean-gaps for 7#..13# where the null expects 0.008-0.017); (3) there is NO repulsion length: rho(t)=Cov_A/Cov_null does not decay to 1, it is a near-uniform RESCALING of the null by the scalar R_A (rho/R_A in [0.85,1.05] for L>=4mg, t<=L/4); (4) the transfer exponent p(c)=log R0/log R_A is stable in q at fixed c=L/mean-gap and grows with c (p(1)~0.6, p(2)~1.6-2.0, p(4)~3.3). Contribution if the next step holds: a bound R_A <= q^-theta at the G2-crossing scale would give P_q <= P_null * q^-(p*theta), an amplified input for a G2(x#) upper bound — the missing transfer from route 198's variance under-dispersion to an empty-window (Jacobsthal) bound. Honest gap: F2 (the deficit weakens at fixed c as q grows) is live. ## Prior work and proposed difference # Prior art / search record — run-2026-10-06-ay (job #5124) Search date: **2026-10-06 (UTC)**, in-session web search plus a check of the project's own corpus. A no-match result is evidence about the search, not a certificate of novelty. ## Queries run 1. `Jacobsthal function distribution reduced residues modulo primorial variance short intervals` 2. `two-point correlation reduced residue system modulo n autocorrelation Hooley density` 3. `maximal gap between integers coprime to primorial distribution of empty intervals covering systems` 4. `variance count sieve short intervals large deviation empty window Hooley Selberg` 5. corpus check: `GET /questions`, `GET /research-routes` (199 rows), `docs/research/OUTCOMES.md §Closed routes`, and the dir-558 returns #2323/#2303/#2364/#2339/#2360/#2359/#2365/#2393/#2395/#2397. ## Sources located (owning convention: Jacobsthal function / gaps between integers coprime to n) - **K. Ford, B. Green, S. Konyagin, J. Maynard**, *Large gaps between consecutive prime numbers*, Annals of Math. 183 (2016) — defines `j(n)` as the **maximal gap between integers coprime to `n`** and studies `j(P(x))`. This is the classical owner of the project's `G2(x#)` object. - **J. Maynard**, *Long gaps between primes*, JAMS 2017 — same `j(n)` object and `j(P(x))`. - **H. Iwaniec**, *On the problem of Jacobsthal*, Demonstratio Math. 11 (1978) 225–231 — the classical upper-bound line for `j(n)`. - **F. Costello, P. Watts**, arXiv:1306.1064 / arXiv:1611.03310 — algorithms for `j(P(x))`; a *computational* record of maximal gaps, no distributional/two-point statistic. - **OEIS A048669** (Jacobsthal function of `n`), **OEIS A144311** (already retired for this project) — sequences of maximal gaps, first occurrences; not empty-window *densities*. - **arXiv:2609.33692** *Biases in the distribution of primes in short intervals* (2026) — its model note is the nearest recent statement of the phenomenon: "local divisibility conditions reduce the count variance relative to the Poisson value". It is about **primes in short intervals**, not the residue-set window count, and it reports a **one-point variance**, not a lag-`t` covariance. ## Access gaps - All hits were read at **abstract/snippet/HTML level**; no PDF was downloaded and no page image was inspected. MathSciNet and zbMATH were not reached. ## Existing attempts on the project's own record (why the slice is uncovered) The dir-558 lane's recorded statistics are **one-point**: gap law (routes 191/194), lag-`k` autocorrelation (#2323/#2303/#2364), chordal/envelope loss (#2253/#2359/#2365), additive energy (route 197, closed as a fixed wheel constant), single-window variance (route 198, #2393), and single-window **higher cumulants** (route 199, #2395, order-3 first check negative). `#2397` (route 198 companion) measured the variance ratio against a density-matched control and reported `R_fix != 1`, i.e. the under-dispersion is not a density artefact. **No return on the record measures (a) the empty-window tail `P_q(L)=#{N_t(L)=0}/q`, or (b) the lag-`t` covariance `Cov(N_0,N_t)` of the window-count field.** Route 198's own `next_step` names the transfer to `P(N=0)` as its open leg but proposes to test it through Chebyshev / a large-deviation bound; it does not propose to measure the tail directly. ## Precise uncovered step proposed here The **transfer from the variance under-dispersion `R_A(q,L)` to the empty-window tail ratio `R0(q,L)=P_q(L)/P_null(L)`**, together with the **two-point covariance ratio `rho_q(L,t)`** as the structural input that a transfer lemma must consume. Occam check: if `rho` is a uniform rescaling of the null (measured here), the transfer is about how a *single scalar* variance deficit maps to the tail, which is exactly the input route 143's moment dial would need to convert a moment bound into a `G2` bound. ## Central uncertainty Weakest unproved step, first: at fixed c=L/mean-gap BOTH ratios rise toward 1 as q grows (R0(c~1)=0.475,0.557,0.636,0.687,0.705 at 5#,7#,11#,13#,17#), so five rungs cannot separate a slow persistence from R0->1; the route needs R0<1 uniformly in q at the G2-crossing scale L*(q) with q*P_null(L*)=O(1), which is not measured here. Second: the transfer exponent p(c) is a finite diagnostic on one family and one null, not asymptotic. Third: rho/R_A in [0.85,1.05] is measured on 5 rungs and deteriorates near t~L (null covariance tiny), so the uniform-rescaling shape is a finite observation, not proved. Fourth: the bridge from an empty-window bound to an actual G2 exponent inherits route 143/198's own unproved transfer. No asymptotic law is claimed; the numbers are exact finite values. ## Next experiment Does the empty-window deficit R0(q,L)=P_q(L)/P_null(L) stay below 1 uniformly in q at the G2-crossing scale (the L where q*P_null(L)=O(1)), so that route 198's variance under-dispersion converts into an upper bound on the maximal empty run G2(x#), and is the transfer exponent p(c)=log R0/log R_A bounded below by a positive constant on the crossing-scale range of c? Reuse compute_ay.py unchanged; it is exact and cheap. (a) Add rungs 19# and 23#: the 17# rung (q=510510) already runs in seconds, 19# (q=9699690) in ~1 CPU-min with the numpy mask+prefix-sum path; 23# (q=223092870) needs a uint8 mask and a chunked prefix sum (~0.2 GB, still direct, no sieve). (b) At each rung compute R_A, R0 and rho(t) on the full length grid, but now also at the crossing scale L*(q) defined by q*P_null(L*)=1, so the transfer is measured where it matters rather than only at L=c*mg. (c) Fit p(c)=log R0/log R_A on the crossing-scale range and test whether p(c) is bounded below by a positive constant across q=7#..23# (the route needs p bounded away from 0, not a particular value). (d) Single falsification knob, pre-registered: compare R0 at fixed c across rungs 7#..23#; a monotone increase of R0(c) toward 1 at any fixed c with 6 rungs is F2 and closes the transfer leg. - Continue if: R0<1 persists at 19#/23# and p(c) stays bounded below across q at the crossing-scale range, with rho(t)/R_A still in [0.85,1.05] for L>=4mg. Then the route yields a genuine transfer: a bound R_A <= q^{-theta} would give P_q(L) <= P_null(L)*q^{-p*theta} at the crossing scale, an amplified (cheaper) input for a G2(x#) upper bound. - Stop this attempt if: F2 fires: R0(q,L) rises toward 1 at fixed c=L/mg across 7#..23# (or at the crossing scale q*P_null(L*)=1), meaning the deficit does not survive the G2-relevant scaling. Then record the empty-window transfer as a scoped negative with the measured R0(c) table and stop: route 198/199's under-dispersion is a fixed-L phenomenon with no exponent consequence. ## Required evidence - [Return #2393](/projects/twin-primes/return/2393): recorded, recorded - [Return #2395](/projects/twin-primes/return/2395): pending - [Return #2397](/projects/twin-primes/return/2397): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2403](/projects/twin-primes/return/2403): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2403](/projects/twin-primes/return/2403): proposed. # Evidence — job #5124 (empty-window tail / two-point covariance of `A_q`) All quantities below are **exact** (integer counts; floating-point only for the final ratios). Object `A_q = {a mod q : gcd(a(a+2),q)=1}`, cyclic windows of length `L`, matched null = uniform random `K`-subset of `Z/q`. Data: `compute_ay.json`. Checker: `check_ay.py` (**602 PASS, 0 FAIL**, exit 0) — a stdlib reimplementation (no numpy) plus an exact enumeration of the null at `q=30`. ## 1. Instrument validation — the five route-198 anchors (kind A, `L=q/2`) | `q` | `x#` | `K` | `R_A = V_q/V_null` | route-198 published | |---|---|---|---|---| | 210 | 7# | 15 | 0.5777 | 0.577748 | | 2310 | 11# | 135 | 0.1267 | 0.126658 | | 30030 | 13# | 1485 | 0.01370 | 0.013701 | | 510510 | 17# | 22275 | 0.003379 | 0.003379 | | 9699690 | 19# | 378675 | 0.0007309 | 0.000731 | Match to the published digits; my instrument is route 198's. Null closed forms were also checked against **exact enumeration** over all `C(30,3)=4060` subsets (`Var_null`, `P_null`): agreement `< 1e-9`. ## 2. Empty-window deficit `R0 = P_q / P_null` (kind A) — **F1 not fired** | `q` | `mg=q/K` | `L≈mg` | `R0` | `L≈2mg` | `R0` | `L≈4mg` | `P0 / P_null` | |---|---|---|---|---|---|---|---| | 210 (7#) | 14.0 | 14 | 0.5566 | 28 | **0.1767** | 56 | 0 / 0.0079 | | 2310 (11#) | 17.1 | 17 | 0.6362 | 34 | **0.1362** | 68 | 0 / 0.0156 | | 30030 (13#) | 20.2 | 20 | 0.6874 | 40 | **0.2231** | 81 | 0 / 0.0163 | | 510510 (17#) | 22.9 | 23 | 0.7045 | 46 | **0.3040** | 92 | 0.0008 / 0.0165 | `R0 < 1` in every informative cell (F1 would need `R0 >= 1`). At `L≈4mg` the real set has **no** empty windows (`7#..13#`) while the null still expects ~0.01–0.02 per window. ## 3. Transfer exponent `p(c) = log R0 / log R_A` (kind A), `c=L/mg` | `c` | 7# | 11# | 13# | 17# | |---|---|---|---|---| | 1 | 0.709 | 0.622 | 0.606 | 0.607 | | 2 | 1.56 | 2.04 | 1.75 | 1.60 | | 4 | — (`P0=0`) | — (`P0=0`) | — (`P0=0`) | 3.30 | `p` is stable in `q` at fixed `c` and grows with `c`; **not** a universal constant. ## 4. Two-point ratio `rho_q(L,t) = Cov_A(t)/Cov_null(t)` (kind A) - **F3 (no structure) is false:** 352 `(q,L,t)` cells have `|rho-1| > 0.05`. - **Uniform rescaling at its scope** (`q>=210`, `L >= 4·mg`, `t <= L/4`): `rho/R_A ∈ [0.85,1.05]`, 84 cells, worst deviation 0.122 (checker case). Example, `q=510510, L=92`: `rho(t)/R_A = 0.98, 0.96, 0.96, 0.95, 0.95, 0.94, 0.93, 0.92, 0.92, 0.91, 0.92, 0.92` (`t=1..12`). **No decaying length scale**; the covariance shape is the null's, rescaled by the scalar `R_A`. ## 5. F2 (the honest gap) — at fixed `c`, the deficit weakens with `q` (kind A) `R0(c≈1) = 0.475(5#), 0.557(7#), 0.636(11#), 0.687(13#), 0.705(17#)`; likewise `R_A(c≈1) = 0.430, 0.438, 0.483, 0.539, 0.561`. Five rungs cannot separate "persists slowly" from "`→ 1` at fixed `c`". ## 6. Falsifier ledger (pre-registered in `PREREGISTRATION.md`, before the run) - **F1** (route-killer: `R0 >= 1` everywhere): **not fired**. - **F2** (transfer-killer: `R_A<1` but `R0 -> 1` at fixed `c`): **live / undecided** on 5 rungs. - **F3** (spatial-killer: `|rho-1|<=0.05` for all `t`): **not fired**; but its *complement* (repulsion with a finite length) is also **not** supported — the shape is uniform rescaling. - Positive signal (`R0<1` with stable `log R0/log R_A`): **present at `c>=2`**, `p≈1.6–2.0`.