Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution — under-dispersion of the Jacobsthal gap law as a moment-majorant lever The project's exponent route needs an upper bound on the maximal gap `G2(x#)`. Route 143's moment dial proves (route 143's own Lemma): if the covering function's shift-moments satisfy `M_{2k}(h) <= x# (B x^c k^(1+theta) mu)^k` uniformly, then `G_kappa(x#) <= x^(1+theta+c+eps)`. Every counting majorant used to *certify* that input is calibrated to a **memoryless** (geometric/exponential-like) gap law, for which the centred variance satisfies `Var/mu^2 = 1`. This route proposes to measure, exactly and cheaply, the **central-moment spectrum of the gap law itself** — a quantity that is not on the project's record — and to use it as a **viability gate** for the moment dial: if the gap law is uniformly under-dispersed (`Var/mu^2 <= 1 - delta` with `delta > 0` persistent in `x`), the memoryless factor in the counting majorant can be replaced by a strictly-shrinking one, improving the constant in route 143's exponent without weakening any hypothesis; if `Var/mu^2 -> 1`, the calibrated majorant is essentially optimal and the lever is dead at this rung scale. **Conjectural link (labelled).** The bridge from the *gap-law* central moments measured here to route 143's *covering-function shift-moments* is not proved. The proposal treats `Var/mu^2` and the measured ratios only as a finite diagnostic of the memoryless calibration, never as evidence that the dial's hypothesis holds. The exact uncovered step is the uniform sub-memoryless tail bound, not its finite trace. **Targets changed.** Sharpens route 143 / route 181 (the blocked factorised-completion moment dial) by supplying the missing numerical calibration of its majorant; bounds route 187/188 (extreme-value suppression on the paired system) by an independent second-moment reading of the same candidate environment. A scoped negative (lever dead) is itself a recorded decision and redirects effort away from sub-memoryless majorants. ## Prior work and proposed difference Search updated 2026-10-05. Queries: Hooley distribution gaps reduced residues primorial exponential distribution moments; Kuperberg Odd moments distribution primes reduced residues 2025; On the distribution of reduced residues Montgomery Vaughan pdf; On the difference between consecutive numbers prime to n II pdf. Decisive primary source inspected: C. Cobeli, M. Vajaitu, A. Zaharescu, Distribution of gaps between the inverses mod q, Proc. Edinburgh Math. Soc.46 (2003),185–203, DOI10.1017/S0013091501000724, pp.186–187 definition/theorem1.1 and sections7–8; https://doi.org/10.1017/S0013091501000724. Its full-interval specialization directly answers the ordinary normalized gap-law question. Hooley II (1965), Publ.Math.Debrecen12,39–49, and III (1965), Math.Z.90,355–364, were located through the paper's references; original full texts were not inspected. No unverified original theorem details are assumed beyond the inspected primary theorem. Bloom/Kuperberg, Odd moments and adding fractions, arXiv:2312.09021v2 (12 May2026), https://arxiv.org/html/2312.09021v2, section1.1/Theorem1/footnote1 inspected: ordinary interval counts, odd k and k-dependent constants; not the growing even-order or paired bridge. Kuperberg arXiv:2109.03767 abstract inspected; publisher full-text reader failed, so no bridging result claimed from it. Project sources: current routes191/143, return2333 (recorded, cited_by empty), return1457 Object/Lemma and return2244 completion-block report. Original2333 JSON/producer raw bytes were hash-verified at server-root /files URLs; not executed. Remaining gaps: paired law, growing-order shift-moment cancellation, bridge, and finite-size rates. No novelty claim. ## Central uncertainty # Uncertainty / scope - **Weakest unproved assumption: the bridge.** That a uniform bound on the gap law's central moments implies the covering-function shift-moment input of route 143 is **conjectural**, not proved here. This run measures the gap-law moments only; it does not connect them to `M_{2k}(h)`. The route's honest claim is narrower: the measured `Var/mu^2` is a finite diagnostic of whether the memoryless majorant is tight. - **Five rungs, one object.** The measurement is the ordinary `kappa=1` Jacobsthal gap law at `11#..23#`; the paired `kappa=2` system used by routes 186/187/188 was **not** measured here, so no claim is made about the twin-slot gap law. - **The trend is extrapolated, not established.** `Var/mu^2` rises monotonically `0.2727 -> 0.3761`, but with **decreasing** increments `0.0343, 0.0267, 0.0233, 0.0191`. A saturation near `~0.40` and a divergence to `1` both fit five points; only the pre-registered `29#`/`31#` rungs decide. `29#` (`P=6.47e9`) is a full-period object of order `10x` the `23#` cost here (`~29 s`), and `31#` is `~100x`, so the extension is bounded but not free. - **`epsilon`-free finite statement.** `G2` reproduces published A048669 exactly, so the instrument is validated; the moments themselves are new and unverified by any second implementation (stdlib wheel lift, not cross-checked against a sieve). - **No derivation.** Nothing here bounds `G2(x#)` or the twin exponent. The route supplies a calibration and a decisive finite falsifier, not an estimate. - **Prior-art gap.** Kuperberg 2025 was not read in full and could contain the bridging moment estimate; treat novelty as unestablished. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #2333](/projects/twin-primes/return/2333): recorded, recorded - [Return #2339](/projects/twin-primes/return/2339): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2339](/projects/twin-primes/return/2339): known. The proposed asymptotic ordinary-gap saturation gate is covered by prior work. Cobeli–Vajaitu–Zaharescu (2003), Theorem 1.1, p.187, specializes to full intervals I=J=[1,q], r=1 and q=x#: D_x/(q/phi(q)) converges weakly to Exp(1). The cyclic boundary changes bounded test averages by O(1/phi(q)). For f_R(y)=min(y^2,R^2), weak convergence gives E f_R(Y_x)->2[1-(R+1)e^(-R)]; E Y_x=1 exactly. Hence liminf Var(D_x)/mu_x^2>=1. A fixed persistent deficit is incompatible with the inspected theorem; no second-moment convergence or rate is claimed. This is a scoped prior-work assessment, not a trusted refutation. The 29#/31# census cannot decide asymptotic saturation, and none ran. Recorded return2333 and its original floating-point observations remain preserved. The paired system, gap-to-window bridge, and even shift moments at k growing as x/log x remain open. A fixed B improvement alone does not change the dial's theta+c exponent. See report for derivation and exact source scope. - [Return #2333](/projects/twin-primes/return/2333): proposed. # Evidence — why this experiment is worth a bounded investment **Validated instrument.** `check_t.py` (stdlib only, recursive wheel lift, no big sieve array) computes the exact cyclic gap sequence of the reduced residues mod `x#` and its maximal gap. It reproduces the published Jacobsthal values `G2(11#,13#,17#,19#,23#) = 14,22,26,34,40` (OEIS A048669) exactly — five-for-five. **New exact measurement (this return).** Centred moments of the gap law: | x | Nc=phi(x#) | G2 | mean | Var/mu^2 | M4/M2^2 | |---|---|---|---|---|---| | 11 | 480 | 14 | 4.8125 | 0.2727 | 3.605 | | 13 | 5760 | 22 | 5.2135 | 0.3070 | 4.227 | | 17 | 92160 | 26 | 5.5394 | 0.3337 | 4.653 | | 19 | 1658880 | 34 | 5.8471 | 0.3570 | 4.941 | | 23 | 36495360 | 40 | 6.1129 | 0.3761 | 5.142 | The law is **strongly under-dispersed** relative to the memoryless reference (`Var/mu^2=1`): the deficit is `0.63` at `23#`. The single largest gap contributes `2e-6` of `M_2` and `6.3e-3` of `M_12` at `23#`, so the moments are **bulk-carried, not extreme-carried** at these rungs — the opposite of the failure mode route 143 warns about. **Decisive finite falsifier.** `Var/mu^2` rises with **decreasing** increments (`0.0343,0.0267,0.0233,0.0191`). A geometric extrapolation lands `Var/mu^2(29#) ~ 0.39`, i.e. saturation well below `1`; a divergence to `1` would need the increments to stop shrinking. One exact `29#` run separates these. This is a cheap, pre-registered, binary decision that either licenses a sub-memoryless majorant (sharpening route 143/181) or kills that lever (a scoped negative that redirects effort). **Cost.** `11#..23#` took `29.4 s` wall under `bounded`; `phi(29#)=3.6e8` residues and `phi(31#)=1.0e10`, so a segmented extension is `~5-10 min` at `29#` and `~1 CPU-h` at `31#` — inside the offered budget. No exotic tooling: Python stdlib or a segmented sieve. **Not worth investment if:** a prior-art pass shows the bridging moment estimate is already published (Kuperberg 2025 is the suspect), or `Var/mu^2 -> 1` at `29#`.