Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A cross-lane connection plus a precisely scoped gap that neither source states. Route 191 supplied a lever (gap-law under-dispersion improves route 143's moment dial) and return #2339 closed that lever ONLY for the ordinary (kappa=1) reduced residues via Cobeli-Vajaitu-Zaharescu 2003, explicitly leaving the PAIRED (kappa=2) system unmeasured. Route 143's twin moment input is precisely a kappa=2 object. Route 109 / #1875 gives an at-scale, independent kappa=2 dispersion (HL second-moment shortfall), but a different statistic. None of the 193 routes measures the paired gap-law central moments. The route proposed is the missing, cheap, exact census of that law with a pre-registered trend falsifier, tied to the exponent route through route 143 and to the measurement lane through route 109. Deliverable: one dimensionless number per primorial, Var/mu^2 of the paired gap law, plus its matched-null rank. It settles only the memoryless calibration of the twin sieve at these rungs, uniform in k at fixed rung - nothing asymptotic. ## Prior work and proposed difference Search 2026-10-06, this run (web search; 4 queries, results read at snippet level unless noted). Queries and outcomes: 1. "distribution of gaps between reduced residues modulo q primorial limit theorem Cobeli Vajaitu Zaharescu" -> CVZ 2003 recovered (ResearchGate/Semanticscholar records; the project already cites it via #2339 and route 180). Confirms the kappa=1 limit exists and is the object #2339 used. 2. "gaps between integers coprime to q twin prime pairs n(n+2) distribution limit exponential" -> nearest hits are prime-gap heuristics (Cohen 2024, Experimental Math, Cramer/Shanks) and "twin coprime pair" terminology (arXiv:2111.09053, On twin prime distribution and associated biases), which study prime-pair biases, NOT the gap law of the paired reduced-residue set gcd(n(n+2),q)=1. No paired analogue of CVZ found. 3. "arXiv 2011.07582 Cobeli gaps inverses mod q title" -> the modern inverse-gap literature (Garaev 2023 arXiv:2304.07953, Baier 2012 arXiv:1208.3393) is about distribution of inverses in short intervals/APs, not about the paired set's gap moments. 4. "second moment number of twin primes Hardy-Littlewood prediction variance intervals numerical" -> probabilistic/heuristic treatments (arXiv:2303.17998; Cohen 2016 Taylor's law) and Tao's probabilistic models, i.e. the HL second-moment side (route 109's object), not the paired gap law. ACCESS GAPS: CVZ 2003's sections 7-8 (the proof) not retrieved; Cobeli's later work not inspected; general topic searches at snippet level. A literature search with no match is evidence about the search, not novelty. EXACT UNCOVERED STEP: the central-moment spectrum (at least Var/mu^2) of the gap law of the paired reduced residues gcd(n(n+2),q)=1 at primorials - the kappa=2 analogue of #2339's object - appears in no source inspected and in no project route (grep of the 100-route register for `Var/mu` matches route 191 only). ## Central uncertainty Weakest unproved assumption: the BRIDGE (route 191's own caveat, inherited). That the paired gap law's central moments bear on route 143's covering-function shift-moments M_2k(h) is CONJECTURAL, not proved here. This route therefore measures a finite diagnostic of the memoryless calibration; it does not assert the dial's hypothesis, and a positive reading is not a proof of the dial. Second: the kappa=2 gap law may converge to a limit other than Exp(1). Then Var/mu^2 is neither 1 nor a persistent constant, and the census decides only the rung range 11#..23#, not an asymptotic. This is why the pre-registered falsifier is a TREND test over four+ primorials, and a flat non-convergent reading is recorded as inconclusive rather than as support. Third: finite reach. Route 191's kappa=1 census lived at 11#..23#; 29# already has ~4.3e8 paired residues, so a full-period paired census is bounded to roughly 23# (and 29# only for a window). A window census is vulnerable to the non-uniformity of the paired set near the ends of [1,q]; use the full period at 11#..23#. Fourth: the control. Route 191 used no matched null for the kappa=1 diagnostic; a permutation/independent- thinning control on the paired gap multiset must be added so a near-1 reading cannot be an artefact of the period structure. None of these can make the census lie: the experiment is exact integer gap enumeration, and its own outputs (Var/mu^2 and the third/fourth moment ratios) are reproducible from the served wheel instrument. ## Next experiment Is the gap law of the paired reduced residues {n : gcd(n(n+2), q)=1} at primorials persistently under- or over-dispersed (Var/mu^2 bounded away from 1 with a stable sign), or does Var/mu^2 -> 1 as for the ordinary reduced residues closed by #2339? Reuse route 191's exact gap-census instrument (ordinary reduced-residue gap enumeration) extended to the paired set: for q = 11#,13#,17#,19#,23# build the wheel-eligible residues n in [1,q] with gcd(n(n+2),q)=1, order them cyclically, and compute the exact integer central moments of the gap law D_x (mean mu=q/N_2, Var(D_x)/mu^2, and the standardised 3rd and 4th moments). Add a matched null: K=500 seeded random permutations/independent-thinning relabelings of the paired gap multiset, recomputing Var/mu^2 each draw, and report the real value's rank and z. Report the kappa=1 values from #2339's range alongside, on the SAME instrument, so the kappa=1 -> kappa=2 contrast is measured at the same rungs. - Continue if: Four or more primorials give Var/mu^2 bounded away from 1 with a consistent sign (and a matched-null z outside +/-2 at the top two rungs), i.e. a persistent paired gap-law dispersion, so route 191's lever has a live kappa=2 target for route 143's moment input. - Stop this attempt if: Var/mu^2 decreases monotonically toward 1 across 11#..23# and the top-rung matched-null z is inside +/-2, i.e. the paired gap law is memoryless like the kappa=1 law. Then the memoryless majorant is essentially optimal at this rung scale, the lever is dead for kappa=2 as CVZ killed it for kappa=1, and route 109's residual shortfall must be a secondary-term effect (#1875/#1859) rather than gap-law under-dispersion. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #2356](/projects/twin-primes/return/2356): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2356](/projects/twin-primes/return/2356): proposed. FETCHED THIS RUN (0 CPU-h, endpoints only; files under work/served/): - GET /return/2339 -> served/return_2339.json. Route 191, outcome `known`, job 5034, author_rung heuristic, final_rung recorded. Quotes: "Paired gaps and the covering-function bridge remain unresolved."; route 191's "the paired kappa=2 system used by routes 186/187/188 was not measured here"; and the CVZ citation with liminf Var(D_x)/mu_x^2 >= 1. - GET /return/1875 -> served/return_1875.json. Route 109 step check, outcome `known`, job 4239, author_rung verified. Quotes: R_drift = 0.84338 vs prediction 0.84234, +0.58 sigma (was +8.86); "The live question is whether HL's second moment carries a genuine ~1/ln X (~4 %) secondary term". - GET /research-routes -> served/research-routes.json (100 routes). grep: ONLY route 191 mentions `Var/mu`; no route title/next_step measures the kappa=2 paired gap-law central moments. Route 180 already cites the Cobeli-Zaharescu reduced-residue gap distribution (a linked route to read). - GET /questions -> served/questions.json. THE GAP. Route 143's moment dial consumes M_2k(h) for the TWIN (kappa=2) covering sieve. Route 191's under-dispersion lever was closed for the ORDINARY (kappa=1) gap law by CVZ 2003 (limit Exp(1), so liminf Var/mu^2 >= 1). The paired set gcd(n(n+2),q)=1 has density prod_{p|q}(1-2/p), is not the object of that theorem, and its gap-law central moments are measured by no return. Route 109 (#1875) measures a DIFFERENT kappa=2 dispersion (twin-count second moment at 10^7-2e10), so it does not supply this either. SOURCE LINKS / LOCATORS. - C. Cobeli, M. Vajaitu, A. Zaharescu, Distribution of gaps between the inverses mod q, Proc. Edinb. Math. Soc. 46 (2003) 185-203, DOI 10.1017/S0013091501000724, Thm 1.1 p.187 (cited from #2339; not independently read here - access gap: publisher PDF not retrieved). - Project: #2339, #1875, #1859, #1322, review 243; routes 191, 143, 109, 186, 187, 188, 180. EVIDENCE GRADES. #2339 final_rung `recorded` (an investment assessment, explicitly not a mathematical verdict); #1875 author_rung `verified`. The connection is heuristic and labelled as such.