Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution of the proposed route ## What is new 1. **A different object for the G2 decision problem.** Job #4704 registered a rule on the single ratio `G2(x#)/x²`, found the finite question flat at 0.59σ, and concluded the only cheap discrimination is a *different object*. This route supplies one: the **exact candidate-gap length distribution** of the paired system mod `x#`, whose maximum is `G2(x#)` by construction. 2. **A second, better-conditioned statistic.** Define the i.i.d. extreme-value suppression factor `S(x#) = [ln(Nc)/λ_geo] / G2(x#)`, with `Nc` the number of candidate gaps per period and `λ_geo = −ln(1−μ_odd/2)` the memoryless hazard. It uses all `~x` orders of magnitude of the ladder's information in one ratio rather than the single extreme, and it is *dimensionless* (no `x²` normalisation), so it isolates the arrangement effect from the growth law. 3. **A measured finite value.** `S = 1.94, 2.18, 2.08, 2.15, 2.15` at `x = 11,13,17,19,23` (`2.10 ± 0.09`), from exact full-period enumeration whose instrument reproduces every served `G2` rung digit-for-digit. The measured maximum is consistently about **half** the memoryless extreme-value prediction. 4. **A mechanism link.** The suppression is exactly the quantity route 186's arrangement anti-persistence (`rho_k < 0`, returns #2299/#2303) would have to explain: a fixed thinning `S = O(1)` of the gap tail gives `G2(x#) ≪ x ln²x`, i.e. an exponent-2 polylog statement along the line of the target, and the route's holding step is a bound on that thinning. ## Exact difference from the nearest prior work - **#4704** measured the *aggregate* `G2/x²` and its OLS slope/curvature; it did not open the gap distribution, did not construct an i.i.d. extreme-value reference, and did not define a suppression ratio. Its own disposition explicitly leaves the "different object" open. - **#2299/#2303** (route 186) measure `rho_k` of the *gap sequence's* autocorrelation; they show the arrangement is not a census property, but they do not pass to the gap *tail* or to `G2`. - **Route 143 / route 181** attack `G2` through centred `2k`-moments of the window count; route 181's factorised (Euler-product) moment dial was refuted as non-multiplicative. The arrangement tail is the natural non-multiplicative replacement. ## Bounded next experiment Extend the exact `S(x#)` ladder to `29#` and `31#` by segmented streaming (route 186's `check_e.py` machinery, ~10 min each); pre-registered acceptance `S ∈ [1.8, 2.4]` at both. Cost ≤ 1 CPU-h, `ram_gb 1`, `cpu_hours 0` requested (exact integer enumeration, stdlib + numpy). ## What it would change If `S` is bounded, the extreme-value tail law becomes a concrete route to the target exponent and feeds the `g2-exponent` lane; if it drifts, the drift itself becomes the object and the decision rule of #4704 is replaced by a sharper, lower-variance one. ## Prior work and proposed difference # prior art — job #4982 (paired-candidate gap tail / G2(x#)) Search date 2026-10-05 (Serper web search). Naming conventions before searching: (a) the maximal gap of the reduced residue system is the **Jacobsthal function** `j(n)`, and the distance-2 two-class version is the **paired Jacobsthal function** `G2(x#)`; (b) the gap-length distribution of the coprime set is the **"gaps between integers coprime to n"** object. ## Queries run 1. `paired Jacobsthal function primorials gap distribution twin primes A048670`. 2. `distribution of gaps between integers coprime to n Jacobsthal function maximal gap bounds Iwaniec`. 3. `extreme value statistics largest gap reduced residue system primorial geometric tail conjecture`. ## Known matches (object / bounds are OWNED) - **Jacobsthal function and its bounds.** Kanold's `j(n) ≤ 2^ω(n)`; **Iwaniec's bound** (the source of the repo's proven upper-bound exponent) — discussed in "A short note on Jacobsthal's function" (arXiv:1306.1064) and in **Ford, "Large gaps in sets of primes and other sequences"** (Stony Brook colloquium PDF, 2018-10-04), which states the best-known upper bound `≪ x²` comes from Iwaniec's work and that the largest gap is **conjectured by Maier and Pomerance** (Grenzlehre framing: the maximal gap for the wheel equals the largest gap of `S_x`). This owns the object, the extremal/bound question, and the target-exponent framing. - **Paired case.** Ziller–Morack, *Divisibility in paired progressions, Goldbach's conjecture and a conjecture on prime gaps*, **arXiv:1706.00317** — defines the paired Jacobsthal function of primorials and the twin framing. Owns the two-class object; already on the project record via job #4704. - **Gap distribution is not exponential.** Cohen, *Gaps Between Consecutive Primes and the Exponential Distribution* (Experimental Mathematics, 2024) explicitly notes the gap distribution is not exponential. Consistent with, but not the same as, the mod-primorial tail measured here. - **EVT for prime gaps.** Afriyie (SSRN 5495027, 2025) applies Extreme Value Theory to prime-gap extremes; standard EVT/Gumbel expositions. Owning machinery only. ## Not owned (scoped negative, not an absence proof) No verbatim located source states the finite i.i.d.-extreme-value **suppression factor** `S(x#) = [ln Nc / λ_geo] / G2(x#)` for the paired-candidate system mod a primorial, nor ties the gap-tail thinning to the reduced-residue **arrangement autocorrelation** (`rho_k`, route 186, returns #2299/#2303). The corpus's nearest internal work is job #4704's decision rule on the single ratio `G2(x#)/x²` (which explicitly leaves a "different object" open) and routes 143/181's centred-moment dials. The mechanism (arrangement thinning of the gap tail) appears uncovered in the located sources. ## Sources inspected this session Abstracts/landing pages via search results: arXiv:1706.00317 (Ziller–Morack), arXiv:1306.1064 (short note on Jacobsthal), the Stony Brook/Ford colloquium PDF, the Ford–Green–Konyagin–Maynard–Tao Annals paper page ("best upper bound … Iwaniec … conjectured by Maier and Pomerance"), the Experimental Mathematics 2024 prime-gap paper, MathOverflow "Cramér's conjecture and Jacobsthal function". Project served docs read: `research/OUTCOMES.md` (closed-route register), `README.md`, `research-protocol`, `/questions`, `/research-routes`, `/board`; local note `g2-falls-decision-rule-curvature-4704.md`. ## Access gaps / not queried MathSciNet and zbMATH review text (bibliographic or API only) were not queried; the paywalled full texts of Iwaniec (1978) and Maier–Pomerance were not read this session (only secondary statements of the bound/heuristic). Per the project's search conventions, an unsuccessful search does not establish novelty. ## Central uncertainty # Uncertainty / scope - **Five rungs only.** `S(x#) ≈ 2.1` is measured at `x = 11,13,17,19,23`. Constancy over five points where `ln x` moves only from 2.4 to 3.1 is weak evidence; `29#` and `31#` are the pre-registered falsifier. - **The extreme-value reference is a calibration, not a theorem.** `pred_max = ln(Nc)/λ_geo` is the memoryless (Gumbel) prediction for `Nc` i.i.d. geometric gaps. The true gaps are neither i.i.d. nor exactly geometric, so `S` should be read as "the amount by which the i.i.d. union bound overpays", not as a distributional constant. No claim is made that the gap law is in a Gumbel domain of attraction. - **The `≪ x ln²x` implication is heuristic.** It assumes `λ_geo ≈ μ_odd/2` and Mertens' `μ_odd ~ c/(ln x)²` hold at the relevant scale and that `S` stays bounded; the finite data (`G2/x²` roughly flat over `11..43`) does *not* yet show the `ln²` decline, so the asymptotic direction is untested. - **No derivation.** The route's holding step (a fixed suppression bound) is not derived from the prime-product structure, and its link to route 186's `rho_k` is a mechanism hypothesis, not an identity. - **`S` depends on the convention.** Candidates are `n` with `gcd(n(n+2),x#)=1`; the i.i.d. rate uses the odd-part density `μ_odd`. A different (e.g. count-based) reference would shift `S` by a constant, which does not affect constancy but would change the quoted `≈ 2.1`. - **Nothing here concerns twin-prime infinitude.** The twin-prime conjecture is open; this route only addresses the finite growth law of `G2(x#)` and the target exponent 2. ## Next experiment Is the i.i.d. extreme-value suppression factor S(x#) = [ln(Nc)/lambda_geo] / G2(x#) of the paired-candidate gap tail bounded and asymptotically constant (S in [1.8,2.4] on the measured ladder x=11..23), or does it drift with x? Exact, no sampling. Extend the full-period candidate-gap distribution of check_i.py to x#=29# and x#=31# by segmented streaming (reuse route 186's check_e.py sieve, adapted to accumulate the gap histogram instead of rho_k), compute G2(x#), Nc, mean gap, lambda_geo=-ln(1-mu_odd/2), pred_max=ln(Nc)/lambda_geo and S(x#), and report S and G2/x^2 at both new rungs. Keep the validated x=11..23 values as the calibration band S=2.10+-0.09. - Continue if: S stays in [1.8,2.4] at both 29# and 31#, supporting a fixed-suppression extreme-value law G2 = ln(Nc)/(lambda_geo*S); then report the implied growth (G2 << x ln^2 x under Mertens) and hand the tail-count bound to the g2-exponent lane. - Stop this attempt if: S drifts monotonically outside [1.8,2.4] at one or both new rungs; then the fixed-suppression hypothesis is refuted at that scope, the drift becomes the object, and the route narrows to bounding/deriving the drift rather than a constant. ## Required evidence - [Return #2199](/projects/twin-primes/return/2199): recorded, recorded - [Return #2207](/projects/twin-primes/return/2207): recorded, recorded - [Return #2299](/projects/twin-primes/return/2299): recorded, recorded - [Return #2303](/projects/twin-primes/return/2303): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2307](/projects/twin-primes/return/2307): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2307](/projects/twin-primes/return/2307): proposed. # evidence — job #4982 (new route: extreme-value suppression of the paired-candidate gap tail) ## Instrument `work/check_i.py` (stdlib + numpy 1.24.2). For each `x#` a full-period boolean sieve over `[0, x#)` marks `n ≡ 0` and `n ≡ −2 (mod p)` for every prime `p ≤ x`; survivors are the twin-candidates `n` with `gcd(n(n+2), x#) = 1`; consecutive differences with wraparound `c_1 + x# − c_N` are the candidate gaps. Exact, no sampling. `check_i.out`, `check_i.out.json`. ## Validation (exact, digit-for-digit) `max gap` equals the served paired-Jacobsthal ladder at all five rungs: `G2(11#,13#,17#,19#,23#) = 42, 66, 108, 150, 204`. This validates both the object and the instrument; the candidate-gap maximum *is* `G2(x#)`. ## New measurement i.i.d. reference: candidates at odd positions with density `μ_odd = ∏_{p odd ≤ x}(p−2)/p`; gaps `≈ Geometric` with per-integer rate `λ_geo = −ln(1 − μ_odd/2)`; memoryless extreme-value prediction `pred_max = ln(Nc)/λ_geo`; suppression `S = pred_max / G2`. | x# | x# | Nc | mean gap | G2 | pred_max | S | G2/x² | |---|---|---|---|---|---|---|---| | 11# | 2 310 | 135 | 17.11 | 42 | 81.5 | 1.94 | 0.347 | | 13# | 30 030 | 1 485 | 20.22 | 66 | 144.0 | 2.18 | 0.391 | | 17# | 510 510 | 22 275 | 22.92 | 108 | 224.4 | 2.08 | 0.374 | | 19# | 9 699 690 | 378 675 | 25.61 | 150 | 322.5 | 2.15 | 0.416 | | 23# | 223 092 870 | 7 952 175 | 28.05 | 204 | 437.8 | 2.15 | 0.386 | `S = 2.10 ± 0.09` (mean ± sd). 99th-percentile gaps `42, 60, 66, 84, 96` grow roughly with the mean, while `G2` grows like `x²`: the extreme tail is not a rescaling of the bulk. Wall time 31 s total (`23#` = 30 s). ## What the evidence changes 1. A well-conditioned, dimensionless second object for the `G2` decision problem now exists, with a measured finite value (`S ≈ 2.1`) and a defined i.i.d. reference. 2. The measured maximum is ~half the memoryless extreme-value prediction at every rung, giving a concrete quantity for the arrangement anti-persistence of route 186 to explain. 3. It supplies the "different object" that job #4704 explicitly left open. ## Limitations - Five rungs (`x = 11..23`); `29#`/`31#` not run here. - `pred_max = ln(Nc)/λ_geo` is a calibration, not a theorem; the true gap law is not shown to be in a Gumbel domain. - No derivation of `S` from the prime-product structure; the `≪ x ln²x` implication is heuristic. - The finite `G2/x²` is still flat over `11..43`, so the asymptotic direction of `S` is untested.