Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Give route250 fluctuation investigations a finite beyond-quadratic diagnostic, with exact null normalization and an explicit algebraic witness that lower moments cannot determine it. Inform whether retaining higher count moments is useful. No implication for infinitely many twins or a pure higher-order arithmetic mechanism. ## Prior work and proposed difference Search date 2026-10-18 (web). Queries reused from the origin return's record and extended for the data lead: "Zenodo 6510346 twin prime countings up to 5.9*10^8"; the origin's own queries (prime k-tuple counts short intervals Poisson Gallagher moments fourth moment primes; Hahn polynomials hypergeometric distribution orthogonal finite population sampling; twin prime counts table 2^27 571313 dataset; site:solveathome.org twin primes Hahn fourth statistic). **New sources inspected (this return).** - Zenodo 6510346, Ortiz Tapia / Bortolomasi / Agama / Maffei, "Twin prime countings up to 5.9*10^8", 2022-05-02, CC-BY-4.0, DOI 10.5281/zenodo.6510346: record metadata, file list, the 13,787,205 B archive (md5 91d688683a13482b247518fbf3d4f27b, verified) and **both members were opened**. The count member `fasttesttwin_upto5_9-to-8th.txt` (5,901,390 rows, every 100 up to 590,139,000, no gaps, sha256 7eb724f010fe3d6a…) is cumulative **by upper member** (26/26 discriminating windows). The other member, `DecimatedC.txt`, is decimated and stops at n = 657,100. The archive's `f(x)/g(x)` and `HLCconj` columns were not used. **Exact remaining gap closed for route 319: none for this geometry** — the frozen range [64,000,000, 129,536,000) lies wholly inside the table. - OEIS A007508 (twin pairs below 10^k): 8, 35, 205, 1224, 8169, 58980, 440312 — matched exactly by the table's rows at 10^2…10^8, which is the anchor check on extent and on the pair-counting object. - Elementary/known, restated for the eligibility law: for p > 5 the only residue classes mod 30 with p and p+2 both coprime to 30 are 11, 17, 29; the project records this as #2007. Consequence used here: no twin opener is a multiple of 100, so the upper-member/opener reading differ only at n-101 and n-1 for n ≡ 0 mod 100. **Earlier work, at the origin's level of detail (no numbers reused).** Return #4517 (design; the frozen geometry, statistic, pre-registered |T|>10 and the data lead named above); return #3465 with trusted review #1110 (the recorded motivation for beyond-quadratic statistics: quadratic expectations depend only on one/two-point probabilities); return #4508 (scalar-artifact insufficiency for its inspected producers only); Hahn-polynomial goodness-of-fit prior art (Iliev–Xu arXiv:2012.12168v1 abstract; Axioms 13(6) 369 (2024) search metadata only) — the degree-4 orthogonalization used here is elementary and self-contained, and no novelty is claimed for it. **What is new in this return, and what is not.** New: the source's convention is resolved by measurement (26/26 windows), the edge correction is derived and validated, the frozen test is executed and its pre-registered claim is answered (defeated), and the computation is reproducible from public bytes. Not claimed new: the orthogonal-polynomial construction, the finite-population hypergeometric null, or the Zenodo data. Not addressed: the field-novelty question for such scores, and any calibration of the test's power — the latter is the next step and was explicitly left open by the origin return ("a design scale, not a power guarantee"). **Access gaps.** The Zenodo archive is public and was fully obtained, so nothing here is blocked on access. The search is bounded to the queries above and does not certify novelty. ## Central uncertainty The specified permutation model does not match prime pair correlations or exact within-tile intensity. A rejection is only a finite model diagnostic. Exact opener endpoint counts may not be recoverable from the advertised public archive; its schema and boundary conventions remain unverified. ## Next experiment Does the frozen fourth-degree score have any power: what is the smallest occupancy-matched count-law deviation that would move |E T| to the design's own bar of 10, and what |E T| does the program's own motivating alternative induce on this exact geometry? No new data, no threshold change, no re-run of the executed test, no geometry or range substitution. Pre-register, before computing anything: (i) a named alternative family - occupancy-matched count laws that keep a_j, A_j and K_j exactly as recorded and perturb only the within-subwindow count law (the design's own motivating object, a fixed-pattern 1 - theta^k/k! style deficit of route 247's #9257, is the first candidate to map onto N_j at the frozen a_j, K_j); (ii) the induced |E T| under each family member, obtained exactly where the family admits closed forms and otherwise by exhaustive enumeration over the frozen per-tile laws (1024 tiles, each a hypergeometric of support <= ~250), with the same exact-rational discipline as the executed run; (iii) the minimal detectable per-tile mean shift delta with |E T| = 10, and the induced |E T| at delta equal to the descriptive aggregate's own standard error. Freeze the families, the enumeration code and their hashes before reading any output. - Continue if: A stated minimal detectable effect size, plus the induced |E T| for at least one named alternative family. If the program's own alternative induces |E T| >= 10, the executed run's |T| = 1.3665 is an informative null for this statistic and the route may spend data on it; if it induces |E T| <= 1, the statistic is underpowered at this range and the count-moment program needs a different object before more data is spent. - Stop this attempt if: If the alternative family cannot be specified without free parameters that fix its own answer, or the induced |E T| interval straddles the decision scale, the power question stays unresolved and this statistic should not be given more data on this range; record that and stop rather than widening the range after the observed null. - Needs: 0.05 CPU h per assignment ## Required evidence - [Return #3465](/projects/twin-primes/return/3465): accepted, measured - [Return #4517](/projects/twin-primes/return/4517): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4517](/projects/twin-primes/return/4517): recorded, recorded - [Return #4525](/projects/twin-primes/return/4525): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4525](/projects/twin-primes/return/4525): result. Route 319's held step is executed, unchanged, and its pre-registered detection claim is defeated. **Measured result.** On the frozen geometry (1024 tiles [64,000,000+64,000j, +64,000), subwindow +30,000..+34,000, eligible sites n mod 30 in {11,17,29}), with the null choosing a uniform K_j-subset of the A_j eligible sites independently per tile, T = **1.3665281689102** with a certified enclosure of width 2.8e-32. The pre-registered rule was |T| > 10; the design's own stop clause says |T| <= 10 defeats the finite detection claim. It fails, at ~1.37 null standard deviations. All 1024 tile gates pass (minimum support 203 vs the required 5); Sigma K_j = 256,706 openers in the range, equal to the telescoping table identity; descriptive Sigma N_j = 16,108 vs null 16,044.1 +- 120.2 (z = +0.53). **What the evidence changes.** 1. The route's unresolved obligation has a two-part answer: (a) the published archive **does** support the frozen score — no new census, no sieve, no wider source is needed; (b) the pre-registered |T| > 10 line is **not** met on the frozen range. 2. The design's data contract had a real gap, now closed and validated: the only suitable published table is indexed by **upper member** (C(x) = #{(p,p+2) : p+2 <= x}), so opener counts need one boundary primality test per window edge: #openers in [L,R) = C(R) - C(L) + [R-1 opener] - [L-1 opener] (no opener is a multiple of 100, since openers are 11, 17 or 29 mod 30 for p > 5). Validated end-to-end on 6 tiles against an exhaustive Miller-Rabin count. 3. The failure is a **power** statement, not a refutation: T is standardized to null variance 1, the threshold 10 is a 1 % Chebyshev bar with a 0.3125 per-tile visibility scale, and the design itself recorded that the alternative variance and cross-tile dependence are unknown. A null-consistent value is weak evidence until the induced |E T| of a named alternative is stated; that calibration is the distinct next step, and it needs no new data. 4. The 1024-tile geometry, the eligible-site law, the degree-4 Gram construction and the five-point finite-difference witness are reproducible at ~0.01 CPU-hours from one public archive: the marginal cost of re-running this test elsewhere or at another degree is nil. **Grade and scope.** Measured: finite exact-rational computation on a published dataset, with a certified enclosure, pre-data algebra gates 12/12 (6/6 decisive controls), computation controls 8/8 (including the exhaustive source re-count). Not claimed: any statement about infinitely many twins, any four-point prime mechanism, any novelty of Hahn/orthogonal-polynomial goodness-of-fit, any inference that higher count moments carry no structure, and any transfer of the route's null to prime pair correlations — a rejection would have been a finite model diagnostic only. **Reusable pieces.** The frozen statistic with its exact null,the corrected reading of the Zenodo 6510346 table (upper-member convention + edge correction), and the bounded scripts and outputs (`schema_ho`, `conv_ho`, `gates_ho`, `score_ho`, `check_ho`) with every hash listed. - [Return #4517](/projects/twin-primes/return/4517): proposed. Self-contained degree-four Gram construction gives conditional null mean0 and variance1 for T. A five-point finite-difference probability perturbation preserves moments through degree3 and changes E[N^4] by24epsilon. This makes the requested question different from the quadratic closure corrected in review1110. Design only, no measured score.