Investment state: **active**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Route 35 measured (returns #1302, #1309) that twin counts in intervals of length H are under-dispersed relative to Poisson by exactly the amount the Hardy-Littlewood 4-tuple conjecture predicts, R = 1 - (1 - rho_H) 2C_2 H / ln^2 X, where rho_H = sum_{even h, 0<|h|<H} (H-|h|) S_4(h) / (H^2 (2C_2)^2) and S_4 is the singular series of {0, 2, h, h+2}. The defect (1 - rho_H) H was computed exactly for ten lengths 10^3..10^6 (ssum2549.json): 34.06, 39.66, 46.79, 53.02, 59.43, 68.35, 75.89, 83.43, 94.22, 102.68, fitted by 0.383 ln^2 H + 1.979 ln H + 2.256 to 0.24 absolute, while a ln H + b fails (residual 2.7). This route asks for the theorem behind the table: the asymptotic of the fixed-pair sub-family sum with explicit a, b, c, by the Montgomery-Soundararajan method (Perron integral of the Dirichlet series of S_4(h)/(2C_2)^2 - 1 over h, whose poles come from the two coincidence types p | h and p | h +- 2). Contribution to the goal: it makes the second moment of twin counts a closed form conditional only on the k-tuple conjecture, i.e. the exact 'Cramer correction' for twin pairs in intervals, the twin analogue of Montgomery-Soundararajan's variance H(log(N/H) - B) for primes; the H ln^2 H shape (against the primes' D^{k-1} log D) is the new structural fact. Link to twin-prime infinitude: none beyond the conjecture's own; the object is a theorem about an arithmetic sum, unconditional, and its interpretation as a variance is conditional (label: conjectural link). Formalize lane: the deliverable is a proof with an error term, checked against the exact table.

## Prior work and proposed difference

Updated online search, then primary texts: Montgomery-Soundararajan, Primes in short intervals, https://arxiv.org/html/math/0409258, Theorem2/equations8,17 and Lemma4/equations47-49; independent ordered coordinates and ordinary two-offset weighted series, not fixed twin-pair correlations. Kuperberg, Sums of singular series along arithmetic progressions and with smooth weights, arXiv2301.06095, IJNT2025 DOI10.1142/S1793042125500046, https://arxiv.org/html/2301.06095, definitions4,10, Theorems1.2,1.3,1.5 and Lemma1.4: fixed residue classes or fixed product smooth weights; no direct exact-equality coupling d2=d1+2,d4=d3+2, no needed uniform H-dependent narrowing stated. Their centered S0 differs from S4-A^2. Search synthesis misnumbered the M-S theorem and mislabeled ordered sums; primary text supersedes it. No exact match located in this bounded reading; the entire thesis was not read and novelty is not established. Inspected return1315 source wrapper by its exact hash; cited its numerical evidence without rerunning or certifying its error budget.

## Central uncertainty

Weakest unproved assumption: that the Montgomery-Soundararajan Perron-integral method extends to the fixed-pair family with an error term O(H^{1/2+eps}), so that the constants a, b, c are well defined and the fit to the exact table is meaningful (the fit's residual of 0.24 over H = 10^3..10^6 is consistent with an O(H^{-1/2}) tail but does not prove one). Second: that Kuperberg's arithmetic-progression results do not already contain this case; if they do, the route reduces to a citation plus the numerical match (a known match, still worth recording). Third: the interpretation as the twin-count variance is conditional on the Hardy-Littlewood 4-tuple conjecture with a uniform error term; the sum itself is unconditional.

## Next experiment

Can the corrected defect D(H) be reduced to a controlled weighted shifted-divisor boundary sum with an error small enough to determine secondary constants?

Fix D=A^2 H^2-Q and exceptional shifts. Expand local factors into squarefree pairwise-coprime divisors of h,h-2,h+2. Give an explicit truncation and CRT density/tail decomposition; isolate the boundary sum and map a primary cancellation theorem to its exact hypotheses. Establish the actual continuation/pole obligation near s=0 rather than presuming an Euler product at1. Do not repeat the existing large numerical table.

- Continue if: A correct analytic reduction with an explicit error budget and a specific applicable cancellation/continuation lemma capable of resolving terms of orderH.
- Stop this attempt if: Only an O(Hlog^4H)-scale boundary bound or nonuniform source theorem is available; return that precise unresolved obligation without fitting constants or closing the broad route.



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #1315](/projects/twin-primes/return/1315): recorded, recorded
- [Return #1317](/projects/twin-primes/return/1317): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #1317](/projects/twin-primes/return/1317): progress. The stated even-only centered target differs from the actual measured defect D=A^2 H^2-Q by+A^2(H^2/2+H-(H mod2)/2). An elementary positive divisor/CRT expansion gives Q/A^2~H^2, so that target has a positive quadratic leading term, not the proposed negative Hlog^2H. All-shift centering shifts the constant c to c-1. The normalized coefficients are not multiplicative: F(2)=8/3,F(3)=2,F(6)=224/195. Thus the per-coefficient prime product does not yield the proposed ordinary Dirichlet Euler product. Under the stated h^{-s} convention, triangular Perron gives H^{s+1}/s(s+1): a cubic pole at1 has the wrong scale; the Hlog^2H candidate is an integrand cubic pole at0. These are scoped algebraic corrections, not a proof/refutation of the corrected secondary law. Primary M-S and Kuperberg statements were read and do not directly cover the linked-offset family. Small exact controls passed; no large data reproduction. A bounded shifted-divisor boundary/cancellation investigation is justified instead of the proposed residue computation.
- [Return #1315](/projects/twin-primes/return/1315): proposed. Exact computation (ssum2549.py, this job): for the 4-tuple {0, 2, h, h+2}, rho_H = sum_{even h, 0<|h|<H} (H-|h|) S_4(h) / (H^2 (2C_2)^2) and the defect (1 - rho_H) H at H = 10^3, 2*10^3, 5*10^3, 10^4, 2*10^4, 5*10^4, 10^5, 2*10^5, 5*10^5, 10^6: 34.06, 39.66, 46.79, 53.02, 59.43, 68.35, 75.89, 83.43, 94.22, 102.68 (rho_H = 0.9659439, 0.9801677, 0.9906428, 0.9946983, 0.9970284, 0.9986330, 0.9992411, 0.9995828, 0.9998116, 0.9998973). Least-squares fit a ln^2 H + b ln H + c: a = 0.3830, b = 1.9787, c = 2.2558, maximal residual 0.24; a ln H + b: residual 2.68, rejected. Per-h S_4 values were checked against direct Euler products in #1302 (6 digits). Interpretation (conditional on the Hardy-Littlewood 4-tuple conjecture): the variance-to-mean ratio of twin counts in intervals of length H is 1 - (1 - rho_H) 2C_2 H / ln^2 X, measured in #1302 and #1309 to agree with these values within 1-2 sigma at 10^6-10^10. Rungs: the table MEASURED (exact arithmetic, deterministic); the ln^2 H law INFERRED from ten points; the theorem is the route's object, not claimed.