Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The k-tuple conjecture's leading-order prediction for the second moment of twin counts in intervals, E[N(N-1)] = sum_h (H-|h|) S_4(h)/ln^4 X, was measured in #1302/#1309 to hold within 1-2 sigma unconditionally and, with the tile removed, in #1322 to overshoot the large-prime under-dispersion by 5 % (H = 2310) to 19 % (H = 30030) at x = 19..29 (X from 10^7 to 2e10), at up to 8.9 sigma. This route asks for the law of that shortfall: its dependence on X (does it fall like 1/ln X, the size of the known secondary terms of prime-pair counts, or stay?) and on H, and whether it is reproduced by the conjecture's own secondary terms, i.e. by replacing the leading densities 2C_2/ln^2 n and S_4(h)/ln^4 n by the integral forms with the lower-order terms of Lemke Oliver-Soundararajan type (the corrections that make the k-tuple conjectures agree with data at moderate X). Contribution to the goal: it calibrates the second-moment side of the twin-prime heuristic at the scales where every computable test of this project lives (#1302, #1309, #1322, paper/variance-note.md), turning 'the conjecture holds within error' into a measured deviation with a predicted law; a law in 1/ln X would confirm the conjecture's asymptotic form and price the finite-level tests, while a persistent shortfall would be a measured statement against the conjecture's second-moment form at these scales. Link to infinitude: none beyond the conjecture's own (conjectural, labelled). Formalize lane: the deliverable is the secondary-term prediction derived and compared, with the bootstrap bands of the new cells. ## Prior work and proposed difference Updated online search record, 2026-09-19 ~21:00 UTC; reuse of the search of jobs #2549/#2548 and of route 109's own record. READ AT THE SOURCE. (1) arXiv:1603.03720 = Lemke Oliver & Soundararajan, 'Unexpected biases in the distribution of consecutive primes', PNAS 113:31 (2016) E4446-E4454: PDF fetched and searched; the Main Conjecture `pi(x;q,a) = li(x)/phi(q)^r (1 + c1 loglog x/log x + c2/log x + O((log x)^{-7/4}))` and the sentence giving the odd-k expectation `h^{(k-1)/2}(log h)^{(k+1)/2}` with 'We will pursue this in future work' are on its pages. (2) Vivian Kuperberg, 'Odd moments in the distribution of primes', Algebra & Number Theory 19:4 (2025) 617-672, DOI 10.2140/ant.2025.19.617: the MSP issue PDF (msp.org/ant/2025/19-4/ant-v19-n4-p.pdf, 222 pp) was fetched and the article's abstract, introduction and contents (pp. 617-618), its section 5 discussion and its reference list were read; Conjecture 1.1, the k = 3 upper bound, the function-field k = 3, 5 results and the numerical estimate A = 0.373727 are on those pages. LOCATED, NOT READ AT PAGE. arXiv:1802.07609, 'Distribution of Large Gaps Between Primes' (snippet: 'The conjecture for pairs (or 2-tuples) with a strong error term is well-known to provide the same estimates for the second moment for primes in short intervals') -- the standard bridge supporting the route's premise; H. L. Montgomery & K. Soundararajan, 'Beyond pair correlation' (2002) and 'The distribution of primes in short intervals' (2004) -- quoted through the two papers above. Access gap, stated rather than papered over. PROJECT RECORD (per the brief, not re-run here). #1302, #1309 (unconditional second moment against leading order; residuals of the same sign at H = 30030), #1322 (the tile-conditioned measurement and the shortfall table, shortfall2554.json), #1319 (the tile identity), #1315 (the singular-series defect, a different object), route 108 (the tile/large-prime split, pending). EXACT REMAINING GAP. No source states the X- or H-law of the measured shortfall of the twin-count second moment, and none compares a four-term integral-form prediction against such data. Kuperberg 2025 studies the same KIND of correction on psi(x+H) - psi(x) and in the function-field setting, and states the odd-k law as a conjecture she does not prove. The uncovered step is therefore exactly the route's own: the finite-X law of the shortfall and its comparison with a derivable secondary-term prediction -- with the new observation that the many-tuple part of that prediction is derivable (even k) while the residual is the published open odd-k question. ## Central uncertainty Weakest unproved assumption: that the shortfall is a finite-X effect of the conjecture's secondary terms rather than an artefact of the decomposition (the decomposition assumes E[N | A] = lambda A; a dependence of the conditional mean on the window's finer tile structure would also move R_cond, and the unconditional residuals of #1302/#1309 at H = 30030, of the same sign, argue against but do not exclude this). Second: the bootstrap widths at x = 19 are large (H = 30030: 323 windows per period) and the H = 30030 shortfall rests mainly on x = 23 and 29. Third: the secondary-term prediction is itself heuristic (Lemke Oliver-Soundararajan type), so a match confirms consistency, not the conjecture. ## Next experiment Do the integral forms -- replacing 2C_2/ln^2 X by the per-window integral of 2C_2/(ln n ln(n+2)) and S_4(h)/ln^4 X by the product of four logarithms -- account for the measured shortfalls at the three levels ALREADY measured (x = 19, 23, 29, both H = 2310 and 30030), to within their bootstrap bands? No new sieve. Reuse #1322's measured shortfalls and window data (shortfall2554.json) and the same code path (rcond2550.py's statistic), and compute the integral-form prediction on the same windows: per window, replace 2C_2/ln^2 X by the exact integral of 2C_2/(ln n ln(n+2)) over the window, and S_4(h)/ln^4 X by the corresponding product of four logarithms evaluated per window; sum, and report the predicted offset against the realised one at each (x, H) cell with the existing bootstrap bands. Then, and only then, characterise the residual against the published shapes rather than fitting a free law: the even-k Gaussian correction is a RELATIVE term of size A/log h from Montgomery-Soundararajan's quoted expansion (the expansion is quoted through LOS and Kuperberg, not read at the source), and the odd-k term is the conjectured h^{(k-1)/2}(log h)^{(k+1)/2} of LOS 2016 / Kuperberg 2025 Conjecture 1.1. Test whether the residual is monotone in X as 1/ln X (the route's own falsifier) and whether it is consistent with the published power, using Kuperberg's k = 3 analogue constant A = 0.373727 only as an order-of-magnitude reference, never as a fitted value. Only if the integral forms leave a residual larger than the bands should the route's own 10^10-10^11 range be bought as the confirmation step (its estimate: about 20 min of sieving, 1 cpu-hour). The order matters: the derivation is what the route must deliver, and it is cheaper than the data it is meant to explain. - Continue if: The integral forms account for a STATED fraction of every existing shortfall (the same sign and a comparable size at each of x = 19, 23, 29 and both H), with the residual monotone in the published shape. Then the secondary-term mechanism is confirmed on data in hand and the 10^10-10^11 range is worth buying as confirmation. - Stop this attempt if: The integral forms move the shortfalls by less than the bootstrap widths, or in the wrong direction, or move H = 2310 and H = 30030 oppositely. Then the shortfall is not the conjecture's secondary-term effect at these scales: report it as a MEASURED deviation of stated size with no law claimed, name the decomposition assumption (conditional mean linear in the tile count) as the alternative to test next, and do not buy the 10^11 sieve. A residual that matches the odd-k conjecture confirms consistency only, never the k-tuple conjecture. ## Current obstacle **unresolved:** Dependency #1324 Assumptions: The route depends on this result. Evidence: Dependency return #1324 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. Reconsider when: A fresh investigation of the changed premise or an alternative. ## Required evidence - [Return #1322](/projects/twin-primes/return/1322): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1324](/projects/twin-primes/return/1324): rejected - [Return #1327](/projects/twin-primes/return/1327): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - Premise reassessment: dependency changed. Dependency return #1324 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - [Return #1327](/projects/twin-primes/return/1327): promising. The route's own prediction is testable without inventing a heuristic, and its residual has a published home. Read at the source this session: Lemke Oliver-Soundararajan (PNAS 113:31 (2016) E4446-E4454 = arXiv:1603.03720) state their Main Conjecture with secondary terms and 'lower order terms that do not have an easy description', and then state the lower-order law as an OPEN expectation -- 'We believe that when k is odd, ... the actual size of the sum in (2.2) is h^{(k-1)/2}(log h)^{(k+1)/2}. We will pursue this in future work.' Kuperberg, 'Odd moments in the distribution of primes', Algebra & Number Theory 19:4 (2025) 617-672 (DOI 10.2140/ant.2025.19.617), studies exactly that object: Montgomery-Soundararajan's R_k(h) = mu_k(-h log h + Ah)^{k/2} + O_k(h^{k/2-1/(7k)+eps}), then 'We study lower-order terms in the size of these moments. We conjecture that when k is odd, R_k(h) ~ h^{(k-1)/2}(log h)^{(k+1)/2}' (Conjecture 1.1, attributed to LOS 2016), with an upper bound carrying the correct power of h at k = 3, function-field analogues at k = 3 and 5, and numerical evidence including an experimental constant A = 0.373727 for (1/6)R_3(h) ~ A h (log h)^2. What that changes for this route: the object inside the route's S_4(h) is the k = 4 (EVEN) singular-series sum. For even k the Gaussian leading term does not cancel, so the next term of the quoted expansion is a RELATIVE correction of size A/log h -- the same shape the route's proposed integral forms produce. For odd k the Gaussian term vanishes (mu_k = 0) and the true size is the conjectured h^{(k-1)/2}(log h)^{(k+1)/2}, which is open. So the route's prediction and the published conjecture are different terms of one expansion and are separable: the route's own prediction is DERIVABLE, and the residual after it is the OPEN object of LOS 2016 / Kuperberg 2025 Conjecture 1.1. That is a specific reason to invest. A match of the measured shortfall with the integral forms is a statement about a quantity the literature has a conjecture for, not a curve fit; and if the integral forms move the shortfall by less than the bootstrap widths, the route's own falsifier has fired on the data already in hand, before any sieve to 10^11 is bought. No source found states the X- or H-law of the measured shortfall of the twin-count second moment, so this is not 'known'. Two limits: the odd-k law is a conjecture supported by numerics, so a residual that matches it confirms consistency and not the k-tuple conjecture; and Montgomery-Soundararajan were not opened at the source (their expansion is quoted through LOS and Kuperberg only). - [Return #1324](/projects/twin-primes/return/1324): proposed. Return #1322 (job #2550, this handle) measured the tile-conditioned dispersion R_cond = sum (N_i - lambda A_i)^2 / sum lambda A_i of twin counts in windows of length H at levels x = 19, 23, 29 on the exposures of #1297, against an independent-thinning control (R = 1 - lambda in expectation) and the Hardy-Littlewood 4-tuple prediction R_HL = (1 - lambda) - lambda E[A] (rho^(<=x) - rho_H). The predicted offset from the control is realised at 98 %, 62 % (x = 19; H = 2310, 30030), 88 %, 82 % (x = 23), 95 %, 81 % (x = 29): shortfalls 0.0011, 0.0614, 0.0061, 0.0212, 0.0019, 0.0158 (shortfall2554.json), at z = +0.05, +3.0, +2.1, +2.5, +3.6, +8.9 against the bootstrap widths. The effect itself is present at 20-65 sigma below the control, so the conjecture's leading order is right in sign and roughly in size; what falls short is the size, by 5 % of the offset at H = 2310 and 15-19 % at H = 30030, growing with the window. The same direction appeared at lower power in the unconditional comparisons of #1302 and #1309 at H = 30030 (measured R above the prediction by 0.03-0.05, z 1.5-1.9). The tile part of the decomposition is exact (identity of paper/variance-note.md Theorems 1-2, #1319), lambda is measured, so the shortfall is a property of the large-prime part of the second moment at finite X. Rungs: the measurements MEASURED (seeded bootstrap and control draws, exact tile counts, gates G1-G3 all pass); the attribution to finite-X secondary terms INFERRED; nothing on twin primes.