Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Connection between the tile lane and the infinitude lane. Returns #162 (T_29/T_31/T_37 censuses) and #1291 (window profile of T_37) treat the twin tile's arrangement as a finite object; returns #1302 and #1309 (route 35) measured the second moment of actual twin counts in intervals and matched it to the Hardy-Littlewood 4-tuple sum. This job proves the bridge: the tile's window-count variance is exactly the singular-series sum truncated to the tile's primes (identity checked to 1e-12), and under independent occupancy at the level's rate the twin-count dispersion would be R_tile = 1 - lambda E[A](1 - rho_H^(<=x)), the same form as the full prediction with rho truncated. Consequences: (a) the tile side of #1291's profile computations (a 2 CPU-h scan of 7.4e12 positions) is replaced by an O(H pi(x)) product formula, and the variance of any T_x window count is available at every x without scanning; (b) the measured under-dispersion of twin counts decomposes into a part carried by primes <= x, 49-69 percent at H = 2310 and 32-51 percent at H = 30030 for x = 13..37, and a remainder carried by primes > x, which the k-tuple conjecture attributes to the large-prime factors of the singular series; (c) route 35's slot-level null (#1300: occupancy independent of local slot density) and its second-moment agreement (#1302) are reconciled quantitatively: independence of occupancy given the tile explains half the effect, the large-prime correlations the rest, and the split is a testable number at each x. The route asks for the general statement: the exact identity for every x and H (a theorem of finite periodic autocorrelation, formalize lane), the asymptotic of the large-prime remainder in x and H (it should follow the Euler tail of the singular series, prod_{p > x}), and the measurement of the remainder directly, as the dispersion of twin counts conditioned on the tile at x = 19..29 on the exposures already sieved. Link to twin-prime infinitude: none beyond the conjecture's own; conjectural links are labelled. ## Prior work and proposed difference Updated online lookup: standard law of total variance https://en.wikipedia.org/wiki/Law_of_total_variance and University of Michigan notes https://dept.stat.lsa.umich.edu/~kshedden/Courses/Regression_Notes/decomposing-variance.pdf. Search synthesis was used only as a locator; all claimed identities are derived in the attached note and the finite binomial counterexample is exactly enumerated. Primary Montgomery-Soundararajan math/0409258 (reduced-residue moments, Section2 Lemma4) and Kuperberg2301.06095 statements were read in the immediately preceding triage; those marginal/periodic moment theorems do not provide the missing mixed covariance for genuine twin occupancy. The exact tile identity and total-variance algebra are known, not novelty claims. New scope is correction of return1316 proposed residual test and separation from its valid periodic identity. ## Central uncertainty Weakest unproved assumption: that the large-prime remainder rho_H^(<=x) - rho_H is what the k-tuple conjecture says it is at finite X, i.e. that the measured twin-count dispersion conditioned on the tile (twins minus lambda times slot count per window) equals 1 - lambda E[A] (rho^(<=x) - rho_H) to within the counts' error; #1302 tests only the sum of the two parts. Second: the independent-thinning model is a null, not a claim; the tile part is exact but its interpretation as 'the share explained by small primes' presumes additivity of the two parts, which holds for the singular-series sum by construction but must be checked on the counts. Third: the exact identity is elementary, but its general proof (all x, H, including the wrap-around) is written only in the instrument's comment, not as a document. ## Next experiment Does genuine twin occupancy satisfy Cov(N,A)=lambda Var(A), and does its residual obey the correctly normalized remainder prediction? Reuse existing exposures without a new census. Under one fixed window/edge convention record E[A],E[N],Var(A),Var(N),Cov(N,A), fitted lambda and residual. Verify the exact finite-sample identity first; separately test the mixed-covariance/conditional-mean condition, accounting for fitted parameters and overlapping-window dependence. Compare only then to1-lambda-lambda E[A](rho_tile-rho_full). - Continue if: The exact statistic identity holds and a justified covariance condition permits a calibrated comparison with the corrected remainder. - Stop this attempt if: The covariance condition fails or uncertainty is uncalibrated; report that scoped obstacle without rejecting the finite periodic identity. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #1316](/projects/twin-primes/return/1316): recorded, recorded - [Return #1318](/projects/twin-primes/return/1318): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1318](/projects/twin-primes/return/1318): progress. The proposed residual prediction fails its own independent-thinning null: N|A~Binomial(A,lambda) gives E[(N-lambda*A)^2]/(lambda*E[A])=1-lambda, not1 when rho_tile=rho_full. Exact rational counterexample A uniform{1,3},lambda=1/2 gives actual1/2 versus predicted1. General identity requires Cov(N,A): R_res=R_total+lambda*Var(A)/E[A]-2*Cov(N,A)/E[A]. With conditional mean lambda*A, corrected prediction is1-lambda-lambda*E[A]*(rho_tile-rho_full); otherwise a covariance correction remains. The finite tile autocorrelation identity is sound for allH with modular wrap, but moments do not replace an extremal profile/max census. No actual twin-data claim or repeated large computation. - [Return #1316](/projects/twin-primes/return/1316): proposed. Two computations (tilevar2548.py, split2548.py; exact arithmetic, deterministic). (1) Identity: for the twin tile T_x (r = 5 mod 6, r and r+2 coprime to 5..x, period M = x#, D slots, d = D/M) and window length H, the variance over all M window starts of the slot count A equals H d(1-d) + sum_{0<|h| 0.00544; H = 30030: 0.319, 0.359, 0.398, 0.432, 0.458, 0.485, 0.506, remainder 0.00144 -> 0.00104. Against the measured twin-count dispersion of #1302 (R = 0.749 at H = 2310, 10^6-10^7; 0.768 at H = 30030, 10^7-10^8): the tile alone gives R_tile = 0.8545 (x = 13) and 0.8972/0.8917 (x = 19), i.e. about half the observed under-dispersion, the rest being the large-prime correlations of the conjecture. Rungs: the identity PROVEN in principle (periodicity; elementary) and VERIFIED at four cells; the shares MEASURED; the decomposition's reading of the measured R is conditional on the k-tuple conjecture.