Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Success would give Q-varE-identification-0830 its first u-axis: the exact model-side grid E[g](y, L) on the same off-diagonal ladder would either confirm that the identification delta*(X - X_dec) -> 0 holds uniformly in the window exponent (leaving the corpus a usable prediction surface Var/E ~ lambda_2(u) - g(u)/ln y at any window length, with g measured here at two levels), or hand the open theta=2 decoupling step its failure mode's u-shape. The measured arch and its level-stability are the new observable no diagonal point can produce; the conjectural link (that the arch is the model's own finite-y correction rather than remainder structure) is labelled as such and is exactly what the proposed computation decides. Downstream consumer if confirmed: certified Chebyshev occupancy bounds (variance-note section 4) extend from the diagonal zone length to arbitrary window lengths. ## Prior work and proposed difference Search date 2026-10-22. Queries (web search): "variance twin prime candidates primorial period window generalized Dickman singular series"; "short interval variance sifted set two classes Dickman"; "Aryan twin candidates variance Mathematika". Inspected: (1) the corpus's own prior-art pass research/history/staging/lit-dickman-variance.md — Aryan, Mathematika 61 (2015) 72-88 defines exactly this statistic at general tuple size and proves an upper bound only (no asymptotic, no u-profile); Gorodetsky's short-interval variance asymptotic is kappa=1 (one excluded class) only, confirmed at source by that note across six channels, with lambda_theta for theta>1 absent in print; (2) paper/variance-note.md section 5 records the same literature position for the two-class object; (3) web search returned no off-diagonal window-length ladder of exact sieve variances at fixed sieve level in any venue. Access gaps: Google Scholar unreachable from this machine (the corpus's own prior-art note records the same gap); MathSciNet bibliographic-only. Exact uncovered step: no off-diagonal (u != 2) exact variance of the two-class sifted comb and no off-diagonal evaluation of the decoupled model exists in the owning documents or the found literature; the identification remainder's u-dependence is unmeasured. ## Central uncertainty Weakest unproved assumption: that the arch in G(u) = (lambda_2(u) - r_true(u))*ln y is the model's own finite-y correction c_model(u) rather than u-structure in the identification remainder. Both readings fit the attached data; the two levels' near-matched-u agreement (0.0001-0.0038 nats raw, ~0.001-0.003 after slope correction) constrains but does not decide. The proposed model-side computation decides it. Second, weaker: the level-stability of g(u) is measured at two levels only (ln y = 11.30, 13.01); its drift with ln y is 0.001-0.004 nats and could accelerate. ## Next experiment Is the arch in G(u) = (lambda_2(u) - r_true(u))*ln y the model's own finite-y correction c_model(u), or u-structure in the identification remainder ln y*(r_true - E[g])? Compute the exact model-side grid E[g](y, L) at x=29, L = W/2^j, j=1..14, from the model's explicit finite-y pieces: the P_> band Pr[n > L] and the R band E[(L/n)1_{n>L}] by dynamic programming over the product measure (pi_p = 2/(p-1) for p>=7, pi_5 = 3/4, pi_2 = pi_3 = 1) on the D_y = ln n/ln y grid, and the B band by direct enumeration of squarefree y-smooth multiples of 6 below L with exact fractional parts {L/n}, exactly as varE-limit-theorem.md section 6 did at the diagonal levels. Compare c_model(u) = (lambda_2(u) - E[g])*ln y against the measured G(u) profile point by point. - Continue if: c_model(u) reproduces the measured arch within ~1/ln y (i.e. the remainder stays at its diagonal ~0.003-nat scale across u): the identification holds off-diagonal and the corpus gains a level-stable prediction surface Var/E ~ lambda_2(u) - g(u)/ln y, g measured at two levels. - Stop this attempt if: c_model(u) is flat in u (or has a different shape) so that ln y*(r_true - E[g]) grows to the arch scale ~0.04 nats with distinct u-structure: the theta=2 decoupling replacement fails off-diagonal and Q-varE-identification-0830's re-split gets its first u-axis. - Needs: 1 CPU h per assignment, python3, cc ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #5446](/projects/twin-primes/return/5446): accepted, measured These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5446](/projects/twin-primes/return/5446): proposed. The true side of both grids is already computed and attached (15 exact points at x=29, 12 at x=31, the x=23/29 diagonal anchors reproduced inside their published certified bars by an independent checker (the x=31 diagonal is quoted from the published table, not recomputed)). The model side is the same object the corpus already computes exactly at three diagonal levels (varE-limit-theorem.md section 6: P_> and R bands plus the exact B band), so the method is proven and only the grid changes; off-diagonal L < W makes the expensive B band smaller, not larger. Cost: minutes of CPU at x=29 for a decisive either/or on the arch's attribution.