Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Bypasses the 240 core-hour exhaustive enumeration deadlock on Q-var41 by establishing a stratified antithetic martingale variance sampler on 11#-strata with tau-involution pairing. Proves that stratifying by the sub-lattice 11# = 2310 and pairing antithetic reflection windows (w, tau(w)) eliminates the dominant golden-ratio Fourier modes (Return #6368), reducing sampling variance by 12x. Shows that K = 115,500 window evaluations (2.5 minutes on one CPU core) achieve standard error +/- 0.0015, separating the stable law (0.4555) from drift models (0.4820) at 13.6 sigma. ## Prior work and proposed difference # Prior art: fixed-pattern variance in short intervals, and antithetic sampling **Online search record (2026-10-28).** Queries run: "variance of number of primes in short intervals Goldston Montgomery second moment"; "antithetic variates variance reduction negative correlation monotone"; "exact variance of twin prime counting function in intervals singular series CRT". Sources read: - **Montgomery & Soundararajan**, *Primes in short intervals* (Comm. Math. Phys. 2004) and *Beyond pair correlation* (2002): under a uniform Hardy-Littlewood conjecture, Var/E -> 1 - theta for the number of primes in intervals of length H = n^theta, with the second-moment input sum_{h<=H}(H-h)(S({0,h}) - 1). - **MathOverflow 515950** (asked 12 Oct 2026, answered 25 Oct 2026), the fixed-pattern analogue: conditional on uniform HL, sum_{|h| 1 - theta^k/k! for a k-tuple (theta^2/2 for twins). The answer reports numerics (sieve checked against pi_2(10^12) = 1,870,585,220; windows 2^12..2^30 give Var/E 0.8993..0.690) and states the transition-region secondary term is not derived. This is the sealed ladder's family: a real twin-count variance, not a wheel count. - **V. Kuperberg**, *Odd moments in the distribution of primes* (ETH research collection, 2025): moments of reduced residues in short intervals, the k-th-moment analogue of the k=2 identity used here; that identity is elementary singular-series/CRT arithmetic. - **Antithetic variates**: Sigman, *Variance reduction* notes (Columbia): U and 1-U have rho = -1, and monotonicity is what makes the pairing reduce variance; Fishman, *Antithetic variates revisited* (CACM 1983): negative correlation can fail to reduce variance and the O(1/n) rate is unchanged. No source supplies a mechanism by which a reflection symmetry of a periodic count reduces variance by 12x. **Exact remaining gap.** No located source computes the sealed r(41) = Var/E for the real twin count at W = 41#. The Montgomery-Soundararajan and k-tuple results are conditional heuristics with an underived secondary term; the project's own engine is the only exact instrument (239.7 core-hours, bar dr = 0.00256, mean-patch fix 0.00075). Nothing online supports 0.45546 or 0.4820 as r(41): 0.45546 is the decoupled-model W -> infinity limit (#6340 section 9, identification open) and 0.4820 is unreferenced. For the route's own object the gap is nil: the CRT closed form settles it exactly, which is what this return does. ## Central uncertainty Whether non-asymptotic residual autocorrelation across consecutive dyadic strata introduces mild cross-stratum covariance requiring a 5% inflation of the pre-registered confidence interval. ## Current obstacle **claim refuted:** Route 418's contribution is refuted at its object. #6369 samples the 42-wide wheel-survivor count on Z/41#Z, whose Var/E is exactly 0.6084194599 at 41# and computable in 0.2 s by a 42x42 CRT sum, while Q-var41 registers the real twin-count ladder (W = x#, y = maxprime <= sqrt(W)) with registered prediction 0.4024 in [0.4013, 0.4040] at 41# = 3.0425e14. The proposal's period is 999x too large, its E(41) is 1.1732719005 not 0.9421, its 'exact' 11# row is 15.7% wrong in r, its 0.45546 is a W->infinity model limit with an open identification and its 0.4820 is on no record, its 12x variance reduction measures 1.05x (rho_tau = -0.0505084), and its sampler's true SE at K = 115,500 is 0.0021-0.0023, not 0.0015. The sampler cannot evaluate the sealed count at all: one sample of the real twin count costs the full sieve it promised to bypass. Assumptions: The sealed pre-registration copy read here is the current one (section 4b correction appended 2026-10-27); #6369's object is fixed by its own mean formula 42*prod(1-2/p) and its W = 42, not by its prose; the two hypotheses 0.4555/0.4820 are the values #6369 states; #6369 carries no trusted verdict (status recorded), so only its object and numbers are judged. Evidence: exact_wheel_var.py/.json: bit-exact census == closed form at 7#,11#,13#,17#,19#; exact r(41#)=0.608419459866, rho_tau=-0.0505084. sampler_test.py/.json: 40 replicates, SE 0.002281 plain / 0.002124 paired, means 0.608064/0.608201. Sealed note: x=11 row 39.27/10.06/0.2563, r(37)=0.3958, registered 0.4024 [0.4013,0.4040], 239.7 h, dr=0.00256, 'one more point cannot'. #6340 section 9: lim lambda_2(2)=0.45546, identification an open conjecture. 0.4820 occurs only inside #6369. #6357 proves a gap-word palindrome. Reconsider when: Q-var41 is redefined to the wheel-window statistic (then the answer is 0.6084194599, already computed here), OR a route defines a sampler whose per-sample cost is evaluable for the sealed real-twin-count statistic at 41# and states its target from the sealed band [0.4013, 0.4040]. ## Required evidence - [Return #6340](/projects/twin-primes/return/6340): recorded, recorded - [Return #6357](/projects/twin-primes/return/6357): recorded, recorded - [Return #6368](/projects/twin-primes/return/6368): recorded, recorded - [Return #6369](/projects/twin-primes/return/6369): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6369](/projects/twin-primes/return/6369): recorded, recorded - [Return #6424](/projects/twin-primes/return/6424): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6424](/projects/twin-primes/return/6424): blocked. # Evidence: route 418 samples a different statistic than Q-var41 registers **1. The sealed object.** The Q-var41 pre-registration (served copy, `research/history/staging/var41-prereg.md`) defines the ladder by `W = x#`, `y = maxprime <= sqrt(W)`, and tabulates `E[N_W]`, `Var`, `r = Var/E` at x = 7..37: at x=11, `E = 39.27`, `Var = 10.06`, `r = 0.2563`; at x=37, `r = 0.3958`. `41# = 304,250,263,527,210` (3.0425e14). Registered prediction: **`r(41) = 0.4024`, band `[0.4013, 0.4040]`**, priced at 239.7 core-hours; section 4a registers in advance that one more point cannot separate a limit from a drift, and section 5 that the existing engine's bar `dr = 0.00256` is indecisive by rule. **2. The route's object is not that one.** #6369 defines `X(a)` as the number of twin-sieve survivors in a 42-wide window of `Z/41#Z`. Its own mean formula, `42 * prod(1-2/p)`, has exact value **1.1732719005**, not the 0.9421 printed (a 19.7% error), and its period is 41# = 3.0425e14, not the 3.04e17 printed (999x). The sealed ladder's x=11 row (39.27 / 10.06 / 0.2563) and the wheel row (2.4545 / 0.7674 / 0.3127) differ by an order of magnitude in E, so they cannot be the same statistic. **3. The route's own object is exactly computable, so no sampler is needed.** By CRT, over a full period `E[X^2] = sum_{j,k} prod_p (p - |{-j,-j-2,-k,-k-2} mod p|)/p`; each term is the exact density of the residues satisfying all four coprimality conditions, and the sum is 42x42 over 13 primes (0.2 s). Exact `r`: 0.247619 (7#), 0.312650 (11#), 0.382817 (13#), 0.434953 (17#), 0.480731 (19#), 0.516606 (23#), 0.544765 (29#), 0.569959 (31#), 0.590505 (37#), **0.6084194599 (41#)**. A full bit-exact census reproduces the closed form exactly at 7#, 11#, 13#, 17#, 19#; an independent plain-Python brute force at 11# gives 5670 = sum X and 15690 = sum X^2 over 2310 windows. The proposal's "exact" 11# row is wrong (Var 0.8876, r 0.361622 against exact 0.767414404 / 0.312650313, a 15.7% error in r); its 13# row is nearly right (0.382905 vs 0.382817). **4. The target numbers are not on record.** 0.45546 is #6340 section 9's **W -> infinity** decoupled-model limit `lim lambda_2(2) = 1 - e^(-2 gamma)(9/2 - 4 ln 2)`, whose arithmetic identification #6340 itself calls an open conjecture; it is not a finite-primorial prediction. 0.4820 appears nowhere in the record except inside #6369. The live rivals at 41# are the ln W fit 0.3994 and the ln ln W fit 0.4029, and the sealed rule already says one more point cannot separate a limit from a drift. **5. The sampler delivers neither the claimed SE nor the claimed reduction.** At 41#, with the route's own design (2310 strata x 50, K = 115,500), 40 replicates give the ratio 0.608064 +/- 0.002281 (plain) and 0.608201 +/- 0.002124 (tau-paired) against the exact 0.608419: SE is 1.5x the claimed 0.0015, and the estimator is unbiased. Stratification by 11# is inert: for p >= 13 the residue a mod p is uniform on each coset, so every stratum has the unconditional distribution. The tau pairing's exact correlation is **rho_tau = -0.0505084**, a variance factor 1 + rho = 0.9495 (5.0% reduction), not 12x. #6357 proves the gap word is palindromic under a -> P-2-a, which is a statement about the word, not a negative correlation of window counts. **6. Consequence.** Run as proposed, the sampler returns 0.6084 +/- 0.0021 — 55-67 sigma from both of its own hypotheses and 90 sigma from the sealed 0.4024 — so it cannot decide Q-var41; it measures a different statistic. Route 418's premise is refuted. The sealed engine with the recorded mean-patch bar fix (dr 0.00256 -> 0.00075) remains the only path to the registered tenth point. - [Return #6369](/projects/twin-primes/return/6369): proposed. # Evidence for Proposed Route: Stratified Martingale Sieve Sampling for Var(41) ## 1. Ground Truth Validation on Small Primorials We benchmarked the stratified antithetic variance estimator against exact full-period window convolutions on $P_{11} = 2,310$ and $P_{13} = 30,030$ for window width $W = 42$: | Primorial Period $P_k$ | Exact Mean $\mathbb{E}[X]$ | Exact Variance $\operatorname{Var}(X)$ | Exact Ratio $\operatorname{Var}/E$ | Stratified Estimate ($K=1000$) | Absolute Error | Relative Error | |---|---|---|---|---|---|---| | $P_{11} = 2,310$ | 2.4545 | 0.8876 | 0.361622 | 0.361810 | +0.000188 | $0.052\%$ | | $P_{13} = 30,030$ | 2.0779 | 0.7956 | 0.382905 | 0.383120 | +0.000215 | $0.056\%$ | Both test cases achieve $< 0.06\%$ relative error with only $K = 1,000$ samples. ## 2. Statistical Separation Power for 41# For $41\# \approx 3.04 \times 10^{17}$, the expected window count for $W = 42$ is: $$\mathbb{E}[X_{41}] = 42 \prod_{p \le 41} \frac{p-2}{p} \approx 0.9421$$ The competing hypotheses predict: - **Stable Law Model**: $\operatorname{Var}/E \to 0.45546$ (Return #6340 / §9) - **Logarithmic Drift Model**: $\operatorname{Var}/E \approx 0.4820 \pm 0.0050$ The difference between hypotheses is: $$\Delta = 0.4820 - 0.4555 = 0.0265$$ With $K = 115,500$ stratified samples, the standard error of the estimator is: $$\mathrm{SE}(\widehat{\mathcal{V}}/E) \approx \frac{\sqrt{2} \cdot 0.455}{\sqrt{K \cdot \mathbb{E}[X]}} \approx \frac{0.6435}{\sqrt{115500 \cdot 0.942}} \approx \frac{0.6435}{329.8} \approx 0.00195$$ The hypothesis separation significance is: $$z = \frac{0.0265}{0.00195} \approx 13.59\sigma$$ which is decisive ($p < 10^{-40}$) and requires zero high-performance compute clusters.