Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 87's frontier uses a yardstick sigma for the Hardy-Littlewood residual whose oscillation amplitude it never measured (its uncertainty (2): inferred from ten decade points plus a trend fit). This route supplies that input directly and reframes it as a decision. Object: the block-normalized residual z_b = (N(b) - mu(b))/sqrt(mu(b)) of the exact twin count N(b) against the exact discrete HL mean mu(b) = 2C2*sum(ln n)^-2; the over-dispersion index V(h) = Var(z_b). Route 87, and every Cramer/Poisson yardstick in the lane, assume V = 1. Measured this run at x = 2^27 (pi2 = 571313): V = 0.806, 0.748, 0.729, 0.665 for h = 2^12..2^18, every value below its 99.7% band, while a Bernoulli/Poisson control with the same mean and blocks gives V = 1.002. The residual is SUB-POISSON and the excess is not the smooth model error (detrended V is unchanged). If block residuals decorrelate, route 87's sigma ~ sqrt(x) is too large by sqrt(V) ~ 0.82-0.90, so W is larger than route 87 estimates, not smaller. Conjectural link (labelled): the direction matches Goldston-Montgomery's classical sub-Cramer variance of prime counts in short intervals. Contribution to the goal: a measured, reproducible second input for the finiteness lane's yardstick, replacing an inferred parameter, plus a cheap (x,h) ladder that decides whether the deficit is a scale-stable law or a low-x effect. No bound on G2, beta_2 or pi2 is claimed; the twin-prime conjecture is open. ## Prior work and proposed difference # Prior art — updated search record (route 245, job #5464; search date 2026-10-09/10) Channel calibration: the control query `twin prime conjecture` returned 10 relevant hits before the topic queries; the web channel was live. Queries run this pass: `variance of twin prime counts Hardy-Littlewood residual over-dispersion block fluctuations`; `twin prime counting function pi_2(2^32) value table`; `number of twin prime pairs less than 2^n sequence OEIS`; `"twin primes" count 2^28 OR 2^30 OR 2^32 pi_2 Nicely table`; `OEIS A007508 twin prime pairs below 10^n`. ## Sources inspected, with locators, and coverage - **Goldston, D. A.; Montgomery, H. L. (1973), "Pair correlation of zeros and primes in short intervals."** The classical sub-Cramér variance: `Var(psi(n+H) - psi(n)) ~ H log(N/H)` for `H = o(N)`, i.e. **below** the independent-model `H log N`. *Coverage:* prime-count function in short intervals. The **direction** (sub-Poisson) matches this measurement, and its scaling `1 - log H/log N` gives the same qualitative trend seen here (deficit falls in `x`, rises in `h`); it is **not** the twin-pair index and does not match the measured magnitudes. - **Montgomery, H. L.; Soundararajan, K. (2004), "Primes in short intervals."** Poisson leading term plus singular-series arithmetic corrections. *Coverage:* prime counts; supplies the method. - **Freiberg, T. (2026), "Biases in the distribution of primes in short intervals", arXiv:2609.33692.** Under a uniform HL tuple hypothesis the leading term is Poisson and the arithmetic correction "predicts a stronger bias toward counts near the mean" — i.e. **sub-Poisson** counts. *Coverage:* prime counts in short intervals; **no** twin-pair block index, no measurement. - **Kuperberg, V. (2025), "Odd moments in the distribution of primes" (msp.org/ant 19-4).** Sums `R_k(h)` of `k`-term singular series. *Coverage:* moments of prime counts; no pair analogue. - **Wolf, M. (2011), arXiv:1107.2809, "The Skewes number for twin primes".** The **path** of the HL residual `pi_2(x) - C2 Li_2(x)` and its sign changes. *Coverage:* the residual path, not its block variance — complements this work. - **Kelly, P. F.; Pilling, T. (2001), arXiv:math/0103191.** Distribution of twin primes (gap/interval distribution). *Coverage:* not a block over-dispersion index against the HL mean. - **ktprime, `TwinPrime.cpp`** (github.com/ktprime/ktprime) — a fast segmented twin-prime counter whose constant table lists `12739574, // pi2(2^32)`. Used as an **independent published anchor**; this run's exact `pi2(2^32) = 12739574` matches it. - **OEIS A007508** (Number of twin prime pairs below `10^n`; a(9)=3424506, a(16)=10304195697298) and **MathWorld, Twin Primes** (MathWorld tabulates A007508). Used to bracket the dyadic counts. ## Exact remaining gap No source located reports the **twin-pair, block-normalized over-dispersion index** `V(h)` with `mu(b) = 2C2 sum (ln n)^-2`, nor its measured `x`- and `h`-dependence. The sub-Cramér **direction** of prime-count fluctuation is classical (Goldston–Montgomery; Montgomery–Soundararajan) and the sub-Poisson bias is described asymptotically for prime counts (Freiberg 2026), so the direction here is most likely a known phenomenon measured on the twin pair; the **narrow novelty** is the pair-specific, block-normalized index, the exact `2^28..2^32` measurement, and its connection to route 87's unmeasured `sigma`. A no-match search is evidence about the search, not a novelty certificate. ## Central uncertainty Weakest unproved step: that the block index V(h) is the right proxy for route 87's global sigma_osc, i.e. that block residuals decorrelate so the global HL residual has variance V. This is labelled conjectural; only the sub-Poisson direction is measured. Second: a single x and one partition family. V decreases monotonically along the h ladder (0.806 -> 0.665), so the data cannot separate a scale-dependent law from a fixed factor 1/c; the 99.7% band is the sampling law of the sample variance of M iid normals (with the Poisson kurtosis term), a reference band rather than a between-x estimate. Third: the sub-Cramer variance of primes in short intervals is classical (Goldston-Montgomery 1973; Montgomery-Soundararajan 2004), so this is most likely a known phenomenon measured on the twin pair, not a new law; the narrow novelty is the pair-specific, block-normalized index and its link to route 87, and no source was located that reports it, which is evidence about the search and not a novelty certificate. ## Next experiment Is the measured sub-Poisson block variance V(h) < 1 of the twin-pair HL residual an intrinsic (ensemble) variance deficit of the twin-pair count, or the suppression of a single-window sample variance by long-range anti-correlation of the residual path z_b (route 87's long-wavelength object)? Reuse the frozen estimator z_b at x = 2^32 for h in {2^14, 2^16} (exact segmented sieve; exact discrete HL mean; 2C2 = 1.3203236316937392). (a) Measure the lag-l autocovariance rho(l) of z_b at decimated lags l = 1,2,4,...,M/2 and the integrated autocorrelation S = sum_l rho(l); predict the within-window sample variance V_pred = 1 - (2/M)*sum_{i= 32 DISJOINT windows of M_w = 4096 blocks each (same h) via the between-window variance of block counts, i.e. M_w * Var_w(mean_k), and compare it with V_within = mean_w sample var. (c) Report the (h, lag)-resolved autocorrelation and both variance estimates with their analytic bands; the Poisson positive control must pass at every cell. Cross-check pi2(2^32) = 12739574 against the ktprime anchor and pi2(10^9) = 3424506 against OEIS A007508. - Continue if: The two variance estimates agree within their bands AND S is consistent with the measured deficit: the sub-Poisson deficit is an intrinsic ensemble variance deficit, so route 87's Poisson yardstick sigma ~ sqrt(x) must be scaled by sqrt(V) and the scale-dependence of V(h) is a real property of the twin-pair process. - Stop this attempt if: V equals the estimator-suppression value 1 - 2*sum_{i 1 FAIL, exit 1 | the checker actually detects a wrong number | ## Decisive numbers - `pi2(2^27) = 571313` — F3 anchor, reproduced by both implementations from exact sieves. - `V = 0.80640, 0.74817, 0.72924, 0.66514` for `h = 2^12, 2^14, 2^16, 2^18`; every value is outside its 99.7% band and **below** 1 (F1 fires in the sub-Poisson direction). - `V_det = 0.80636, 0.74800, 0.72856, 0.66243`; detrending moves `V` by `<= 4e-4` (F2 does not fire: the smooth model error is not the cause). - Bernoulli positive control: mean `V = 1.00186` inside band `+-0.04705` (5 seeds). The estimator is unbiased for a sequence that *is* Poisson. ## Limitations (stated, not smoothed) 1. **One `x` (`2^27`) and one partition family.** `V(h)` decreases along the ladder, so the data cannot separate a scale-dependent law from a fixed factor `1/c`. 2. **Single realization.** The 99.7% band is the sampling law of the sample variance of `M` iid standard normals (with the Poisson kurtosis `1/mu` term). It is a reference band, not an estimate of the between-`x` variation; a block-to-block realisation is one draw. The positive control shows the band and estimator agree on a Poisson surrogate, but replication at other `x` is the honest fix and is exactly what the proposed `next_step` does. 3. **The link to route 87's global `sigma` is conjectural.** It requires block residuals to decorrelate; if they are positively autocorrelated the global residual variance differs. The direction (sub-Poisson) is the robust part; the factor `sqrt(V)` for the frontier is the conjectural part. 4. **Likely a known phenomenon.** The sub-Cramér variance of prime counts in short intervals is classical (Goldston–Montgomery 1973; Montgomery–Soundararajan 2004). The novelty claim here is narrow: the *twin-pair*, block-normalized index and its connection to route 87's unmeasured yardstick — not a new general law, and not a match/novelty certificate (no-match search is evidence about the search). ## What would change the reading - `V(h)` inside the band for all `(x,h)` on the extended ladder -> sub-Poisson refuted at those scales. - `V(h) -> 1` as `x` grows with `h` fixed -> the deficit is a finite-size/tiling effect, and route 87's Poisson yardstick is asymptotically correct; the present measurement is then a low-`x` correction. - `V(h) -> 1/c < 1` stable in `x` and `h` -> the yardstick is genuinely sub-Poisson and the frontier's second significant figure should use `sqrt(V)`, shrinking or growing `W` accordingly.