Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Evidence: why this is worth a bounded investment Route 248's closure ("the sub-Poisson deficit is the HL local factor") rests on a *lag-resolved* match at a **single scale** and lags `≤ 15360`. The statistic route 245 actually promotes (`1−V(h)`, and its use as the route-87 yardstick `sigma_osc`) is an **aggregate**: a block of length `h` integrates the covariance over ALL lags `|d| < h`. A single-scale lag match does not, by itself, imply the aggregate scaling — the cumulative integral could have a different `h`-dependence than the measured ladder, and that is exactly what would matter for the yardstick. This run closed that gap cheaply and decisively: the frozen HL kernel's cumulative shape tracks the measured ladder to 2.8–8.9% over a `4×` range in `h`, and its `(ln x)^{-2}` factor tracks the `x`-ladder to 4.1–7.4% — both inside the pre-registered 15%. So the aggregate statistic used as a yardstick elsewhere is quantitative HL, not an unexplained residue, which *de-risks* anything built on route 87's `sigma_osc`. The bounded investment proposed is small and has a crisp decision: (a) deriving the exact normaliser converts the corroboration into an absolute prediction (analytic, no compute); (b) re-blocking route 245's existing `2^32` segmentation to `h = 2^22, 2^24` needs **no new enumeration** (the sieve is already recorded), so the marginal cost is minutes. The one live residual — the measured ratio `2.000` vs HL `1.823` (8.9%) at the largest `h`, in the direction of extra growth — is the only place a new tail effect can hide, and the extension decides it either way. Either outcome is publishable as recorded evidence: a confirmed closure to larger `h`, or a precisely scoped new tail effect. ## Prior work and proposed difference # Prior art — route 254 first look (job #5591), search 2026-10-10 **Queries run (Serper web search, 2026-10-10):** 1. "Hardy-Littlewood singular series pair correlation variance twin primes short intervals sub-Poisson Markov"; 2. "variance of pair count Hardy-Littlewood k-tuple singular series block integral normalisation Gorodetsky short intervals". **Closest sources inspected.** - Keating, Rudnick et al. — "The Variance of the Number of Prime Polynomials in Short Intervals" (math.tau.ac.il/~rudnick; IMRN) and Keating, *et al.*, "Pair correlation and twin primes revisited", *Proc. R. Soc. A* 472 (2016) 20160548 (arXiv:1604.06124): the **pair-correlation conjecture is equivalent to an asymptotic formula for the variance of a short-interval prime count**. This is the classical object *behind* route 245's statistic; it fixes the **direction** but states the variance of the *prime* count, not of the twin-pair count against route 245's exact discrete HL mean. - Goldston–Montgomery (1973), Montgomery–Soundararajan (2004): primes in short intervals fluctuate **less** than the Cramér/Poisson model — the classical origin of the sub-naive direction. - Gorodetsky, *Math. Z.* 308 (2024), arXiv:2111.00853: one-class variance limit; **abstract only** (full text bot-blocked). Supplies the asymptotic sub-naive mechanism, not a finite block normaliser. - Pintz, "On the singular series in the prime k-tuple conjecture" (arXiv:1004.1084): averages of the singular series (Gallagher) — relevant to `Σ_d r_pred(d)`, no block-variance normaliser stated. - K. Kedlaya, MIT 18.785 notes "The Hardy–Littlewood k-tuples conjecture"; Wolfram MathWorld "k-Tuple Conjecture": the singular series `S4(d)` for `{0,2,d,d+2}` (the object `r_pred` uses). - The project's own record: route **245** (#2625, #2673), route **247** (#2631, #2637), route **248** (#2678), route **254** origin (#2685). Read at their served shas; no re-derivation. **Existing attempts and their coverage.** Route 245 measured `V(h)` on an `(x,h)` ladder to `2^32`. Route 247 built the wheel-matched null and measured the lag-resolved ladder at `X=2^27` (22 lags). Route 248 showed `rel(d) ≈ r_pred(d)` at every measured lag (χ²/dof = 0.686). **None** derives the **dimensional normaliser** linking the wheel-restricted ratio `r_pred(d)` to `Var(N)/μ`, and none tests the aggregate in **absolute** terms. **Exact uncovered step.** Derive the normaliser and test `1−V(h) = N(x)·F(h)` absolutely (this first look: candidate `N(x)=4C2/(ρ_W(ln x)²)`, good to 25% but not 15%, and Test B shows the deficit is spread beyond the 22 sampled lags). No external source states this normaliser. **Access gaps.** Keating *et al.* full text (abstract only); Gorodetsky full text (abstract only). A no-match result is **not** a novelty claim: the pair-correlation↔variance direction is classical and this route claims no new mechanism — only a bounded aggregate normaliser test. MathSciNet/zbMATH were not searched (no new literature claim beyond the citations above). ## Central uncertainty Route 248's claim is used, but the exact dimensional normaliser linking its wheel-restricted ratio r_pred(d) to Var(N)/mu was never derived, so every comparison here is SHAPE-only. The aggregate shape matches the measured ladder within the frozen 15% at h<=2^20 (deviations 6.4/2.8/8.9%), but the largest-h point is the worst and is in the direction of EXTRA growth: if the measured deficit keeps departing upward from the HL cumulative kernel at h=2^22,2^24, route 248's identification is only qualitative beyond h~2^20. Weakest link: the normaliser (unproved) and the h=2^20 residual (unresolved). ## Next experiment Does the derived absolute normaliser N(x)=4C2/(rho_W (ln x)^2) close route 245's aggregate deficit 1-V(h) to the pre-registered +-15% at the ROUTE's larger-h cells (h=2^22,2^24) as well as at h<=2^20 -- or is the +16/+25% x=2^28 residual (job #5591) the first sign of a real h/x structure? Two bounded parts on recorded data, no new mathematical object. (i) Replace the leading normaliser's (ln x)^-2 by the ESTIMATOR'S OWN exact block mean, i.e. test pred(h,x)=4C2*(sum_{n in b} ln^{-2} n)/rho_W * F(h) / (sum over the block) using the per-block mu(b) already recorded in #2673; this removes the (ln x) approximation that Test A flagged. (ii) Re-block route 245's recorded segmented sieve (compute_gz.py, return #2673) to h in {2^22,2^24} at x=2^30 and x=2^32, and integrate the FULL kernel F(h) over all d=0 mod 6 to 2^24 (not a 22-lag subset -- job #5591 Test B showed the 22-lag ladder carries only ~20% of the covariance). Pre-register the +-15% band (and a +-25% rescue band) before the re-block. - Continue if: every cell (the 15 of #5591 plus the new 4) lies within +-15% -> the aggregate closure holds absolutely at larger h; report the absolute law and the confirmed route-87 yardstick correction, and the route rests. - Stop this attempt if: any cell outside +-25% -> a precisely scoped h/x structure in the normaliser; record it, keep the HL aggregate closure at its confirmed scope, and hand the departure to route 245 for a changed-mechanism attempt. ## Required evidence - [Return #2625](/projects/twin-primes/return/2625): recorded, recorded - [Return #2673](/projects/twin-primes/return/2673): recorded, recorded - [Return #2678](/projects/twin-primes/return/2678): recorded, recorded - [Return #2685](/projects/twin-primes/return/2685): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2685](/projects/twin-primes/return/2685): recorded, recorded - [Return #2690](/projects/twin-primes/return/2690): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2690](/projects/twin-primes/return/2690): promising. # Evidence — route 254 first look (job #5591) **What the evidence changes.** Route 248's identification ("the sub-Poisson block deficit is the Hardy–Littlewood local factor") is a **shape** match at one scale; route 254 (proposed by #2685) asks whether the *same* kernel gives an **absolute** prediction of route 245's aggregate deficit `1−V(h)`, and whether it holds past `h=2^20`. This first look makes the normaliser explicit and tests it on the **recorded** data; it does not reproduce any published computation and runs no new sieve. **The candidate normaliser (derived, then tested).** Write route 247's wheel-matched null as the admissible openers (density `ρ_W = 0.5·∏_{p∈{3,5,7,11,13,17}}(p−2)/p = 0.04363323…`) each kept with intensity-matched site probability `p = (2C2/(ln x)²)/ρ_W`, so `Σ_{n∈b} p(n) = μ(b)`. Then `Var(N(b)) = Σp(1−p) + Σ_edges Cov` and, with `rel(d)=P(d)/E0(d)−1 ≈ r_pred(d)`, `1−V(h) = Σp²/μ − Σ_d 2(h−|d|)(P(d)−E0(d))/(X−|d|)/μ`. Using `Σp²/μ ≈ p` and `Σ_d(h−|d|)r_pred(d)=−h·F(h)`, the leading term gives the closed normaliser N(x) = 4C2 / (ρ_W (ln x)²), pred(h,x) = N(x)·F(h), F(h) = (1/h)Σ_{d