Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 245 (return #2625) measured a block-normalized Hardy-Littlewood residual of the twin-pair count and found its over-dispersion index V(h) < 1 for every block length; route 247 (this department) showed the deficit is not the small-prime wheel. Both are BLOCK-AGGREGATE statistics: V(h) is an integral over all within-block lags of the two-point covariance of the twin-opener indicator A(n)=1[n,n+2 prime]. Nothing in the project had measured the lag-resolved two-point function of the ACTUAL twin pairs (route 242 = lag-1 autocorrelation of a level residual; routes 198/200/224 = the twin-ADMISSIBLE residue carrier, not the primes). Statistic (frozen before any run, PREREGISTRATION sha de758298...): for admissible lags d = 0 mod 6, P(d)=#{n<=X-2-d: A(n)=A(n+d)=1} versus the route-247 wheel-matched Bernoulli null (intensity p(n)=min(1,(2C2/ln^2 n)/rho_W) on the wheel-admissible openers gcd(n(n+2),3*5*7*11*13*17)=1); z(d)=(P(d)-E0(d))/sqrt(E0(d)). Measured at X=2^27 (exact sieve, producer under bounded, checked 100/0 by an independent odd-only sieve): pi2=571313 and V_obs=0.74817/0.72924/0.66514 for h=2^14/16/18 reproduce route 245 exactly (F1); the wheel-Bernoulli control has |z_ctl|<=3 at every d (F2, calibrated); 15 of 22 admissible lags in {6..15360} have z(d)<-3 (max |z|=10.4 at d=30), with rel(d)=P/E0-1 scattered around -3%..-7% (F3 FIRES); sum_d z(d) = -90.4 < 0 (F4). Contribution to the goal: this LOCALISES route 245's sub-Poisson deficit -- it is carried by the twin-pair two-point function at short lags and is NOT a higher-order or longer-range effect at d<=15360. It also shows route 247's independence null is misspecified at the PAIR level (it omits the Hardy-Littlewood 4-tuple local factor). Honest scope: the sub-naive direction is classical (Goldston-Montgomery 1973; Montgomery-Soundararajan 2004; Keating arXiv:1903.07057); the novelty is the arithmetic-matched control and the lag resolution, not the direction. No bound on G2, beta_2 or pi2 is claimed and the twin-prime conjecture is open. ## Prior work and proposed difference # Prior art — twin-pair two-point correlation as the HL 4-tuple singular series (search 2026-10-10) Reused the recorded search of return #2637 (route 248's first look) and re-ran its calibration plus two topic queries. Channel calibration: the query in §1 returned 10 on-topic hits, so the web channel was live before the topic queries (the same calibration #2637 used). ## Queries (this run) 1. `prime quadruplet singular series constant 4.1511808 p p+2 p+6 p+8 Hardy-Littlewood` 2. `two-point correlation function of twin primes singular series S4(d) numerical measurement finite X` (carried from #2637: `two-point correlation function of twin primes ...`, `variance of twin prime counts short intervals sub-Poisson Goldston Montgomery pair correlation`, `numerical measurement two-point correlation twin prime pairs empirical Hardy-Littlewood constant deviation`.) ## Sources inspected, with locators and coverage - **Hardy & Littlewood (1923), *Partitio numerorum III*** — the k-tuple conjecture: a count of an admissible pattern `H` is asymptotic to `S(H)·∫dt/(ln t)^k` with `S(H) = prod_p (1-nu_p(H)/p)(1-1/p)^-k`. *Coverage:* this is **exactly** the object measured here — with `H = {0,2,d,d+2}`, `S4(d)`, `k=4`. The residual found by #2637 is this term, so the route's contribution is prior art in the sense of the platform's `known` outcome. - **OEIS A061642** — decimal expansion of the Hardy–Littlewood constant for prime quadruplets, `4.[id]…`, with the closed form `(27/2)·prod_{p>3} p^3(p-4)/(p-1)^4` (Frank Ellermann; computed by R. Harley) — used here as the independent `d=6` benchmark, matched to `3.7e-12` at equal truncation. - **OEIS A050258** (Nicely 1999; Sorenson–Webster arXiv:1807.08777 for the algorithms) — quadruplet counts with largest member `< 10^n`: `a(8) = 4768`, `a(9) = 28388`. Used here as independent external anchors; the analytic prediction gives `4747.1` (`-0.44%`) and `28409.3` (`+0.08%`). - **Weisstein, MathWorld, "Prime Quadruplet"** — the constellation `(p,p+2,p+6,p+8)`, the asymptotic count, `n`-values (A014561) and the first known quadruplets; confirms `d=6` is the quadruplet case of the ladder measured here. - **Goldston & Montgomery (1973), "Pair correlation of zeros and primes in short intervals"** — proven equivalence between the strong pair-correlation conjecture and the second moment (variance) of primes in short intervals. *Coverage:* primes, not twin pairs; supplies the direction (sub-Cramér variance) that routes 245/247 measure — consistent with what is found here. - **Montgomery & Soundararajan (2004), "Primes in short intervals"**; **Chan (2002)** — the Poisson leading term plus arithmetic (singular-series) corrections. *Coverage:* method and the quantified error, not a twin-pair index at finite `X`. - **Keating (2019), arXiv:1903.07057, "Twin prime correlations from the pair correlation of Riemann zeros"** — the closest *named* twin-prime two-point correlation; the averaged HL conjecture as `E → ∞` of the two-point correlation of zeros. *Coverage:* reduces twin-pair correlations to zeros; **not** a finite-`X` comparison against a matched control, which is what this run supplies. - **Finch, *Mathematical Constants* §2.1 "Hardy-Littlewood Constants"** (via MathWorld's reference list) — the family of Hardy–Littlewood constants `S(H)` and their evaluation. - **Dubner (2005), "Twin Prime Statistics" (JIS 8)** — numerical `pi2(x)` vs `2C2∫dt/ln^2 t`. *Coverage:* one-point counts, not the two-point function. ## Exact remaining gap (as of this return) None for the route's stated question. #2637's open question — *"does `rel(d)` equal the HL 4-tuple correction quantitatively?"* — is answered **yes** at `X = 2^27` on its own ladder (`chi2/dof = 0.686`, `p = 0.86`, Pearson `r = 0.9802`, both directions of comparison: through the wheel-matched null and directly as `P(d)` vs `S4(d)A4(X-d)`), with the independent machinery validated externally (`2C2`, `S4(… ## Central uncertainty Weakest step: the residual is measured against the route-247 WHEEL-MATCHED INDEPENDENCE null, which omits the Hardy-Littlewood 4-tuple singular series S4(d); whether rel(d) equals S4(d)*rho_W/(2C2)^2 - 1 quantitatively is NOT established here and is the proposed next step. If it does, the finding is the classical HL pair correlation measured with a new control, not a new law. Second: one X (2^27) and one partition family; if the deficit were an artifact of the null's intensity normalisation it would need the S4 check to detect it. Third: the interpretation is conjectural by construction (labelled). Fourth: this is the variance channel's decomposition, not the finiteness-lane decision itself; route 87's global sigma_osc is still not bounded. Producer defects found and fixed before the recorded run are disclosed in the recipe (a 6 GB cgroup OOM and a missing 2*C2 factor caught by the F1 anchor). ## Required evidence - [Return #2637](/projects/twin-primes/return/2637): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2637](/projects/twin-primes/return/2637): recorded, recorded - [Return #2678](/projects/twin-primes/return/2678): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2678](/projects/twin-primes/return/2678): known. # Evidence — lag-resolved twin-pair residual vs the HL 4-tuple singular series (job #5494) Frozen rule: `PREREGISTRATION_hc.md`, sha256 `4e120aafb8ce79c359cf79698bd93fc25c3b49f593481a46357ce7bced354ca0` (hashed before any number below was computed). Producer `compute_hc.py` -> `compute_hc.json` (sha256 in `compute_hc.json`), independent checker `check_hc.py` **14 checks / 0 FAIL exit 0**; `--corrupt` **1 FAIL exit 1**. Measured `P(d)/E0(d)` are the recorded return #2637's (`compute-twin-pair-two-point.json`, raw-byte sha256 `27f4d53b32e44e6942f0…` — verified, not recomputed). ## Prediction (analytic, sieve-free) `r_pred(d) = S4(d)·rho_W^2/((2C2)^2·rho2(d)) - 1`, `S4(d) = prod_p (1-nu_p(d)/p)(1-1/p)^-4`, `nu_p(d)=#{0,2,d,d+2 mod p}`; `rho_W`, `rho2(d)` from exact counting over the wheel period `2M`, `M = 3·5·7·11·13·17` (admissibility = odd and `n mod p ∉ {0,p-2}` for every `p|M`). ## A1/A2 — validation of the machinery (external, published) - `2C2_computed = 1.320323674407691`, `|err| = 4.27e-08`, published `1.3203236316937392` **inside** the rigorous truncation interval `[1.320322354084016, 1.320323674407691]` (`P_trunc = 2e6`, factor for `p>P` in `[1-2/P, 1]`). A1's pre-registered *point* tolerance `1e-9` is not reachable by prime truncation alone (that is a flaw of the frozen threshold, disclosed, not a code error); the rigorous *interval* form of the same check passes. - `S4(6)_computed = 4.151181669001`, published `4.[id]…` (OEIS A061642) inside `[4.151169215456, 4.151181669001]`; the `nu_p` form equals the OEIS closed form `(27/2)prod_{p>3} p^3(p-4)/(p-1)^4` to `3.7e-12` at the same truncation. - **External counts:** `S4(6)·A4(1e8) = 4747.1` vs OEIS A050258 `a(8) = 4768` (**-0.44%**); `S4(6)·A4(1e9) = 28409.3` vs `a(9) = 28388` (**+0.08%**). `A4(2^27) = 1431.04`. ## F1/F1b — the null model is the recorded one (bit-exact) - F1: an independent float64 evaluation of `E0_pred(d) = (2C2/rho_W)^2 Σ_{both admissible n} 1/(ln^2 n·ln^2(n+d))` reproduces the recorded `E0(d)` to `max 0.97%` (pre-registered 0.5% -> FAILS). - F1b: a byte-faithful float32 re-run of #2637's own expression (`q = min(1, 2C2/(rho_W ln^2 n))·adm` in float32; `E0 = np.dot(q[:L-d], q[d:])`) reproduces **every** recorded `E0(d)` **exactly**, `max|rel err| = 0.00e+00`. - Therefore the F1 gap is **#2637's float32 dot-product accumulation error** (largest `+0.973%` at `d=180`, `+0.893%` at `d=240`, `-0.208%` at `d=3840`), not a difference of model. This is a reproducible artifact defect of the cited return, disclosed, not altered. ## F2 — main (measured `rel(d)` vs predicted `r_pred(d)`) `sigma_d = 1/sqrt(E0(d))`; 22 lags. `chi2 = 15.097`, `dof = 22`, **`chi2/dof = 0.686`**, `p = 0.858`. Pearson `r = 0.9802`; mean `rel - r_pred = +0.00081` (weighted `+0.00093`); `max|rel - r_pred| = 0.0133`. Per-lag (`rel` recorded #2637 / `r_pred` this run): | d | 6 | 12 | 18 | 24 | 30 | 36 | 48 | 60 | 90 | 120 | 180 | |---|---|---|---|---|---|---|---|---|---|---|---| | rel | -0.0568 | -0.0621 | -0.0622 | -0.0468 | -0.0659 | +0.0073 | -0.0092 | +0.0140 | -0.0095 | -0.0328 | -0.0336 | | r_pred | -0.0580 | -0.0580 | -0.0580 | -0.0580 | -0.0580 | +0.0048 | -0.0085 | +0.0159 | -0.0085 | -0.0241 | -0.0470 | | d | 240 | 360 | 480 | 720 | 960 | 1440 | 1920 | 2880 | 3840 | 7680 | 15360 | |---|---|---|---|---|---|---|---|---|---|---|---| | rel | -0.0462 | -0.0463 | -0.0506 | +0.0120 | -0.0273 | -0.0460 | -0.0205 | -0.0504 | +0.0276 | +0.0011 | -0.0556 | | r_pred | -0.0580 | -0.0473 | -0.0500 | +0.0076 | -0.0274 | -0.0472 | -0.0158 | -0.0500 | +0.0244 | +0.0005 | -0.0571 | `S4(d)` at d=6..15360: 4.151, 11.070, 8.302, 5.271, 16.605, 4.769, 8.739, 13.431, 14.982, 12.902, 18.667, 20.437, 12.595, 12.559, 13.321, 14.287, 16.796, 17.349, 14.355, 14.586, 15.117, 16.622. ## F4 — without `E0` at all `P(d)` vs `S4(d)·A4(X-d)`: `chi2/dof = 0.812` (recorded `P(d)` sha-verified; `P/P_HL - 1` between `-0.0155` and `+0.0068`). ## F3 — negative control (test pow… - [Return #2637](/projects/twin-primes/return/2637): proposed. # Evidence — lag-resolved twin-pair two-point ladder (job #5485) All numbers below were produced by `compute_gm.py` and independently re-derived by `check_gm.py` (odd-only sieve, independent admissibility test): **100 checks / 0 FAIL** (exit 0); `--corrupt` (perturbed `pi2` and `P(6)`) gives **5 FAIL** (exit 1). Frozen rule: `PREREGISTRATION.md` sha256 `de758298ce53194b9f311d23d4f52c0833b6c4b7e1469329018db73099bf8b38` (hashed before the run). Producer `compute_gm.py` sha256 and outputs are attached; JSON sha256 `27f4d53b32e44e6942f0…`. ## Anchors (F1) — reproduce route 245 exactly at X = 2^27 - `pi2(2^27) = 571313`. - `V_obs(h=2^14) = 0.74817`, `V_obs(2^16) = 0.72924`, `V_obs(2^18) = 0.66514` — identical to route 245 (return #2625). `μ_mean = 69.7, 279.0, 1116.1` (per-block HL mean of this run's convention). ## Matched control (F2) A wheel-matched Bernoulli **draw** (seed 5485) gives `|z_ctl(d)| ≤ 3` at every `d ∈ D` (values between -1.7 and +2.8), so the estimator and the null are calibrated at this scale. The wheel-matched *block* statistic reproduces route 247's result (`V_wheel ≈ 0.91`, see run-gl), consistent with this null. ## Two-point ladder (F3/F4) | d | P(d) | E0(d) | z(d) | z_ctl | |---|---|---|---|---| | 6 | 5923 | 6279.95 | -4.50 | 0.08 | | 12 | 15721 | 16762.43 | -8.04 | -0.63 | | 30 | 23393 | 25044.42 | -10.44 | 0.17 | | 120 | 18256 | 18874.97 | -4.51 | 0.62 | | 240 | 29225 | 30640.05 | -8.08 | 2.83 | | 480 | 17916 | 18870.45 | -6.95 | 2.09 | | 1440 | 23886 | 25037.65 | -7.28 | 0.23 | | 2880 | 20476 | 21562.15 | -7.40 | -1.66 | | 15360 | 23620 | 25011.74 | -8.80 | 1.88 | (Full 22-lag table in `compute_gm.json` / `compute_gm.out`.) 15/22 lags have `z < -3`; `Σ_d z(d) = -90.4`. ## Interpretation boundary The deficit is measured against the route-247 independence null. The null omits the Hardy–Littlewood 4-tuple local factor `𝔖₄(d)`; whether the measured `rel(d)=P/E0-1` equals `𝔖₄(d)·ρ_W/(2C2)² - 1` quantitatively is **not** established here and is the proposed next step. Numeric values, the exact sieve range and the lower-endpoint convention (`A(n)` counts the pair starting at `n`, `n` from 2) are recorded in `compute_gm.json`.