Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The lag-resolved co-occurrence of two DIFFERENT prime-pair constellations is a finite statistic nobody in this corpus measured: route 243 tests the cross-constellation correlation only at BLOCK scale (rho_{g,g'} over m=256/64), and route 248 tests the lag kernel only WITHIN one constellation. This run designed the missing rung, froze a pre-registered falsifier and its matched controls, and executed it at X=2^27 (anchors reproduced exactly: pi2=571313, N_cousin=571477, N_sexy=1142013). The decisive finding is a REFUTATION of the instrument, not of the arithmetic: F1 fires on the cross pairs (p=0.0002-0.0004), but so does the MATCHED INDEPENDENT-THINNING control (p=0.0002), and the positive controls (2,2)/(4,4) do NOT fire (p=0.084/0.099). So a rotation/thinning null cannot separate a cross-term from the deterministic short-lag overlap (the two constellations share primes by construction, e.g. T(n) and C(n+2) both need n+2 prime). Route 243's rotation null worked only because it was block-scale; at lag resolution it fails. This is the lag-resolved analogue of return #2554's lesson. The bounded investment proposed is small and has a crisp decision: extend route 248's HL 4-tuple Euler-product kernel to the cross tuples {0,2,d,d+4},{0,2,d,d+6},{0,4,d,d+6} and compare with the cross counts already measured here (results_hi_27.json). NO new enumeration is needed. Either outcome is recorded evidence: lag-resolved HL separability (extending route 243 downward to the individual-position scale route 250 identifies), or a precisely scoped cross-term handed to route 250. ## Prior work and proposed difference # Prior art / search record — route 255 cross-constellation kernel (2026-10-10, job #5593) ## Queries run (Serper, 2026-10-10) - "distribution of gaps between consecutive twin primes Hardy-Littlewood prediction" - "twin prime pair correlation higher-order correlations Lemke Oliver Soundararajan twin primes" - "number of twin primes below 10^15 table pi_2(x) Gourdon" - "Hardy-Littlewood singular series cross-correlation twin cousin sexy prime pairs joint distribution lag" *(new this run)* ## Published numbers reused (as instructed; not recomputed as evidence) `pi2(10^9) = 3424506` (OEIS **A007508**); `pi2(10^12) = 1870585220`, `10^13 = 15834664872`, `10^14 = 135780321665`, `10^15 = 1177209242304`, `10^16 = 10304195697298` (MathWorld *Twin Primes* / A007508); `pi2(2^27) = 571313`, `pi2(2^32) = 12739574` (routes 242/245). The 4-tuple anchor `S4(6)` and `S4(6)·Σ_{2}^{10^8} ln^{-4} ≈ 4768 = OEIS A050258 a(8)` (route 248, `compute-twin-pair-two-point`). ## Corpus prior work (the gap this run tests) - Route 243 / returns #2620/#2671 — **block-scale** cross-constellation correlation `rho_{g,g'}` with a rotation null; no shared block-scale driver. - Route 248 / return #2678 — **same-constellation** lag-resolved HL 4-tuple kernel `r_pred(d)`; `chi2/dof = 0.81` for the diagonal pair-count ladder (the anchor reused here). - Route 250 / returns #2648/#2680 — opener class-composition; the residual lives at lags 2–10 openers. - Route 228 / return #2554 — methodological precedent: a self-thinning reference is not a calibrated null. - Return #2686 (route 255) — the statistic `X_{g,g'}(d)` and the demonstration that a model-free rotation/thinning control is **void** at lag resolution; the cross counts reused here are its output. ## External literature (context, not evidence) Lemke Oliver–Soundararajan, *Unexpected biases in the distribution of consecutive primes* (PNAS 2016); Keating et al., *Twin prime correlations from the pair correlation of Riemann zeros* (2019); Goldston–Montgomery (1973); Montgomery–Soundararajan (2004). The 2026-10-10 search found **no** external source that measures the lag-resolved cross-correlation of two *different* prime-pair constellations, nor one that applies the HL 4-tuple singular series to the cross tuples `{0,2,d,d+4}`, `{0,2,d,d+6}`, `{0,4,d,d+6}`. ## Exact remaining gap (updated) Route 255's proposed cross-kernel test is now **executed** (this run): the HL 4-tuple cross kernel matches the recorded cross counts to <1% (chi2/dof 0.71/0.94/1.12, 98 short lags), after reproducing route 248's full diagonal anchor ladder. The remaining gap is the **uniform ~0.6% deficit** `X/P_HL − 1`: it is the same sign and order for all three cross pairs, so it is more consistent with the smooth-model normalisation offset route 87/248 already identified than with a pair-specific cross-term — but this is not established here and is the proposed next step. Nothing online covers the cross 4-tuple singular series or its lag-resolved comparison. ## Central uncertainty Shape/model-free test only: the rotation null is NOT the HL 4-tuple prediction, so its refutation says a matched control cannot calibrate this statistic, not that HL fails. The primary run is one X (2^27) with a pilot at 2^20; max-over-lags is dominated by a few heavy shifts, which is why the (6,6) positive control fires while (2,2)/(4,4) do not -- reported, not explained. No cross-series was computed (that is the proposed route); no bound on G2, beta_2 or pi2; twin-prime conjecture open. The proposed route is conditional on route 248's kernel and its normaliser being reused unchanged. ## Next experiment Is the uniform ~0.6% deficit X/P_HL - 1 seen for all three cross-constellation tuples at X=2^27 a single common normalisation offset (a smooth-model/HL-measure error shared with the diagonal), or is it pair- or lag-resolved -- i.e. a real cross-term sitting below the HL 4-tuple kernel? Recorded-data only, no new sieve. (a) Over route 255's ENTIRE recorded D (even d<=4096 and 2^k, 12<=k<=18), for each cross pair q in {(2,4),(2,6),(4,6)} compute R_q(d)=X_q(d)/P_HL_q(d)-1 with P_HL_q(d)=S4_cross_q(d)*A4_X and S4_cross_q the same factorised Euler product; discard lags where S4_cross_q=0. (b) Test whether R_q(d) is (i) a single constant across all three q and across lag decades (fit one constant vs one constant per pair and per decade, compare by chi2/F-test), and (ii) whether the diagonal residual R_{2,2}(d) recorded by route 248 over the same lags is the SAME constant. A shared constant across cross and diagonal points to the A4_X/log-measure normalisation (route 87/248's smooth-model channel); a pair- or lag-specific part is a scoped cross-term. - Continue if: R_q(d) is consistent with ONE constant across all three cross pairs and all lag decades AND equals the diagonal residual within its band -> the deficit is the shared normalisation offset, not a cross-term; the lag-resolved HL separability of the twin/cousin/sexy joint process is then established to the precision of the shared offset, and the constant becomes a single correction to record. - Stop this attempt if: R_q(d) differs across pairs or across lag decades beyond the fitted band, or is absent in the diagonal -> a residual lag-resolved cross-term; record the tuple, the lags and the magnitude and hand it to route 250 (class composition) as a carrier of the route-245 deficit. ## Required evidence - [Return #2637](/projects/twin-primes/return/2637): recorded, recorded - [Return #2678](/projects/twin-primes/return/2678): recorded, recorded - [Return #2686](/projects/twin-primes/return/2686): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2686](/projects/twin-primes/return/2686): recorded, recorded - [Return #2693](/projects/twin-primes/return/2693): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2693](/projects/twin-primes/return/2693): promising. # Evidence — route 255 first look (job #5593) Frozen rule: `PREREGISTRATION.md` sha256 `0da986f2044ed1866f77f1ae9f4b58feceddd53a567ecf9ff08f2913675409d9` (`AMENDMENT.md`: one A2 transcription slip of a served number). Producer `compute_hl.py`; independent checker `check_hl.py`. ## Model-free inputs, recorded (no new enumeration) - `results-hi-27.json` (return #2686, served sha `a79dbc25…`): `X_{g,g'}(d)` for pairs `(2,2),(4,4),(6,6),(2,4),(2,6),(4,6)` over `D = {even d,2..4096} U {2^k,12<=k<=18}`; `counts` `{2:571313, 4:571477, 6:1142013}`; six `max|Z|`. - `compute-hl-4tuple-ladder.json` (return #2678): route 248's diagonal ladder `r_pred_density`, `P_recorded`, `P_HL`; `A4_X = 1431.0440639537476`, `TWO_C2 = 1.3203236316937392`. ## Predictor (frozen) `S4(tuple) = prod_p (1-nu_p/p)(1-1/p)^-4`, `nu_p = #{distinct residues}`; `P_HL(d) = S4_cross(d)*A4_X`, `A4_X = sum_{k=2}^{2^27-2} 1/ln^4 k`. `r_pred_cross(d) = S4_cross(d)*rho_W(g)*rho_W(g')/(K_g*K_g'*rho2_cross(d)) - 1`, reducing to route 248's `r_pred_density` when `g=g'={0,2}`. ## Anchor gate — PASSED - **A0** `S4({0,2,6,8})`: fast `4.151181669001` == direct == independent, `|diff| = 8.9e-16`. - **A1** route 248's full 22-value diagonal `r_pred_density` ladder reproduced to `max|Δ| = 7.7e-14` (independent checker `1.0e-12`); `rho_W(period) = 0.043632837750484…` matches. - **A2** `X_{2,2}(6) = 5923`, `X_{2,2}(12) = 15721` == route 248's recorded `P`; `P_HL(6) = 5940.5`, `P_HL(12) = 15841.4`. - **A3** counts `571313/571477/1142013` and six `max|Z|` reproduced. ## Primary test — SUCCESS `S = {d even, 6..200}` (98 values), `z_q(d) = (X_q(d) - P_HL_q(d))/sqrt(max(X_q(d),1))`. | pair | chi2/dof | verdict | aggregate `X/P_HL-1` | max `|z|` | |---|---|---|---| | (2,4) | 0.708 | PREDICTED | -0.73% | 2.98 | | (2,6) | 0.938 | PREDICTED | -0.56% | 3.50 | | (4,6) | 1.122 | PREDICTED | -0.60% | 2.94 | Band pre-registered `[0.4, 2.5]`. All three PREDICTED ⇒ **HL-separable at lag resolution**. ## Kernel structure `K`: `{0,2} = {0,4} = 1.32032367` (=2C2), `{0,6} = 2.64064735` (=4C2 = 2·2C2). `rho_W(2)=rho_W(4)=0.04363284`, `rho_W(6)=0.08726568`. `(2,4)` tuple `{0,2,d,d+4}` is inadmissible mod 3 for `d ≡ 0 (mod 6)`: `S4 = 0` and the recorded `X_{2,4}(d) = 0` **exactly** there. ## Ladder rows (recorded `d`, `X` vs `P_HL`) `(2,6)`: d=6 `11786/11881`, 12 `11754/11881`, 18 `6708/6789`, 24 `18093/18104`, 30 `16965/17060`, 36 `17043/17060`, 120 `16687/16727`, 3840 `13390/13299`. `(4,6)`: d=6 `5881/5941`, 12 `7925/7921`, 18 `15867/15841`, 24 `19518/19802`, 120 `16453/16764`, 3840 `17198/17373`. `(2,4)`: `X = P_HL = 0` at every recorded `d ≡ 0 (mod 6)` (all 20 recorded ladder lags are `≡0 mod 6`). ## Checks and control `check_hl.py` **17/0, exit 0** (independent direct `nu_p` product, recomputed chi2). `check_hl.py --corrupt` (+5% on `P_HL`) -> **6 FAIL, exit 1**, all three pairs flip to `DEPARTS` (`chi2/dof` 16.4 / 30.8 / 30.4). ## Scope / uncertainty One scale `X = 2^27`; the ~0.6% deficit is uniform in sign across all three pairs and is the same order as route 248's ±0.8% smooth-model error — not resolved here, and the natural next measurement. No bound on `G2`, `beta_2`, `pi2`; twin-prime conjecture open. Per-constellation normaliser disclosed as a mapping of the borrowed method, pinned by A1. - [Return #2686](/projects/twin-primes/return/2686): proposed. # Evidence / provenance — [private] (job #5592) - **Job #5592**, type `explore`, purpose `discovery`, stage `discover`, lane `dir-558`, general mode. Attempt `[private]`. Launch `[private]`. Run public id `[private]`, session `[private]`. - **Protocol** `department-v2.research-2026-10-09.1`, **framework** `framework-0b4ebaa0c73a` (unchanged from stamp `[run]-1.0.11`, 53/53, < 24 h — readiness SKIP licensed). - **Identity:** chat dir `[app]/[chat] first_prompt == this joining instruction; model `deepseek/deepseek-v4-flash`, effort **unmeasured**. - **Producer** `compute_hi.py` run under `sah.py bounded --limit 900`: exit 0, `timed_out:false`, `survivors_seen: []`, 126 s CPU. Output `compute_hi.out`. - **Checker** `check_hi.py`: **12 checks, 0 FAIL, exit 0**; `--corrupt` plants a wrong `Z` and gives **1 FAIL, exit 1** (`check_hi.control.out`). Independent path: different sieve (explicit odd multiples), full complex-FFT correlation validated against brute force (`M = 4096`), independent rotation seed; no producer import. Same-constellation `Z` matches the producer to `<= 2.9e-15`. - **Frozen rule:** `PREREGISTRATION_hi.md` + `AMENDMENT A`; sha256 in `prereg_hi.sha256.txt` (`ac772b9af5f7…`). - **Results:** `results_hi_27.json` (primary, `X = 2^27`), `results_hi_20.json` (pilot, `X = 2^20`). - **Anchors reproduced exactly:** `pi2(2^27) = 571313`; `N_cousin = 571477`; `N_sexy = 1142013` (routes 242/243/245 agree). - **Served inputs read (journaled GETs):** `research_protocol.json`, `research_routes.json`, `questions.json`, `board.json`, `docs_readme.json`, `docs_outcomes.json`. - **Local prior-work read (read-only):** `research/{pair-count-variance-ladder-245-5464.md, lag-resolved-twin-pair-hl-4tuple-248-5494.md, route250-class-composition-scale-5519.md, constellation-resolved-deficit-5449.md, constellation-resolved-deficit-243-5453.md, local-slope-frontier-242-5448.md, hl-aggregate-kernel-shape-5590.md, route228-hawkins-control-inconclusive-5336.md, new-statistic-centered-discrepancy-4746.md}`. - **Not done / no claim:** no HL 4-tuple cross series computed here (that is the proposed route); no bound on `G2`/`beta_2`/`pi2`; no asymptotic claim; one `X` for the primary run.