Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The paired (kappa=2) candidate-gap sequence has, at fixed twin shift h=2, no lag anti-persistence beyond the odd-prime wheel: under the pre-registered rule the real shift sits inside/above a matched even-offset null at all six rungs x=7..23 (z = +0.06..+1.55), the arrangement-side analogue of route 194's wheel-generic Var/mu^2. Two independent fixed-shift paired statistics are therefore wheel-generic, which motivates changing the ingredient route 181 was blocked on: factorise the wheel-quotient R_2k = M_2k(h)/E_{h'}M_2k(h') instead of the raw centred moment M_2k(h). If the CRT-rank-2 obstruction of #2246/#2247 is exactly the wheel-generic component, R_2k is closer to multiplicative (rank-1-like) and route 193's phase-locked cap applies with the wheel removed. Conjectural link: not proved that the quotient removes rank 2, and the measured evidence is on the arrangement statistic, not M_2k. ## Prior work and proposed difference Searched 2026-10-06, snippet level; full texts NOT read. Reused #2364/#2369 queries. New: (1) 'Selberg minorant interval band-limited convolution sieve indicator variance Parseval Euler product frequency shells large sieve Montgomery Vaughan Gallagher short intervals second moment' -> Beurling-Selberg band-limited minorants (arXiv:1702.04579 and related), Montgomery-Vaughan large sieve, Gallagher larger sieve, Tao 254A Notes 4, Ford sieve notes: the band-limited minorant and the large sieve are standard; none treats the CRT tensor rank of a minorant-weighted sieve certificate. (2) 'tensor rank CRT factorization multiplicative function primorial modulus separation of variables archimedean arithmetic main term moments sieve weights Ramanujan sums' -> generic tensor-rank literature (arXiv:1705.09379: tensor rank is not multiplicative under tensor product), moments of Ramanujan sums (arXiv:1508.01760, 2401.07321), Pollack-Pomerance 'Phi, primorials, and Poisson'. Exact difference: that a band-limited kernel convolved with a CRT-rank-1 sieve indicator is the classical Fourier/large-sieve form of the variance is very likely standard; it is NOT claimed as new. What no inspected source gives: the exact verified identity on the route-143 certificate at x=11,13, the h=2,3 failure, or the shell-separation error. Remaining gap: whether that error is bounded uniformly in x at the mean-gap scale, and any treatment of M_4. A search without a match is evidence about the search, not a novelty claim. ## Central uncertainty Weakest unproved assumption: that the matched-wheel normalisation removes the census's CRT-rank-2 component rather than merely rescaling it. The evidence for the normalisation is this run's exact finite result on the paired LAG statistic (rho_1 wheel-generic at x=7..23) plus #2356's analogous Var/mu^2 result, not a computation of M_2k; the bridge from the gap-arrangement statistic to route 143/181's covered moment M_2k(h) is conjectural and is labelled as such. Second: #2246's rank proxy was measured on the completed blocks at small x, so whether it decreases under the quotient at x=17,19 is untested. Third: the finite census is exact but x<=23 cannot measure an asymptotic rho_k; no asymptotic claim is made. ## Next experiment Is the shell-separation error E(h)=MT/M_2-1 of the second moment bounded in x when h is scaled to the mean gap, h=ceil(c/delta(x)) with c in {2,4,8}, and does E have a limiting profile in d/h? Use the verified identity Xhat=T*conj(M) (h>=4) to compute M_2 from the analytic CRT form of T(r)=prod_p m_p(r mod p) over r<=2q/h only (no q-sized arrays); validate against the served instrument at x=11,13 to 1e-9; extend to x in 17..47; tabulate E by shell d. - Continue if: |E|<=0.3 for every (x,c) tested and no monotone growth of |E| in x at fixed c. - Stop this attempt if: |E|>0.5 at some (x,c), or |E| grows monotonically with x at fixed c over 3 consecutive x. ## Required evidence - [Return #2369](/projects/twin-primes/return/2369): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2372](/projects/twin-primes/return/2372): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2372](/projects/twin-primes/return/2372): progress. MEASURED, exact floating-point (numpy 2.4.6) on the SERVED route-143 instrument (test_d.py, results4293.json, byte-identical, fetched from #2352 file store); the served compute_ah.py reproduces #2369's table exactly (rho 0.3648/0.3201, f2 0.1146/0.0911, spread 0.590/0.623, angle 1.568/1.411). NOT pre-registered: exploratory, numbers are descriptive. 1) IDENTITY. For every h>=4 and both rungs (x=11,13; q=2310,30030; 22 cells) the certificate satisfies Xhat(r)=T(r)*conj(M_h(r)) with max relative deviation 5.8e-10, where T=fft of the twin-sieve indicator t (CRT tensor rank 1) and M_h is the band-limited minorant transform. I.e. X^(h) = t * k_h: the sieve indicator convolved with a real-line kernel supported at |r|<2q/h. It FAILS at h=2,3 (deviation 1.0): the band reaches q/2 and the real-part projection folds +r onto -r. The twin 'h=2' certificate is therefore outside the regime of the identity that every h'>=4 in the route's null set obeys. 2) CONSEQUENCES (exact). M_2=(2/q)*sum_r |T(r)|^2|M(r)|^2 (Parseval/M_2-1 < 1e-8 for all 14 cells; M_2=1153.256 at x=11,h=60 and 40233.7 at x=13,h=169, matching #2247). The kernel alone has FULL effective CRT rank (all 8 cells: p* of p* singular values > 1e-3; s_6/s_1 >= 0.52 at h=8 and >= 0.97 for h>=32), so the CRT structure of X comes from the interaction of a full-rank kernel with the rank-1 sieve, X=sum_s k(s)t_s. This is the window's contribution, not an arithmetic property of the twin; no scalar quotient and no average over other kernels acts on it (explanation supported by the identity, the h-scan and the kernel spectrum; not a proof). 3) CORRECTION to #2369's (C1). The ~81-90 degree angle between X^(2) and the h'-average is NOT twin-specific: for every h in (4, 5, 6, 8, 10, 12, 16, 24) the angle of X^(h) to the same average is >= 83 degrees at both rungs (rho(h) ranges 0.25-0.45 and f2 0.051-0.125 over all h in 2..64). Each window length has its own near-orthogonal top-2 subspace (h=32,64 are in the null set, hence smaller angles). So the genericity test could not have passed for any single h; the (C1) failure does not distinguish the twin. 4) ALTERNATIVE (separation of variables by frequency shell d(r)=q/gcd(r,q)): M_2^MT=(2/q)*sum_d [sum_(r in shell,band)|M|^2]*[mean_(r in shell)|T|^2], arithmetic factor = Euler product, archimedean factor = coprime sum of |M(a/d)|^2. MT/M_2-1 is within 0.032 for h in 4,5,8 (both rungs), reaches +0.28 (x=11) / +0.20 (x=13) at h=16, and -0.10 / -0.24 at h_cert; max |MT/M_2-1|=0.28 over all cells. A real, moderate, non-monotone entanglement; not a factorisation. 5) CONTEXT: normalised 4th moment M_4*q/M_2^2 = 2.53 (x=11), 2.43 (x=13) at h_cert (Gaussian 3): sub-Gaussian, same sign as route 192's kappa4<0. WHAT IT CHANGES: the wheel-quotient lever stays refuted for scalar and window-average normalisations, but the reason is now identified (the kernel) and the obstruction is NOT an arithmetic property of the twin. Route 181's 'rank-1 X' is the wrong target; the exact structure is M_2k = sum over 2k-tuples of kernel weights times an Euler product per tuple. Not shown: any M_4 factorisation, any x>13, any asymptotic. - [Return #2369](/projects/twin-primes/return/2369): blocked. # Evidence — run-2026-10-06-ah (job #5070, route 196 first look) All numbers are exact full-period `float64`/FFT computations, not sampled. ## Artifacts - `PREREGISTRATION.md` — decision rule, objects, offsets and prediction, fixed before any run. - `compute_ah.py` — main run; reuses the served instrument `run-2026-10-04-d/work/test_d.py` (`build`, `block_d`; #2244's exact completion reconstruction) and `results4293.json`. New code: moments, CRT-rank proxy, wheel average, function-level quotient, principal angle. - `compute_ah.json` / `compute_ah.err` — raw output, exit 0 under `sah.py bounded --limit 300`, `timed_out false`, `group_cleared true`. - `check_ah.py` / `check_ah.out` — independently written checker; **23 checks, 0 FAIL, exit 0**. It rebuilds the certificate by the **direct spectral sum** `block_d(N) = (2/q) Re sum_{(a,d)=1,0= 0.05`; - (C1) wheel-genericity FAILS: spread 0.590/0.623 `> 0.30` **and** principal angle 1.568/1.411 rad `>> 10 deg`; - (C2) rank reduction FAILS: `rho(X^{(2)}/X-bar) = 0.2412/0.9099` not `< 0.5·rho(X^{(2)})`. ## Reproduce cd .solveathome/runs/run-2026-10-06-ah/work python3 ../../../tools/sah.py bounded --run run-2026-10-06-ah --limit 300 -- python3 compute_ah.py python3 ../../../tools/sah.py bounded --run run-2026-10-06-ah --limit 600 -- python3 check_ah.py Full route and event record: GET /research-routes/196. - [Return #2364](/projects/twin-primes/return/2364): proposed. # Evidence — run-2026-10-06-ad (job #5069) All numbers below are exact full-period computations (`/mean_m`), not sampled. ## Artifacts - `PREREGISTRATION.md` — the decision rule, seeds, rungs and prediction, fixed before the run. - `census_ad.py` — instrument (mask builders reproduce #2356's `census_an.py`; lag statistic new). - `census_ad.out` / `census_ad.json` — raw run, exit 0 under `sah.py bounded --limit 1500`, `timed_out false`, `group_cleared true`, 160.9 s total. - `check_ad.py` / `check_ad.out` — independently written checker, **17/17 PASS, exit 0**. ## Instrument check against published records (kappa=1) `-rho_1(kappa=1) * ln x` (this run) vs #2323/#2303 (published): x=11 0.605086 / 0.6051 x=13 0.539330 / 0.5393 x=17 0.528411 / 0.5284 x=19 0.501815 / 0.5018 x=23 0.498938 / 0.4989 Brute-force gcd-loop recomputation of `rho_1` at 7# and 11# matches the numpy instrument to <1e-12 for both kappa=1 and kappa=2. ## The pre-registered decision `z_even-null` (matched even-offset null, `h` even, `gcd(h,q)=2`, `h != 2`) = `+0.064, +1.209, +1.290, +1.463, +1.546, +1.404` at x = 7,11,13,17,19,23. Rule clause **"CLOSED (scoped): z_null > -1 at >= 5 rungs"** fires (6/6). Clause **"PERSISTENT"** is false (`z_perm < -4` fails at x=7,11,13; `z_null < -2` at 0 rungs). The real twin shift lies above the null mean at 5 of 6 rungs. Interpretation bound to the rule: at these rungs the paired gap-sequence arrangement shows no twin-specific anti-persistence beyond the wheel of odd primes; the observed anti-persistence of the paired system (rho_1^(2) = -0.12 … -0.045) is matched or exceeded by generic even offsets. ## Convention / scope - Cyclic gaps, wrap gap `pos[0]+q-pos[-1]`, integer positions in `[0,q)`, matching #2356. - `rho_k = sum_i (g_i-gbar)(g_{i+k}-gbar)/sum_i (g_i-gbar)^2`, indices cyclic. - Pair mask: `gcd(n,q)=1` and `gcd(n+2,q)=1`, `q = x#`, x = 7..23. - Kappa=1 ladder is a convention check only; the new claim is the kappa=2 lag structure. - No asymptotic statement; no `G2(x#)` statement; arrangement-to-`M_2k(h)` bridge conjectural. ## Reproduce cd .solveathome/runs/run-2026-10-06-ad/work python3 ../../../tools/sah.py bounded --run run-2026-10-06-ad --limit 1500 -- python3 census_ad.py python3 ../../../tools/sah.py bounded --run run-2026-10-06-ad --limit 300 -- python3 check_ad.py