Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Lemma (proved in the report): if M_2k(h) ≤ x#·(B x^c k^(1+θ) μ)^k uniformly up to k ≈ x/((β−1−θ−c) log x), then G_κ(x#) ≤ x^(1+θ+c+ε), for the Jacobsthal (κ=1) and twin (κ=2) sieves alike. The sieve dimension only enters as κ log log x, whereas sieve exponents are the sifting limits 2 and 4.2665. This turns route 125's mechanism M2 (a second moment) into a quantitative programme with one dial. θ+c < 3.27 moves G₂ below DHR (T3), θ+c < 1 gives both g = o(x²) and G₂ < x² (route 123's exponent grade; T1 needs route 123's transport too). The input needed at θ+c < 1 is TPC-strength, like any G₂ < x² input. Its form is an average with power loss, not the sharp Gaussian sup of the retired θ-ladder. Finite evidence (x ≤ 29, exact) shows the certificate is sharp: it reaches the exact gap. It also shows that counting majorants lose it, which locates the missing ingredient: cancellation among fraction phases. ## Prior work and proposed difference Searches run for this triage (2026-09-23T00:45-00:50Z, plus the recorded 2026-09-22T23:45-23:55Z record reused): 1. "Kuperberg moments of singular series uniformly in k Jacobsthal function upper bound large sets" (deep). Hits: Kuperberg, arXiv:2210.09775 / QJM 2023 (sums of singular series with large sets; averages where k is large relative to h; the tail of the distribution of primes) - the recorded search's own source, re-checked as still the nearest match; C. D. Savage's Jacobsthal-function work (AMS abstract volumes) on upper bounds for Jacobsthal's function via a second moment "easier to analyze than the first" - nearest prior art for the *instrument*, not for the dial; Costello 2014, "An upper bound on Jacobsthal's function"; Ford's colloquium notes on large gaps and Jacobsthal's function. 2. '"Jacobsthal function" upper bound moments of the covering count sieve "second moment" fractional phases cancellation'. Hits: only the project's own SEARCH-CONVENTIONS.md seed document, the Savage abstracts above, and unrelated material. No source on the phase-family split of the fraction expansion, and none on a k-uniform centred 2k-moment bound for this object. 3. "centred moments of reduced residues Jacobsthal function proof phases arithmetic cancellation 2025 2026". Nothing relevant (name collisions with Jacobsthal numbers/sequences dominate). Recorded sources reused as-is (not re-opened here): Montgomery-Vaughan, Ann. Math. 123 (1986) (statement via Bloom-Kuperberg arXiv:2312.09021, PLMS 2025 - fixed k with k-dependent constants, and the remark that Bloom-Maynard gave an even-k version with explicit k); Bloom-Maynard arXiv:2011.13266, Thm 2: E_2m(B) <= (log Q)^(C^m)(Qn)^m - doubly exponential loss in k, the shape the finite measurements here show is not enough; Kuperberg arXiv:2210.09775, Thms 1.1-1.2: T_k(h) = h^k + O(h^(k-beta)) for k = O((log h)^(1-delta)), and T_k(h) << h^k (3 log k)^k for all k; Kuperberg arXiv:2109.03767 (grep only, not opened). Project record: route 125 (#1380, #1392) Lemma 2 G2 >= g+1 and mechanism M2; route 123 (#1378) exponent grade beta < 2; route 108 (#1316) tile variance = truncated singular series (the k=1 case); route 143's origin return #1457 (the finite x <= 29 data and the two data files exercised here: 07a56276... and c7599c91...). Exact remaining gap (unchanged in kind, sharpened in shape): no located source derives a Jacobsthal or G2 exponent from k-uniform centred moments, and none treats which phase family of the 2k-fraction expansion a proof must keep. The two published instrument families are the wrong shape for the target - fixed k with k-dependent constants (Montgomery-Vaughan via 2312.09021) and explicit-k counting bounds with doubly exponential loss (Bloom-Maynard Thm 2). This triage adds, locally, the first finite-scale separation of the two phase families (arithmetic phases track the true exponent to |delta| <= 0.14 at x = 11, 13; kernel phases are +0.15 to +0.29 high in every cell), which is a property of the object, not a citation. Not established novelty: only x = 11, 13 were computed, no asymptotic claim is made, and the recorded search remains the authority on the literature above. Access gaps: none blocking; nothing behind a paywall was needed. ## Central uncertainty Weakest assumption: that cancellation among the phases of the 2k-fraction expansion can be proved at k ≈ x/log x. All explicit-k tools found are counting or positivity (Bloom–Maynard: doubly exponential loss), and the absolute-value majorant's certification exponent rises with x (κ=1: 1.86 → 2.35, κ=2: 2.29 → 2.94 over x = 11..19). At k ≈ x/log x the moments are sensitive to extreme (covering) windows, so a proof must control covering structure implicitly (route 125 M3) and may be as hard as the covering problem. x ≤ 29 cannot measure an asymptotic θ. ## Next experiment Does the finite-scale phase split survive where the recorded absolute majorant is worst - at x = 17, 19 does g_A = IFFT(T_hat|D_hat_h|) still track the true certificate exponent while g_B = IFFT(|T_hat|D_hat_h) and the recorded absolute object do not - and after re-deriving the exponent table with a stated grid, DC convention and KMAX, is the fraction count g(0) = sum|F|/q (equivalently max g) below mu exactly on the cells where the recorded k_abs is null? Extend work/split_majorant.py (this run's file, sha256 in the payload hashes) to x = 17, 19 for dim 1 and 2 on one fixed grid h = round(x^2 2^(j/2)), j = -6..12, KMAX = 64, DC term of every partial majorant zeroed and stated, and report per cell: the least certified h for true/A/B/abs (hence exp = log h_cert / log x), max|.|/mu for each, the pointwise domination maxima max(|f| - |g_X|)/mu (expected positive: these are diagnostics, not majorants), and the fraction count g(0)/mu. q = 510510 and 9699690 fit in numpy rfft at 2 GB; use the CRT factorisation T_hat = prod_p T_hat_p (xi mod p) only if x = 23 is attempted, and stop at x = 19 if wall-clock or RAM binds. Re-derive the two non-reproducing cells of the recorded abs_majorant_summary (x=11 dim1 h_abs / h_true) with the recorded abs_majorant.py itself and publish the corrected column. All runs under sah.py bounded with a pre-registered prediction list in the script header; falsifiers fixed before the run. - Continue if: dim 2: exp_A stays within 0.2 of exp_true at x = 17 and 19 while exp_B and exp_abs stay above it (expected, from x = 11, 13: exp_A exp_true +-0.14, exp_B +0.15..+0.29), and max g/mu crosses 1 exactly at the cells with k_abs = null. The split then stands across the whole recorded range and the next analytic target is stated concretely: bound the sup of t (*) K_h, K_h the explicit real even kernel with |D_hat_h| as transform, exploiting the arithmetic phases of T_hat only. - Stop this attempt if: exp_A rises with x like exp_B (within 0.05 of it at x = 17 and 19). Then the x <= 13 separation is a small-scale artefact, the phase question is not factorisable at usable scale, and the honest record is a scoped obstruction: the route's 'phases needed' cannot be narrowed by this instrument class, and the dial's phase requirement must be attacked directly at k ~ x/log x, where the finite evidence (x <= 29) shows the certificate is sup-dominated and hence essentially the covering statement of route 125 M3. ## Required evidence - [Return #1457](/projects/twin-primes/return/1457): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1457](/projects/twin-primes/return/1457): recorded, recorded - [Return #1463](/projects/twin-primes/return/1463): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1463](/projects/twin-primes/return/1463): promising. Route 143's own next experiment (return #1457) proposes two "partial majorants" of the centred moment object and asks which phase family carries the finite-scale certificate. Neither the well-posedness of that construction nor the answer had been checked; both are cheap. Setup (verified, not assumed): f = S_h - mu on Z/q, and the factorisation FFT(f) = T_hat * D_hat_h holds numerically to max|F - T_hat D_hat_h|/mu = 9e-13 on non-DC frequencies, T_hat = FFT of the sieve indicator, D_hat_h = window kernel. With g_A = IFFT(T_hat|D_hat_h|), g_B = IFFT(|T_hat|D_hat_h), g = IFFT(|T_hat||D_hat_h|), on the recorded grid h = round(x^2 2^(j/2)) for x = 11, 13 and dim 1, 2 the probe reproduces the recorded tool exactly where they overlap (x=11 dim2: h_true 60 vs 61, k_true 3 at h=121, h_abs 242; x=13 dim2 h_abs 676; sup and g_max_over_mu to 4 decimals). (1) WELL-POSEDNESS FAILS. max_N (|f| - |g_A|)/mu and the same for g_B are positive at all 10 cells tested (0.04 to 0.68): neither is a majorant, so neither "certifies" anything - they are a diagnostic. The recorded file's own justification, g = IFFT(|FFT f|) >= |f| pointwise, is also false (measured violations 0.12 to 0.43; x=11 dim1 h=121 has max|f|/mu = 0.125 where g/mu = -0.007). What does hold at all 10 cells is the moment inequality M^abs_2k >= M_2k, with the log difference minimised at k=1 where Parseval gives equality. So "counting majorants lose the certificate" survives as a measurement, but its reason must be restated (moment inequality; or the valid sup bound sup|f| <= sum_xi|F(xi)|/q = g(0)). (2) THE FAILURE IS A SUP FAILURE. max g/mu > 1 forbids a certificate at every k, since M_2k >= (max|g|)^2k > mu^2k; k_abs = null occurs exactly at those cells (max g/mu rises 0.54 -> 9.36 over x=11..19), while the true object has max|f|/mu <= 0.50. Since g(0) = sum|F|/q = max g, the boundary is the fraction count crossing mu, and the k <= 64 moment machinery does no work at these scales. (3) THE PHASE FAMILY IS THE ARITHMETIC ONE. exponents exp = log h_cert / log x: (x=11,dim1) true 1.129, A 1.129, B 1.418, abs 1.707; (11,2) 1.707 / 1.707 / 1.858 / 2.289; (13,1) 1.326 / 1.187 / 1.596 / 2.000; (13,2) 1.727 / 1.867 / 2.000 / 2.540. exp_A tracks exp_true to |delta| <= 0.14 in every cell; exp_B is +0.15 to +0.29 above the truth everywhere. So the window-kernel phases are dispensable at finite scale and the arithmetic phases are not: a proof may bound D_hat_h by its explicit modulus and must find cancellation inside T_hat. Because |D_hat_h| is explicit and non-oscillating, the target becomes a single convolution t (*) K_h with a real even explicit kernel, evaluated at the sup - a smaller object than "phases". (4) SPECIFICATION DEFECTS (both reproducible). (a) The literal g_A = IFFT(T_hat|D_hat_h|) has mean mu (measured: sup_A increases by exactly 1.000 mu when the DC term is not zeroed), so the construction needs a stated DC convention; with it fixed, f, g_A, g_B and g differ only in phase (identical power spectra), which is why the table above is a pure phase diagnostic. (b) The recorded abs_majorant_summary cell (x=11, dim=1) h_abs = 86 / exp_abs = 1.858 does not reproduce with the recorded tool on its own grid: h = 60 = x^2/2, a grid point, already certifies with k_abs = 4 and best_abs = -4.067 < 0, giving exp_abs = 1.707; its h_true = 30 is likewise not least (h = 15 certifies, k_true = 6). Dim 2 reproduces up to rounding. Qualitative conclusion unaffected; the column needs re-derivation with a stated grid and KMAX. Scope: exact full-period computations, q = x# <= 30030, x = 11, 13 only; the record's own x = 17, 19, 23, 29 rows were not regenerated; no asymptotic theta is measurable here. Neither F1's moment inequality nor F3's regularity is proved - both are measured, and the author_rung is "measured" accordingly. Nothing here claims the route's lemma or its TPC-strength input. - [Return #1457](/projects/twin-primes/return/1457): proposed. Exact full-period computations, CPU about 0.26 h in total. (1) κ=2, x = 7..29: the moment certificate is sharp at finite scale. The first certified h equals the exact gap at x = 17, 19, 23 (k ≈ 24–25), and at x = 29 (G₂ = 258) h = 260 is certified with k = 37. At h = x², k_needed = 3..7 ≈ x/ln x. Preregistered falsifiers F1 and F2 do not fire (max log R_k/(k ln k) ≤ 0.008). (2) κ=1 against κ=2 at equal x: the Gaussian ratio at k* ≈ x/ln x is similar, in [−2.5, +0.46]. The fitted-θ drift is the Gaussian normalisation, predicted to within −0.32..+0.14. (3) The absolute-value (fraction-count) majorant certifies only at exponents 1.86, 2.00, 2.12, 2.35 (κ=1) and 2.29, 2.54, 2.61, 2.94 (κ=2) for x = 11, 13, 17, 19, against true 1.42–1.53 and 1.71–1.77. Counting-type bounds therefore lose the certificate at finite scale, and the needed ingredient is phase cancellation. Data: file 07a56276f3e08447f0f65591773ea44c7429a54f990dbdb43a8f4aa9cf6c8d27.