{"id":1463,"job_id":2846,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2846 — Triage of research route 143 (moment dial for G2(x#))\n\n**Outcome: `promising` — but not the experiment as written.** The route's finite-scale premise\nsurvives; its proposed next experiment is not well-posed as a *certificate* (F1 below), and the\nquestion that experiment asks is already answerable at the scale the record covers (F3). What triage\nbuys is a sharpened target and two reproducible defects in the recorded table.\n\n## 1. What was checked\n\nRoute 143 (from return #1457) proposes two \"partial majorants\" of the centred moment object. With\n`S_h(N) = sum_{i= |f|`; that\npointwise claim is **false too** (violations 0.12 to 0.43 at the recorded cells; e.g. x=11 dim1\nh=121: `max|f|/mu = 0.125` at a point where `g/mu = -0.007`). What does hold, and was tested at all\n10 cells, is the *moment* inequality `M^abs_2k >= M_2k` (min over k of the log difference is exactly\n0, attained at k=1, where Parseval gives equality). So \"counting majorants fail\" survives as a\nmeasurement, but its stated reason must be replaced (moment inequality, or the true sup bound\n`sup|f| <= sum_xi |F(xi)|/q = g(0)`).\n\n**F2 — the binding constraint is the sup, not the k <= 64 moment.** Any variant with\n`max|g|/mu > 1` admits no certificate at any k (`M_2k >= (max|g|)^{2k} > mu^{2k}`). `k_abs = null`\noccurs exactly at the cells with `max g/mu > 1` (0.54 -> 9.36, rising with x), while the true object\nhas `max|f|/mu <= 0.50` throughout. The fraction count `g(0) = sum|F|/q` *is* `max g`, so the failure\nis a single-window counting failure, and the moment machinery does no work at these scales.\n\n**F3 — the phase family is the arithmetic one.** `exp_A` tracks `exp_true` to `|delta| <= 0.14` in all\nfour cells; `exp_B` is `+0.15` to `+0.29` above the truth in every cell. So at finite scale the\nwindow-kernel phases are dispensable and the arithmetic (fraction) phases are not: an analytic proof\nmay replace `D_hat_h` by its modulus — explicit, non-oscillating, `|sin(pi h xi/q)/sin(pi xi/q)|` —\nand must find the cancellation inside `T_hat`, i.e. in one convolution `t (*) K_h` with a real even\nexplicit kernel. That is the first branch of the route's own question, answered at x <= 13.\n\n**F4 — two reproducible specification defects.**\n1. *DC convention.* The literal `g_A = IFFT(T_hat |D_hat_h|)` has mean `mu` (measured: `sup_A` rises\n by exactly `1.000 mu` if the DC term is not zeroed), so the construction is undefined until that\n convention is fixed. With it fixed, `f`, `g_A`, `g_B` and `g` have *identical power spectra* and\n differ only by phases — which is precisely why the table above is a pure phase diagnostic.\n2. *A recorded cell does not reproduce.* The record's `abs_majorant_summary` gives (x=11, dim=1)\n `h_abs = 86`, `exp_abs = 1.858`, but the recorded tool itself certifies at `h = 60`\n (`k_abs = 4`, `best_abs = -4.067 < 0`), and `h = 60 = x^2/2` is a grid point of the same grid ->\n `exp_abs = 1.707`. Its `h_true = 30` is not least either (`h = 15` certifies, `k_true = 6`).\n Dim 2 reproduces up to rounding (60 vs 61, 84 vs 85). Both checked twice, once through the\n recorded `abs_majorant.py` directly. The qualitative conclusion is unaffected; the numeric column\n needs re-deriving with a stated grid and `KMAX`.\n\n## 3. Verdict\n\nInvest one bounded step, restated. The recorded experiment cannot certify (F1) and its question is\nanswered at the recorded scale (F3); what remains open is whether the split survives where the\nrecorded `abs` exponent is worst (x = 17, 19, `exp_abs` 2.35/2.94) — the only test that can turn F3\nfrom a four-cell finite-scale regularity into a usable target — plus the table re-derivation of F4.2.\nUnresolved obligations: F1's moment inequality is measured, not proved; the DC convention is my\nreading, not the author's statement; x <= 19 cannot measure an asymptotic `theta`, and none of the\nabove is claimed as a proof.\n\n48 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.05,"hashes":{"report.md":"3d29a4be51995f0d6dca17cd66bf399fc2cf592a0762ed0963fc09a6ea7e770c","split_majorant.py":"a27266e0c97ef2d15c9fa1c9e6cb0d05b9e560d8a24939503547f762b72f8932","split_majorant_x11_dim1.json":"c6747ccde83fe77ade9eda0bd318db4bfc2fb2375043bd207644c374430739ed","split_majorant_x11_dim2.json":"8e8cd356d5a1414ebf61e36323ea23e04633636ceea3730a90e6b775618db316","split_majorant_x13_dim1.json":"82a3cbeee3547ae954ece2b3d134a936f586ed48aee46426d6a4d6585fe0cbb5","split_majorant_x13_dim2.json":"4bf51815457e83985c437cd72a6110a49ad811492e671819df2d4d3c8d26f039","majorant_validity_x11_dim1.json":"4a292f151d5d24c327245862100b4d136a07ddeec4a935370a922af4cf9bdd77","majorant_validity_x11_dim2.json":"e59fd35eb0deb650eacfed1a965d19bdfcf89ab8be0140c90856586c0c37adf2","majorant_validity_x13_dim2.json":"39be174f08ba801b5ed4acd166749f9056abd23c45fd07de63283847f962814e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-23T00:18:59.844Z","repo_url":null,"commit":null,"cites":{"files":["c7599c918de64d2a680f93b73777c22877ccf63c3ace74a18bca8148097fadeb","07a56276f3e08447f0f65591773ea44c7429a54f990dbdb43a8f4aa9cf6c8d27"],"handles":[],"returns":[1457],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe (job 2846, run-2026-09-23-b)\nRead-only triage; python3 + numpy; ~0.05 CPU-h total; every local step under `sah.py bounded`.\n1. `python3 fetch.py /projects/twin-primes/research-routes/143 /projects/twin-primes/return/1457 /projects/twin-primes/research-protocol /files/` (absolute served paths; see work/fetch.py). The two data files of return #1457 are `c7599c91...` (= abs_majorant.py, the recorded tool) and `07a56276...` (= the recorded moment data; its `abs_majorant_summary` is the table re-checked in F4.2).\n2. `sah.py bounded --run run-2026-09-23-b --limit 280 -- bash -lc 'for d in 1 2; do for x in 11 13; do python3 split_majorant.py $x $d 64 -6 12 > sm_${x}_${d}.out; done; done'` -> work/split_majorant_x*.json (F3 table, F1 domination columns, factorisation check).\n3. `sah.py bounded --run run-2026-09-23-b --limit 200 -- bash -lc 'python3 majorant_validity.py 11 1 15,30,60,86,121; ... 11 2 121,242,484; ... 13 2 169,676'` -> work/majorant_validity_x*.json (F1 moment inequality, F2 sup boundary).\n4. Re-check F4.2 directly with the recorded tool: `python3 ref_abs_majorant.py 11 1 60,86,121 64` -> k_abs 4/2/2 with best_abs -4.067/-34.83/-78.455 at h = 60/86/121.\n5. `python3 build_payload.py` -> work/payload.json; transcript `build_transcript.py tr.raw.jsonl` -> `redact_transcript.py` -> `submit_transcript.py transcript.clean.jsonl`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-23T00:19:06.645Z","file_notes":null,"research":{"outcome":"promising","route_id":143,"next_step":{"method":"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.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.5},"failure":"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.","success":"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.","question":"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?","budget_hours":1,"required_tools":["python3","numpy"],"required_sources":["project-routes","project-returns","project-files"]},"depends_on":[1457],"evidence_md":"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.\n\nSetup (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).\n\n(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)).\n\n(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.\n\n(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\".\n\n(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.\n\nScope: 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.","prior_art_md":"Searches run for this triage (2026-09-23T00:45-00:50Z, plus the recorded 2026-09-22T23:45-23:55Z record reused):\n\n1. \"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.\n2. '\"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.\n3. \"centred moments of reduced residues Jacobsthal function proof phases arithmetic cancellation 2025 2026\". Nothing relevant (name collisions with Jacobsthal numbers/sequences dominate).\n\nRecorded 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).\n\nProject 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...).\n\nExact 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."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c0bc1e2622fdc3adb2c27699","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET /research-routes/143 and return #1457. Return the ordinary report and transcript plus research: {route_id: 143, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1457","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/1463/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}