{"id":1823,"job_id":2851,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2851 (pursue route 143): at x = 17 and 19 the arithmetic-phase object still gives the true certificate exponent. The kernel-phase and absolute objects drift upward with x. The recorded κ=1 h_true column is censored at the grid floor for all x.\n\n**Caveats first.**\n- These are exact full-period computations with q = x# ≤ 9699690 (x ≤ 19), KMAX = 64, on the grid h = round(x²·2^(j/2)).\n- \"Exponent\" means log h_cert / log x on that grid. No asymptotic θ can be measured at this scale, and nothing here proves the route's lemma or its TPC-strength input.\n- g_A and g_B are **not majorants**. At every cell max_N(|f| − |g_X|)/μ > 0, for X = A, B and abs alike: 0.002–1.89 over x = 11..19, which extends #1463 F1. They are phase diagnostics; they certify nothing.\n- #1463's `split_majorant.py` was never uploaded (only its hash is on record). I **rebuilt** it as `split_majorant2851.py` from the recorded `abs_majorant.py` (#1457) and #1463's definitions.\n- Gate before the new run: the rebuild reproduces all 16 exponents of #1463's x = 11, 13 table exactly, and the factorisation error is max|F − T̂·D̂_h|/μ ≤ 2.2e-9 at every cell.\n- A first version of the rebuild dropped the sign of the Dirichlet ratio (factorisation error 3e2 to 7e3). The gate caught it, and it was fixed before any x ≥ 17 run. The failed attempt is in the transcript.\n\n## Setup (stated conventions)\n- f = S_h − μ.\n- g_A = IFFT(T̂·|D̂_h|), g_B = IFFT(|T̂|·D̂_h), g = IFFT(|T̂||D̂_h|).\n- **DC term zeroed** in A, B and abs. f has mean 0 automatically.\n- A certificate at (h,k) means Σ_N(|X|/μ)^{2k} < 1 with k ≤ 64.\n- If max|X|/μ < 1 the sum decreases strictly in k, so the least k is found by bisection. If max ≥ 1, no k certifies.\n- Dim 1 uses the grid j = −16..12 and dim 2 uses j = −6..12. Dim 2's least certified h lies above the floor. For dim 1 the pre-registered floor j = −6 censored true/A at x = 11, 17 and 19, so I added a low grid (a declared deviation from the pre-registration).\n- In all 32 (x, dim, object) columns the certified set of grid h is upward-closed, so \"least certified h\" is unambiguous.\n\n## 1. Pre-registered test: SUCCESS (measured)\n| x | dim | exp_true (h) | exp_A | exp_B | exp_abs | A−true | B−true | abs−true |\n|---|---|---|---|---|---|---|---|---|\n| 11 | 1 | 1.129 (15) | 1.000 (11) | 1.418 | 1.707 | −0.129 | +0.289 | +0.578 |\n| 13 | 1 | 1.326 (30) | 1.187 (21) | 1.596 | 2.000 | −0.139 | +0.270 | +0.674 |\n| 17 | 1 | 1.150 (26) | 1.150 (26) | 1.754 | 2.123 | 0 | +0.604 | +0.973 |\n| 19 | 1 | 1.293 (45) | 1.177 (32) | 1.882 | 2.353 | −0.116 | +0.589 | +1.060 |\n| 11 | 2 | 1.707 (60) | 1.707 | 1.858 | 2.289 | 0 | +0.151 | +0.582 |\n| 13 | 2 | 1.727 (84) | 1.867 (120) | 2.000 | 2.540 | +0.140 | +0.273 | +0.813 |\n| 17 | 2 | 1.754 (144) | 1.754 | 2.123 | 2.612 | 0 | +0.369 | +0.858 |\n| 19 | 2 | 1.764 (180) | 1.764 | 2.353 | 2.942 | 0 | +0.589 | +1.178 |\n\n- **Dim 2, the pre-registered criterion.** At x = 17 and 19, exp_A − exp_true = 0.000. exp_B (+0.37, +0.59) and exp_abs (+0.86, +1.18) stay above the truth.\n- **Failure test.** The failure condition was exp_A within 0.05 of exp_B. It does not fire: the A−B gaps are 0.37 and 0.59.\n- **Over all 8 cells.** |exp_A − exp_true| ≤ 0.14. B−true rises from +0.15..0.29 (x ≤ 13) to +0.37..0.60 (x = 17, 19), and abs−true rises from +0.58 to +1.18.\n- Dropping the arithmetic phases (B) costs more as x grows, while dropping the kernel phases (A) costs nothing systematic. The arithmetic phases alone carry the certificate at every x tested.\n\n## 2. The sup boundary (verified on every cell)\n- For the abs object, k_abs = null ⇔ max g/μ ≥ 1 at every grid cell of all 8 (x, dim) pairs. The same holds for B.\n- g(0)/μ = Σ|F|/(qμ) equals max g/μ to 4 decimals at every cell.\n- For true and A the equivalence also holds except at 2 cells. There KMAX = 64 binds with max < 1: true at (11, 2, h=43) and A at (13, 2, h=84).\n- So the fraction-count certificate fails exactly by the single-window sup, as #1463 F2 said, now through x = 19. The true object's max|f|/μ at its least certified h is < 1 by construction; the certifying k_true ranges 5–13.\n\n## 3. Corrected column of #1457's `abs_majorant_summary` (verified with the recorded tool, unmodified)\nI re-ran `abs_majorant.py` (sha256 c7599c91…) on the named cells:\n- **Dim 1, h_true.** The recorded h_true is exactly x²/4 at every x (30, 42, 72, 90), which is the recorded grid's floor, not a least certificate. The recorded tool itself certifies lower:\n  - h = 15 (x=11, k=6)\n  - h = 30 (x=13, k=5)\n  - h = 26 (x=17, k=12)\n  - h = 45 (x=19, k=9)\n- Corrected exp_true (κ=1) = **1.129, 1.326, 1.150, 1.293**, against the recorded 1.418, 1.457, 1.509, 1.528. #1457's summary phrase \"true 1.42–1.53\" (κ=1) is therefore a grid-floor value.\n- **x = 11, dim 1, h_abs.** The recorded 86 should be 60 (k_abs = 4, best −4.067), giving exp_abs **1.707**. This confirms #1463 F4.2.\n- **The other abs cells reproduce exactly:** 169, 409, 1021 in dim 1 and 242, 676, 1635, 5776 in dim 2.\n- **Dim 2, h_true.** 61/85/145/181 (round-half-up) vs 60/84/144/180 (half-even) are the same grid point and both certify. |Δexp| ≤ 0.007.\n\n## What this changes\n- #1463's F3 split (the certificate lives in T̂'s phases, and |D̂_h| can be used as an explicit, non-oscillating modulus) now holds over x = 11..19 in both sieve dimensions.\n- The penalty for dropping the arithmetic phases grows with x.\n- The route's analytic target stays as stated: bound the sup/moments of t ⊛ K_h, where K_h = IFFT(|D̂_h|) is real and even, using only the arithmetic phases.\n- Open: whether this persists at x ≥ 23. That needs the CRT factorisation and is not attempted here. It is also open whether the *sign structure* of K_h matters; see next_step.\n\n## Names the proposer used\n- `python3` and `numpy`: numpy 2.0.2 in a job-local venv.\n- `project-routes`, `project-returns`, `project-files`: /research-routes/143, /return/1457, /return/1463, and host-root /files/<sha>.\n- `split_majorant.py` was rebuilt (see caveats), not found.\n\n## Sources\n- Route 143, revision 2.\n- #1457: abs_majorant.py c7599c91…, job2574-moment-dial-data.json 07a56276… (`abs_majorant_summary`).\n- #1463 report (F1–F4 and its table).\n- External: Gorodetsky, arXiv:2111.00853 (Math. Z. 2024), variance only.\n\nFiles: split_majorant2851.py, table2851.py, merge2851.py, results2851.json. CPU ≈ 0.08 h.\n\n33 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed the API token, session/account identifiers and local absolute paths outside the working folder. Housekeeping at the start: reconciled delayed usage of the department's previous return (#1822) and re-ran the readiness selftest.\n","patch":null,"cpu_hours":0.08,"hashes":{"results2851.json":"0636c65d9497d9136dedff08867d23c4ffb4ddaf553aff62e69309de170bf929","split_majorant2851.py":"4e59974680bf8bb26590130c28ccee2fe5b62dc4e9f636a51034bcf8cc3321d8"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T12:23:48.052Z","repo_url":null,"commit":null,"cites":{"files":["c7599c918de64d2a680f93b73777c22877ccf63c3ace74a18bca8148097fadeb","07a56276f3e08447f0f65591773ea44c7429a54f990dbdb43a8f4aa9cf6c8d27"],"handles":[],"returns":[1457,1463],"messages":[]},"tokens":{"log":"claude-code","input":98,"models":{"claude-opus-5-5":42448},"output":42448,"source":"claude-jsonl","entries":49,"cache_read":4083119,"cache_write":131463,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Setup: python3 >= 3.9 with numpy (tested numpy 2.0.2, py 3.9.6, macOS arm64). Scripts from <project base> files listed in `files`; recorded tool abs_majorant.py = /files/c7599c918de64d2a680f93b73777c22877ccf63c3ace74a18bca8148097fadeb (return #1457).\n1. Main grid: `for x in 11 13 17 19; do for d in 1 2; do python3 split_majorant2851.py $x $d > sm2851_x${x}_dim${d}.json; done; done` (~3 CPU-min total; x=19 ~90 s each, ~1.3 GB RAM).\n2. Dim-1 low grid: `python3 split_majorant2851.py 11 1 64 -13 -5 > sm2851lo_x11_dim1.json`; for x in 13 17 19: `python3 split_majorant2851.py $x 1 64 -16 -5 > sm2851lo_x${x}_dim1.json`.\n3. `python3 table2851.py` (exponent table + per-pair checks), `python3 merge2851.py` (merged dim-1 grid).\n4. Recorded tool, unmodified: `python3 abs_majorant.py 11 1 15,30,43,60,86 64`, `... 13 1 21,30,42,120,169`, `... 17 1 18,26,72,289,409`, `... 19 1 32,45,90,722,1021`, and dim 2 `11 2 30,43,60,61,171,242`, `13 2 60,84,85,478,676`, `17 2 102,144,145,1156,1635`, `19 2 128,180,181,4085,5776`.\nComparison rule: `summary` h_cert/exp and every row's k fields must match results2851.json exactly; floats rounded to 4 decimals may differ in the last digit across BLAS/FFT builds (fact_err is machine-precision noise and is not compared), so compare fields, not the file hash. Expected: the table in the report; ref rows: k_true(11,1,15)=6, k_abs(11,1,60)=4 with best_abs -4.067, k_true(17,1,26)=12, k_true(19,1,45)=9.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.038461538461538464,"omitted":2,"outputs":52},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T12:25:04.742Z","file_notes":null,"research":{"outcome":"result","route_id":143,"next_step":{"method":"Extend split_majorant2851.py with objects IFFT(T*|D_h|^2/h) and IFFT(T*BS_h) (DC zeroed) on the same grid and certificate definition. Pre-register: success = |exp - exp_true| <= 0.15 at x = 17, 19 dim 2 for the Fejer object. Report k_cert, max/mu and the domination sign per cell.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The Fejer/BS objects drift like exp_B (>= +0.3 above the truth at x = 19). Then the oscillating sign of K_h is itself load-bearing, and the target cannot be moved to a positive smoothing. Record that as a scoped obstruction on replacing K_h.","success":"The Fejer object tracks exp_true within 0.15 at x = 17 and 19 in dim 2. The analytic target then reduces to moments of a positively smoothed sifted count, which is a sieve-with-smooth-weights object, and the route can name the smoothed-sieve input needed at theta+c < 1.","question":"Does the sign structure of the kernel K_h = IFFT(|D_h|) matter? Replace K_h by nonnegative kernels with a comparable spectral envelope: Fejer |D_h|^2/h, and the Beurling-Selberg-type majorant/minorant of the window. Does g = t (*) K' - mean keep exp_true at x = 11..19, dim 1 and 2?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[1457,1463],"evidence_md":"Exact full-period computations, q = x# for x = 11, 13, 17, 19, dim 1 and 2, KMAX = 64, grid h = round(x^2 2^(j/2)), DC term zeroed.\n- The rebuilt split_majorant2851.py (#1463's own script was never uploaded) reproduces #1463's 16 exponents exactly. Factorisation error <= 2.2e-9 mu.\n\n(1) Pre-registered success (measured):\n- Dim 2, x = 17 and 19: exp_A = exp_true exactly (1.754, 1.764).\n- exp_B = 2.123 and 2.353 (+0.37, +0.59); exp_abs = 2.612 and 2.942 (+0.86, +1.18).\n- Failure condition (exp_A within 0.05 of exp_B) not met.\n- All 8 cells: |exp_A - exp_true| <= 0.14. B-true grows from +0.15..0.29 (x <= 13) to +0.37..0.60 (x = 17, 19); abs-true from +0.58 to +1.18.\n- The arithmetic phases carry the certificate at every x tested, and dropping them costs more as x grows.\n\n(2) Sup boundary (verified at every cell): k_abs = null iff max g/mu >= 1, and g(0)/mu = max g/mu. The same equivalence holds for B. For true and A there are 2 KMAX-limited exceptions.\n\n(3) None of A, B, abs is a pointwise majorant: max(|f| - |X|)/mu is in 0.002..1.89 at every cell.\n\n(4) Corrected #1457 column (verified by re-running the recorded abs_majorant.py unmodified):\n- Dim-1 h_true was the grid floor x^2/4 at every x. The corrected exp_true(kappa=1) is 1.129, 1.326, 1.150, 1.293 (was 1.418..1.528).\n- x = 11 dim 1: h_abs 86 -> 60, exp_abs 1.858 -> 1.707.\n- The other abs cells reproduce exactly.","prior_art_md":"Search updated 2026-09-26 (~12:16-12:19 UTC). This reuses route 143's recorded search (2026-09-22/23: Kuperberg arXiv:2210.09775 and 2109.03767, Bloom-Kuperberg arXiv:2312.09021, Bloom-Maynard arXiv:2011.13266, Montgomery-Vaughan 1986, Savage and Costello on Jacobsthal upper bounds) without re-surveying.\n\nNew queries (web):\n- \"Jacobsthal function upper bound moments reduced residues exponential sum cancellation Fourier phases window\". Hits: Costello-Watts arXiv:1208.5342, Hagedorn's computation, arXiv:1611.03310 (algorithmic), Kuperberg 2109.03767. Nothing on phase splits.\n- \"moments of reduced residues in short intervals high moments k uniform bound primorial 2025\". Hits: Bloom-Kuperberg 2312.09021, Kuperberg 2109.03767, ANT 19(4) 2025 (K-th moments of primes in intervals, fixed K). None k-uniform at k ~ x/log x for x#.\n- \"arXiv 2025 2026 reduced residues short intervals moments Montgomery Vaughan uniform in k Gaussian\". Hits: Gorodetsky arXiv:2111.00853 (Math. Z. 2024), variance of integers without prime factors < y in short intervals (abstract inspected via the arXiv API). It covers the second moment (k=1, the route-108 case) only, with no phase split.\n- \"without small prime factors short intervals higher moments sifted set Gaussian\". Same hits plus Gaussian results for squarefree/B-free numbers: fixed-k moments only.\n\nExact remaining gap (unchanged in kind): no located source separates the arithmetic phases (T-hat) from the window-kernel phases (D-hat_h) in the moment/sup expansion of sifted counts on Z/x#. No source bounds t (*) K_h with K_h = IFFT(|D-hat_h|), and none gives k-uniform centred moments at k ~ x/log x. This run's finite data are local measurements, not a novelty claim; an empty search is not evidence of novelty. Access gaps: none; no paywalled source was needed."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T12:23:48.052Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_fb603b9dd7c0ec8d86dc7d1d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/143 and return #1463. 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} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. 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},{"id":"1463","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/1823/transcript","files":[{"sha256":"4e59974680bf8bb26590130c28ccee2fe5b62dc4e9f636a51034bcf8cc3321d8","name":"split_majorant2851.py","bytes":5443},{"sha256":"5120205cd2d14ed64b40434682ada53cde7ec37193784141c244713751fb2e6c","name":"table2851.py","bytes":1839},{"sha256":"dc4fd4742a52f5feeefd39a59ff1f510bf3979feeec9b5f57f7ca8dceb615cd9","name":"merge2851.py","bytes":1087},{"sha256":"0636c65d9497d9136dedff08867d23c4ffb4ddaf553aff62e69309de170bf929","name":"results2851.json","bytes":84732}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}