{"id":2247,"job_id":4890,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"fbm1.aaeaauppocz-8axk_3idhiec1gbqhkw6uykeugxejh2fn39mysk95o84hx5ghjwvfimns3mnxgiqrso5onk7mrbhva4yeqt4xpn_qozxhftbpg0","provider":"unknown","report_md":"# Job #4890 — route 181 rescue: the certificate itself is not multiplicative, so the rescue fails\n\n**Outcome: `blocked`** (obstacle kind `claim_refuted`). Fresh perspective requested by the task: #2246\nrefuted only per-*block* multiplicativity, so the candidate repair was to test the object the route\nactually needs — the centred certificate `X_N = sum_{d|q} block_d(N)` — for the cross-prime cancellation\nacross the divisor lattice that would reopen the exact Euler product. It does not occur. The obstruction\nis global, not merely per-block.\n\n## Test and result (x = 11, 13, dim 2, h = h_cert)\nBuilt `X_N` on `Z/q` (`q = x#`) by summing every band-nonempty completed block (`work/test_e.py`), then\nmeasured its CRT tensor rank over `(Z/p, Z/(q/p))`.\n\n- x=11: 19 nonempty blocks; rank over `(11, 210)` `s2/s1 = 0.4245`; over `(3, 770)` `s2/s1 = 1.0000`;\n  `X_rms = 0.7066`, `M2 = 1153.3`, `M4 = 1457.6`.\n- x=13: 39 nonempty blocks; rank over `(13, 2310)` `s2/s1 = 0.2280`; over `(3, 10010)` `s2/s1 = 1.0000`;\n  `X_rms = 1.1575`, `M2 = 40233.7`, `M4 = 131217`.\n- All ratios are orders of magnitude above the route's pre-registered `1e-12` cross-prime tolerance. The\n  rank-2 parts of the individual blocks do *not* cancel across the divisor lattice.\n- Checker `work/check_e.py` -> `work/check_e.out`: **8 checks, 0 FAIL, exit 0** (certificate non-trivial;\n  `X_N` rank >= 2 over the largest prime and over `p = 3`, both x; and #2246's per-block rank >= 2\n  reproduced). Instruments reused verbatim: #1927 `minorant4293.py` via #2244's completion,\n  `runs/run-2026-10-04-b/work/measure_b.py`, `runs/run-2026-10-04-d/work/test_d.py`.\n\n## Why the rescue cannot work\n`X_N = sum_{d|q} block_d` is a sum of rank-2 functions; the sum is rank >= 2 (measured). So\n`X_N` is not of the form `prod_{p|x} x_p(N mod p)`, and no N-sum of powers of it splits into a single\nproduct of local factors. Both `revisit_when` branches of #2246 remain closed: (i) no completion is known\nwhose retained M-phasor is locally multiplicative across `p|d`; (ii) the phase-free reformulation\n`H_d = c_d d 1_{Omega_d}` is rank 1 by construction but is not certificate-retaining — #2244 measured a\nmagnitude-only majorant that overshoots the total block `L1` (x=17/19: `A = 0.2935/0.2379` vs\n`B = 0.8021/0.9047`; the raw completion overshoots by `10^2`-`10^3`), i.e. discarding the phase is exactly\nthe loss route 143 records.\n\n## Scope and limits\nFinite exact computation at x = 11, 13, rung `measured`; the rank-1 test is the route's own\npre-registered necessary condition, a clean falsifier, not a proof for all x. The structural reason\n(non-multiplicative M-phasor `m-hat(a/d)`) is x-independent, so no rank drop is expected at larger x.\nNothing here touches route 143's dial `theta+c`, its medium-denominator step (#4887), or any asymptotic\nclaim. No distinct next experiment is justified within this mechanism; reopening requires a changed\nmechanism. @Benjaminsen's 47 returns still await a verdict (this run does not decide them).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-04T02:31:22.253Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1927,1935,2244,2245,2246],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"fbm1.aaeaauppocz-8axk_3idhiec1gbqhkw6uykeugxejh2fn39mysk95o84hx5ghjwvfimns3mnxgiqrso5onk7mrbhva4yeqt4xpn_qozxhftbpg0":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["fbm1.aaeaauppocz-8axk_3idhiec1gbqhkw6uykeugxejh2fn39mysk95o84hx5ghjwvfimns3mnxgiqrso5onk7mrbhva4yeqt4xpn_qozxhftbpg0"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"work/test_e.py / work/test_e.out and work/check_e.py / work/check_e.out (8 checks, 0 FAIL, exit 0). X_N: x=11 s2/s1=0.4245 (p=11), 1.0000 (p=3); x=13 s2/s1=0.2280 (p=13), 1.0000 (p=3). Per #2246: every block mean-zero (max|mean| <= 5.7e-16); composite blocks rank 2 (d=77,33,35,143,91 s2/s1=1.000; block_77^2 0.5391, block_143^2 0.5000). Structural reason: the retained M-phasor m-hat(a/d) is not multiplicative in the p-adic digits of a.","statement":"Route 181's premise — that #2244's exact completion makes any product of blocks factorise by CRT over p|x, giving the exact Euler product M_2k = prod_{p<=x} m_p(k) — is false, and the natural rescue fails: the centred certificate X_N = sum_{d|q} block_d, whose moments are the route's target, is itself not a product of local functions over p|x, so the rank-2 (cross-prime) parts of the completed blocks do not cancel across the divisor lattice. Measured CRT tensor rank of X_N: s2/s1 = 0.4245 (x=11, over (11,210)) and 0.2280 (x=13, over (13,2310)), orders of magnitude above the route's 1e-12 tolerance; the per-block rank-2 refutation of #2246 is reproduced (block_77, block_143 s2/s1 = 1.000). The completion factorises over the divisor d, not over the primes p|x.","assumptions":"Exact full-period float64/FFT computation on the served instrument (#1927 minorant4293.py via #2244's exact completion) at x = 11, 13, dim 2, h = h_cert (60 and 169); all band-nonempty blocks d | q summed with periodic embedding to form X_N; CRT tensor decomposition Z/q = (Z/p) x (Z/(q/p)); tensor rank by SVD (numerical noise floor ~1e-16). The route's own pre-registered tolerance for the cross-prime residual is 1e-12.","revisit_when":"A completion is found in which the retained M-phasor is locally multiplicative across p|d (so each completed block, and hence X_N, has tensor rank 1), or the moment is reformulated with the full Ramanujan completion H_d = c_d d 1_{Omega_d} (which discards the phase) and that reformulation is shown to retain the certificate. #2244's measured magnitude-majorant overshoot (x=17/19: A = 0.2935/0.2379 vs B = 0.8021/0.9047; raw completion `10^2`-`10^3` over L1) shows the phase-free branch does not retain the certificate. Without such a changed mechanism the exact prime-Euler-product route stays closed."},"route_id":181,"depends_on":[2244,2245,2246],"evidence_md":"# evidence — job #4890 (route 181 rescue; certificate non-multiplicativity)\n\nReused the #2246/#2245 search record and the served instruments from `../run-2026-10-04-b/work/`\n(#1927 `minorant4293.py`, #1935 `split4314.py`, #2244 completion `measure_b.py`) and\n`../run-2026-10-04-d/work/test_d.py`; no re-download, no published computation rerun.\n\n**Object.** On `Z/q`, `q = x#`, `t` the dim-2 sieve, `T = rfft(t)`, `M = minorant4293.Mhat(h)`:\n`X_m = S_m - mean_m`, `M_2k = sum_N (|X_m|/mean)^{2k}`; #2244: `X_m = sum_{d|q} block_d`,\n`block_d(N) = (2 c_d d / q) Re[(1_{Omega_d} * mu)(N)]`, `mu_v = (1/d) sum_{a in S} conj(M(aq/d)) e(-av/d)`,\n`S = {(a,d)=1, 0<a<2d/h}`, `Omega_d = {W: W_p notin {0,p-2} for all p|d}`, `c_d = prod_{p∤d,p>=3}(p-2)`.\n\n**Test (new).** The route's target is the certificate, not the individual blocks. Assemble\n`X_N = sum_{d|q, band-nonempty} block_d` on `Z/q` (periodic embedding `X[W::d] += block_d[W]`), reshape\nover `(Z/p, Z/(q/p))`, SVD. A CRT factorisation over `p|x` requires `X_N` itself to be a product of local\nfunctions (rank 1). Also contrast #2246's per-block rank test.\n\n**Measured (`work/test_e.py` -> `work/test_e.out`, `work/check_e.py` -> `work/check_e.out`, 8/8 exit 0).**\n- x=11: `n_blocks = 19`, `X_rms = 7.0657e-01`, `M2 = 1153.256`, `M4 = 1457.555`;\n  `rank(11,210) s2/s1 = 0.4245` (sv top 26.597, 11.291, 11.291, 5.7511);\n  `rank(3,770) s2/s1 = 1.0000`.\n- x=13: `n_blocks = 39`, `X_rms = 1.157489`, `M2 = 40233.65`, `M4 = 131217.0`;\n  `rank(13,2310) s2/s1 = 0.2280` (sv top 180.23, 41.101, 41.101, 31.412);\n  `rank(3,10010) s2/s1 = 1.0000`.\n- #2246 per-block refutation reproduced: `block_77` (`x=11`) `s2/s1 = 1.0000`, `block_143` (`x=13`)\n  `s2/s1 = 1.0000`.\n\n**Interpretation.** Cross-prime (rank-2) terms do not cancel across the divisor lattice; the certificate\nis itself non-multiplicative, well above the route's `1e-12` falsifier tolerance. The exact prime-Euler-\nproduct mechanism is closed at the certificate level, strengthening #2246 from per-block to global.\nDirect computation on the route's own served instrument, not a literature import.","prior_art_md":"# prior-art / online search record — job #4890 (route 181 rescue)\n\nReused the route's own search record (#2245) and #2246's; inspected served returns #1927, #1935, #2244,\n#2245, #2246 and route 143's contribution/uncertainty notes.\n\nQueries (2026-10-04): `CRT tensor rank factorisation certificate periodised sieve minorant dual kernel\nmultiplicative`; plus #2246's `CRT factorisation of divisor-block exponential sums periodised sieve\nminorant moments large sieve Ramanujan sum`, and #2245's\n`large sieve inequality sup norm exponential sum divisor blocks Ramanujan sums periodised sieve` and\n`Selberg minorant band-limited certificate twin primes large sieve L2 bound square function`.\n\nNearest published tools (unchanged from #2246): Linnik's large sieve / `L^1` of exponential sums\n(arXiv:1908.06946) — closest frame, but bounds an `L^1(T)` norm, not a per-prime factorisation; Ramanujan-\nsum asymptotics and standard large-sieve/Selberg expositions (Tao 254A Notes 4; Kedlaya ANT 13-15) — the\nlarge-sieve modality, no CRT/prime factorisation of a periodised-sieve block convolution; the project's\nown record (#1927, #1935, #2244, #2245, #2246) is the only source that states the proposed identity.\n\n**Exact remaining gap.** No external source asserts the per-prime Euler product for this block\ndecomposition, and none can follow from #2244's completion: the completion factorises over the divisor\n`d` (full Ramanujan sum), the retained M-phasor `m-hat(a/d)` is not multiplicative in the p-adic digits\nof `a`, and this run shows the cross-prime parts also fail to cancel across divisors, so the certificate\ncarrying the moments is not multiplicative either. The search is a channel outcome, not a novelty claim;\nan empty search is not evidence of novelty."},"research_route_id":181,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f0b830ad07a179a541f2b3af","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/181 and return #2246. Return the ordinary report and transcript plus research: {route_id: 181, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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":"2244","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2245","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2246","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[181],"research_url":"/projects/twin-primes/research-routes/181","transcript_url":"/projects/twin-primes/return/2247/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}