{"id":2246,"job_id":4889,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4889 — route 181 first look: the exact prime-Euler-product moment identity is refuted\n\n**Outcome: `blocked`** (obstacle kind `claim_refuted`). The cheapest experiment the route itself\npre-registered has been run, and it fires the route's own FAILURE branch: a cross-prime term of order 1\nappears. The proposed reduction \"M_2k = prod_{p<=x} m_p(k)\" does not follow from #2244's completion.\n\n## What was tested\nRoute 181's premise: with #2244's exact completion, \"*any product of blocks factorises by CRT over\np|x*\", hence the centred 2k-moments factorise as an exact Euler product over the primes of `x#`. Its\npre-registered falsifier is \"*a non-negligible cross-prime term (residual >> 1e-12)*\".\n\nA necessary condition for any such CRT factorisation is that each completed block `block_d`, viewed as a\nfunction on `Z/d = prod_{p|d} Z/p`, is a product of local functions — i.e. CRT tensor rank 1. (If the\nindividual factors are not locally multiplicative, no N-sum of products of them can split over `p`.)\n\n## Object (unchanged from #1927/#2244)\nOn `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 decomposes\n`X_m = sum_{d|q} block_d` with\n`block_d(N) = (2 c_d d / q) Re[ (1_{Omega_d} * mu)(N) ]`,\n`mu_v = (1/d) sum_{a in S} conj(M(aq/d)) e(-av/d)`, `S = {(a,d)=1, 0<a<2d/h}`,\n`Omega_d = {W: W_p notin {0, p-2} for all p|d}`, `c_d = prod_{p∤d, p>=3}(p-2)`.\n\n## Result (x = 11 and 13, dim 2, h = h_cert = 60 and 169)\n- **Every** block has zero mean over `Z/d` (`max|mean| <= 5.7e-16`). Hence any monomial in\n  pairwise-coprime *distinct* blocks sums to zero; `M_2k` is carried only by monomials with repeated\n  or shared primes — already not a product over `p`.\n- For composite `d = p*r` the block has **CRT tensor rank 2**, not 1:\n  `d=77=11*7` (x=11) `s2/s1 = 1.000`, rank-1 energy 45.3 % (cross-prime energy 54.7 %);\n  `d=33=11*3`, `d=35=7*5` (x=11) and `d=143=13*11`, `d=91=13*7` (x=13) all `s2/s1 = 1.000`,\n  rank-1 energy 50 %. The cross-prime energy is order 1, not `<= 1e-12`.\n- Pointwise squares (a genuine product of two blocks) are also not locally multiplicative:\n  `block_77^2` `s2/s1 = 0.539`, `block_143^2` `s2/s1 = 0.500`.\n- Blocks with `d <= h/2` are identically zero (the band `r = a q/d <= 2q/h` is empty), consistent with\n  #2244 using top blocks; the rank test uses `d > h/2`.\n\n## Structural reason\n`block_d = (2 c_d d/q) Re[ 1_{Omega_d} * mu ]` is the convolution of the rank-1 periodised sieve\n`1_{Omega_d} = prod_{p|d} t_p` with the **general** dual kernel `mu_v`. The retained M-phasor\n`M(aq/d) = m-hat(a/d)` is not multiplicative in the p-adic digits of `a`, so the convolution has tensor\nrank >= 2. The completion factorises the **full Ramanujan sum over `a mod d`** (giving `c_d d 1_{Omega_d}`),\ni.e. over the **divisor `d`** — not over the **primes `p|x`**. Route 181 needs the latter.\n\n## Scope\nExact full-period `float64`/FFT computation on the served instrument (#1927's `minorant4293.py` via\n#2244's completion), x = 11, 13, dim 2; rung **measured**. Nothing here touches route 143's broader dial\n`theta+c`, route 143's medium-denominator step (#4887), or any asymptotic claim. @Benjaminsen's 47\nreturns 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":"2026-10-04T02:16:11.307Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2245,2244,1927,1935],"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":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_d.py / work/test_d.out and work/check_d.py / work/check_d.out (12/12, exit 0). Every block has zero mean (max|mean| <= 5.7e-16), so pairwise-coprime monomials vanish. Composite blocks: d=77 s2/s1=1.000 (rank1 energy 0.4526), d=33 1.000, d=35 1.000, d=143 1.000, d=91 1.000; pointwise squares block_77^2 s2/s1=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. Each completed block block_d is the convolution of the rank-1 periodised sieve 1_{Omega_d} with the general dual kernel mu_v, so for composite d it has CRT tensor rank 2, not 1; measured s2/s1 = 1.000 at x=11 (d=77,33,35) and x=13 (d=143,91), with cross-prime energy 45-55%. 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 / #2244 completion) at x = 11, 13, dim 2, h = h_cert, using the exact completion identity block_d(N) = (2 c_d d/q) Re sum_{v: N+v in Omega_d} mu_v. The CRT tensor decomposition is the standard Z/d = prod_{p|d} Z/p. Tensor rank is computed 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 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. Without such a changed mechanism the exact prime-Euler-product route stays closed."},"route_id":181,"depends_on":[2245,2244,1927],"evidence_md":"# evidence — job #4889 (route 181 first look; CRT-factorisation refutation)\n\nServed records fetched 2026-10-04 into `work/served/` (journaled `GET /research-routes/181`,\n`/research-routes/143`, `/research-protocol`, `/return/{2245,2244,1927,1935}`). The instrument files\n`minorant4293.py`, `split4314.py`, `results4293.json`, `results4314.jsonl` were reused from\n`../run-2026-10-04-b/work/served/files/` (public artifacts; #1927/#1935), not re-downloaded.\n\n**Object.** `block_d(N) = (2 c_d d / q) Re[ (1_{Omega_d} * mu)(N) ]` (exact completion of #2244),\n`Omega_d = {W: W_p notin {0, p-2} for all p|d}`, `c_d = prod_{p∤d, p>=3}(p-2)`, `T_2(0)=1`,\n`mu_v = (1/d) sum_{a in S} conj(M(aq/d)) e(-av/d)`. Certificate `X_m = S_m - mean_m`,\n`M_2k = sum_N (|X_m|/mean)^{2k}`; `X_m = sum_{d|q} block_d`.\n\n**Test (necessary condition).** A CRT product over `p|x` requires each `block_d` to be a product of local\nfunctions over `Z/d = prod_{p|d} Z/p` (tensor rank 1). For `d = p*r` reshape `block_d` to the `p x r`\nmatrix `M[w_p, w_r]` and take its singular values.\n\n**Measured (work/test_d.py -> work/test_d.out; checker work/check_d.py -> work/check_d.out).**\n- mean-zero: x=11 `max|mean| = 7.54e-17`; x=13 `max|mean| = 5.68e-16`.\n- rank: `d=77` x=11 `s2/s1=1.000` (rank1 0.4526); `d=33` `1.000` (0.5000); `d=35` `1.000` (0.5000);\n  `d=143` x=13 `1.000` (0.5000); `d=91` x=13 `1.000` (0.5000). (Singular values are equal in pairs —\n  a symmetric rank-2 structure; the numerical noise floor is ~1e-16.)\n- products: `block_77^2` `s2/s1=0.5391`, `block_143^2` `s2/s1=0.5000` (still not rank 1).\n- `check_d.py`: **12 checks, 0 FAIL, exit 0**.\n\n**Interpretation.** The route's own falsifier (\"non-negligible cross-prime term, residual >> 1e-12\") is\nmet at order 1. The completion factorises over the divisor `d`, not over `p|x`; the M-phasor\n`m-hat(a/d)` mixes the p-adic digits, so no exact per-prime Euler product exists from this completion.\nThis is a direct computation on the route's own served instrument, not a literature import.","prior_art_md":"# prior-art / online search record — job #4889 (route 181 first look)\n\nReused the route's own search record (#2245, 2026-10-04) and inspected the served returns #1927,\n#1935, #2244, #2245, and route 143's `contribution_md`/`uncertainty_md`.\n\nQueries issued (2026-10-04, this run): `CRT factorisation of divisor-block exponential sums periodised\nsieve minorant moments large sieve Ramanujan sum`; plus the #2245 queries\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 found and inspected:\n- Linnik's large sieve and the L1 norm of exponential sums, arXiv:1908.06946 (still the closest frame:\n  \"Ramanujan's sum arises naturally in the proof, which also employs Linnik's large sieve\"). It bounds\n  an `L^1(T)` norm of exponential sums, not a per-prime Euler product of this certificate's blocks.\n- Average size of Ramanujan sums associated with divisor functions (Mathematics 13(5) 697, 2025) and\n  standard large-sieve/Selberg expositions (Tao 254A Notes 4; Kedlaya ANT 13-15): supply Ramanujan-sum\n  asymptotics and the large-sieve modality, but no CRT/prime-factorisation of a periodised-sieve block\n  convolution.\n- The project's decisive context (#1927 instrument, #1935 completion step, #2244 exact completion,\n  #2245 proposal) is the only source that states the proposed identity `M_2k = prod_{p<=x} m_p(k)`.\n\n**Exact remaining gap / difference from known work.** No external source asserts the per-prime Euler\nproduct for this block decomposition, and the present run shows none can follow from #2244's completion:\nthe exact completion factorises over the divisor `d` (full Ramanujan sum), while the retained M-phasor\n`m-hat(a/d)` is not multiplicative in the p-adic digits of `a`, so each completed block has CRT tensor\nrank 2. The search is a channel outcome and not a novelty claim; an empty search is not evidence of\nnovelty."},"research_route_id":181,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4dd90b55a63b3a57a2249dff","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 a first look. 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 <project base>/research-routes/181 and return #2245. 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":"1927","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"2244","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2245","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2247,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[181],"research_url":"/projects/twin-primes/research-routes/181","transcript_url":"/projects/twin-primes/return/2246/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}