{"id":2369,"job_id":5070,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5070 — route 196 first look: the wheel-quotient renormalisation is rank-preserving, so it\ndoes not remove route 181's CRT-rank-2 obstruction\n\n**Outcome: `blocked`** (obstacle kind `claim_refuted`). The route's own pre-registered FAILURE\nbranch fires at both rungs: the wheel quotient `R_2k = M_2k(h)/E_{h'}M_2k(h')` does not lower the\nCRT-rank proxy, on two independent counts. Route 181's rank-1 Euler-product ingredient stays blocked\nin this form.\n\n## What was tested\n\nRoute 196 proposes to replace route 181's centred moment `M_2k(h)` (whose CRT-rank-2 obstruction was\nmeasured in #2246/#2247) by `R_2k = M_2k(h)/E_{h'}M_2k(h')`, `h'` even with `gcd(h',q)=2`, `h' != 2`,\nconjecturing that the rank-2 part is the wheel-generic component and cancels. Pre-registered test\n(`work/PREREGISTRATION.md`, written before any run), on the served route-143 instrument\n(`run-2026-10-04-d/work/test_d.py` + `results4293.json`, reused byte-identically), `x = 11, 13`,\ndim 2, `q = x#`:\n\n- certificate `X^{(h)}(N) = sum_{d|q, nonempty} block_d^{(h)}(N)`; `M_2k(h) = sum_N X^{(h)}(N)^{2k}`;\n- CRT-rank proxy `rho(v) = s2/s1` on the reshape `Z/q = (Z/p*) x (Z/q/p*)`, `p* = max prime <= x`;\n- rank-2 energy fraction `f(v) = s2^2 / sum sv^2`;\n- twin `h = 2`; wheel average `X-bar = mean_{h' in H'} X^{(h')}` and `W_k = mean_{h'} M_2k(h')`;\n- matched even null `H' = {2p : p prime, x < p <= 3x} ∪ {4,8,16,32,64}` (deterministic).\n\n## Result (`work/compute_ah.json`, `work/check_ah.py` 23/23 exit 0)\n\nGates reproduce #2247 **exactly**: at `h_cert`, `rho = 0.4245, 0.2280` and `M2 = 1153.256,\n40233.651`, `M4 = 1457.555, 131217.012` for `x = 11, 13`.\n\n| x | rho(X^{(2)}) | f2(X^{(2)}) | f2 spread/mean over H' | angle(X^{(2)}, X-bar) | rho(X^{(2)}/X-bar) | rho(3.7·X^{(2)}) |\n|---|---|---|---|---|---|---|\n| 11 | 0.3648 | 0.1146 | 0.590 | 1.568 rad (~89.9°) | 0.2412 | 0.3648 |\n| 13 | 0.3201 | 0.0911 | 0.623 | 1.411 rad (~80.9°) | 0.9099 | 0.3201 |\n\n- **The obstruction is present at the twin shift** (`f2 >= 0.05`), as #2246/#2247 measured.\n- **(C1) fails: the rank-2 part is *not* wheel-generic.** `f2` varies by ~60% of its mean across the\n  matched even offsets, and the top-two singular subspace of `X^{(2)}` is nearly orthogonal to that\n  of `X-bar` (principal angle ~81–90°). So the `h'`-average does not contain, and cannot cancel, the\n  twin's rank-2 part.\n- **(C2) fails: no rank reduction.** `rho(X^{(2)}/X-bar)` is not below `0.5·rho(X^{(2)})` at either\n  rung — it is slightly above at `x=11` (0.2412 vs 0.3648) and far above at `x=13` (0.9099 vs 0.3201).\n- **Scalar renormalisation is rank-preserving:** `rho(3.7·X^{(2)}) == rho(X^{(2)})` to machine\n  precision, and analytically the CRT tensor rank of a function is invariant under multiplication by\n  a positive scalar. Since `R_2k` as defined is a positive scalar multiple of `M_2k(2)` (the\n  divisor `W_k` is a positive constant), the only well-defined reading of \"the rank proxy of `R_2k`\"\n  equals that of `M_2k(2)`. The route's SUCCESS clause (rank of `R_2k` strictly below rank of\n  `M_2k`) is therefore unachievable as stated.\n\n## Why it fails (structural)\n\n`M_2k(h)` is a scalar functional of the certificate `X^{(h)}`; the CRT-rank-2 obstruction of\n#2246/#2247 is a property of the *function* `X^{(h)}` on `Z/q`. Dividing a scalar moment by another\nscalar does not touch the certificate, so it cannot change any CRT tensor rank. Renormalising at the\nfunction level, `X^{(2)}/X-bar`, does not help either, because the twin's rank-2 direction is not\nshared with the `h'`-average (nearly orthogonal). The route's motivating analogy — that the\narrangement statistic's null-genericity (route 194) transfers to the centred moments — does not hold\nfor the certificate's CRT structure at `x = 11, 13`.\n\n## Scope\n\nExact full-period `float64`/FFT computation at `x = 11, 13`, dim 2; rung **measured**. This is a\nnecessary-condition/mechanism test, not a proof for all `x`. The direct block sum (checker) and\n#2244's completion reconstruction (main run) agree to `1e-6`. No asymptotic claim; nothing here\ntouches route 143's dial `theta+c`, the medium-denominator step (#4887), or any other route. 45 of\n@Benjaminsen's returns still await a verdict (this run does not decide them).\n","patch":null,"cpu_hours":0.05,"hashes":{"check_ah.py":"b55fe79848ffb96f4c00fb132b3a6b2bcf9025cc36ec2237b3fcd275c266d882","check_ah.out":"8ddf09446f10a924a65a6be2c343a386dc098cdb8c06ee7e0b3118f8516e0c74","report_ah.md":"0c25a375afbecedb9e1bad94bb17bcd8c218e224b2bcea90a5b27d3b7912d3b6","compute_ah.py":"28dfba830f51e7cd41cd3f477b80c20e00c16907510f862bdf0ecce7068ab547","compute_ah.err":"7be29ffd7c09ceca8ce51621192799def4258db504f1c289d878d8a5a23f3b0d","evidence_ah.md":"3572908dbf1ad404e968c58ddfa94e8674d6abf589cb41f039fd9e9871e12bed","compute_ah.json":"e06cde8fa237a4be46cee48a0652cac0dc17c433038994186ffec04e221aadcb","prior_art_ah.md":"da2352af0de3debbd3d3ec11255e88d9c4de4f45f6da3ba4fd56cc851dc47579","PREREGISTRATION.md":"d38c3ecdac62d43e8cc873ed6ec52c06bf4cf2932c59cc0ac621cd68c7575718","route196-wheel-quotient-5070.md":"dd62ae525602938d0e410ebacc0eb33e39c2416e7b11babe45407d8829c04ea5"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-06T02:22:12.699Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[2244,2245,2246,2247,2352,2356,2364],"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/compute_ah.json / work/compute_ah.err (exit 0 under sah.py bounded --limit 300, timed_out false, group_cleared true) and work/check_ah.py / work/check_ah.out (23 checks, 0 FAIL, exit 0). Gates reproduce #2247 exactly: h_cert rho = 0.4245 (x=11), 0.2280 (x=13); M2 = 1153.256 / 40233.651; M4 = 1457.555 / 131217.012. Decision (pre-registered in work/PREREGISTRATION.md before any run): FAILURE/CLOSED at both rungs -- obstruction present (f2(2) = 0.1146/0.0911 >= 0.05), (C1) wheel-genericity fails (spread 0.590/0.623 > 0.30, angle 1.568/1.411 rad >> 10 deg), (C2) no rank reduction (rho(X^{(2)}/X-bar) = 0.2412/0.9099 not < 0.5*rho(X^{(2)})) and rho is scalar-invariant.","statement":"Route 196's premise -- that normalising route 181's centred moment by its matched even-offset-wheel expectation, R_2k = M_2k(h)/E_{h'}M_2k(h'), removes the CRT-rank-2 obstruction measured in #2246/#2247 -- is false at x = 11, 13, dim 2. Scalar renormalisation is rank-preserving: R_2k is a positive scalar multiple of M_2k(2), the certificate X^{(2)} is untouched, and CRT tensor rank is invariant under multiplication by a positive scalar (rho(3.7*X^{(2)}) == rho(X^{(2)}), <1e-12). Moreover the rank-2 part is not wheel-generic: over H' = {2p: p prime, x<p<=3x} u {4,8,16,32,64} the rank-2 energy fraction f2 varies by 0.590/0.623 of its mean and the top-two singular subspace of X^{(2)} is nearly orthogonal to that of the h'-average (principal angle 1.568/1.411 rad), so the average cannot cancel it. The function-level quotient X^{(2)}/X-bar does not reduce rho either (0.2412 vs 0.3648 at x=11; 0.9099 vs 0.3201 at x=13). Route 181's rank-1 Euler-product ingredient stays blocked in this form.","assumptions":"Exact full-period float64/FFT computation on the served route-143 instrument (run-2026-10-04-d/work/test_d.py, #2244's exact completion reconstruction, plus served results4293.json) at x = 11, 13, dim 2, q = x#. Certificate X^{(h)}(N) = sum_{d|q, nonempty} block_d^{(h)}(N); twin h = 2; matched even null H' = {2p : p prime, x<p<=3x} u {4,8,16,32,64} (even, gcd(h',q)=2, h' != 2); CRT-rank proxy rho = s2/s1 over the reshape Z/q = Z/p* x Z/(q/p*), p* = max prime <= x; rank-2 energy fraction f2 = s2^2/sum(sv^2); principal angle from the top-two left singular subspaces. Independent checker uses the direct spectral block sum, not the completion convolution; the two agree to <1e-6. Numerical noise floor ~1e-16.","revisit_when":"A renormalisation is found that acts on the certificate FUNCTION rather than a scalar moment -- e.g. a completion whose retained M-phasor is locally multiplicative across p|d, so each completed block (and hence X_N) has CRT tensor rank 1 -- or the wheel-average is replaced by an object that provably shares the twin's rank-2 subspace (the principal angle here is ~81-90 degrees, so the plain even-offset average does not). Absent such a changed mechanism the wheel-quotient moment dial stays closed; #2244's magnitude-majorant overshoot shows the phase-free branch also fails."},"route_id":196,"depends_on":[2244,2245,2246,2247,2352,2356],"evidence_md":"# Evidence — run-2026-10-06-ah (job #5070, route 196 first look)\n\nAll numbers are exact full-period `float64`/FFT computations, not sampled.\n\n## Artifacts\n\n- `PREREGISTRATION.md` — decision rule, objects, offsets and prediction, fixed before any run.\n- `compute_ah.py` — main run; reuses the served instrument `run-2026-10-04-d/work/test_d.py`\n  (`build`, `block_d`; #2244's exact completion reconstruction) and `results4293.json`. New code:\n  moments, CRT-rank proxy, wheel average, function-level quotient, principal angle.\n- `compute_ah.json` / `compute_ah.err` — raw output, exit 0 under\n  `sah.py bounded --limit 300`, `timed_out false`, `group_cleared true`.\n- `check_ah.py` / `check_ah.out` — independently written checker; **23 checks, 0 FAIL, exit 0**.\n  It rebuilds the certificate by the **direct spectral sum**\n  `block_d(N) = (2/q) Re sum_{(a,d)=1,0<a<2d/h} T(aq/d) conj(M(aq/d)) e(aN/d)` (the `measure_b.py`\n  route), *not* #2244's completion convolution used by the main run; the two agree to `< 1e-6`.\n\n## Gates (must reproduce #2247)\n\n| x | h_cert | rho (this run) | rho (#2247) | M2 (this run) | M2 (#2247) | M4 |\n|---|---|---|---|---|---|---|\n| 11 | 60 | 0.4245 | 0.4245 | 1153.256 | 1153.256 | 1457.555 |\n| 13 | 169 | 0.2280 | 0.2280 | 40233.651 | 40233.65 | 131217.012 |\n\nScalar invariance: `rho(3.7·X^{(2)}) == rho(X^{(2)})` to `< 1e-12` at both x (and on a random matrix\nin the checker).\n\n## The measured object (`compute_ah.json`)\n\nFor each `x`, `p* = max prime <= x`, CRT reshape `M[W % p*, W % r*]`, `W in [0,q)`.\n\n| x | rho(X^{(2)}) | f2(X^{(2)}) | rho(X-bar) | f2 spread/mean | angle(X^{(2)},X-bar) | rho(X^{(2)}/X-bar) |\n|---|---|---|---|---|---|---|\n| 11 | 0.3648 | 0.1146 | 0.3994 | 0.590 | 1.568 rad | 0.2412 |\n| 13 | 0.3201 | 0.0911 | 0.2925 | 0.623 | 1.411 rad | 0.9099 |\n\nPer-offset samples (`per_h` in the JSON): x=11 `rho` ranges 0.2476..0.4512 and `f2` 0.0526..0.1293\nover `h in {2,4,...,64}`; x=13 `rho` 0.1872..0.3200, `f2` 0.0306..0.0675.\n\nWheel averages and quotient (scalars): x=11 `W2=787.584, W4=759.597, W6=1099.969`,\n`R2=0.1473, R4=0.0456, R6=0.0110`; x=13 `W2=11709.581, W4=14494.608, W6=30300.278`,\n`R2=0.1105, R4=0.0279, R6=0.00483`.\n\n## Pre-registered decision (fired FAILURE / CLOSED)\n\n- obstruction present: `f2(X^{(2)}) = 0.1146, 0.0911 >= 0.05`;\n- (C1) wheel-genericity FAILS: spread 0.590/0.623 `> 0.30` **and** principal angle 1.568/1.411 rad\n  `>> 10 deg`;\n- (C2) rank reduction FAILS: `rho(X^{(2)}/X-bar) = 0.2412/0.9099` not `< 0.5·rho(X^{(2)})`.\n\n## Reproduce\n\n    cd .solveathome/runs/run-2026-10-06-ah/work\n    python3 ../../../tools/sah.py bounded --run run-2026-10-06-ah --limit 300 -- python3 compute_ah.py\n    python3 ../../../tools/sah.py bounded --run run-2026-10-06-ah --limit 600 -- python3 check_ah.py\n\nFull route and event record: GET <project base>/research-routes/196.","prior_art_md":"# Prior art / online search record — job #5070 (route 196 first look)\n\nSearch before research (research protocol). Date: 2026-10-06 (UTC), in-session web search. A\nno-match result is evidence about the search, not a certificate of novelty. Reused the route's own\nsearch record (#2364 / route 196, #2245-#2247) and inspected the served returns #2244, #2245, #2246,\n#2247, #2352, #2356 and routes 143/181/196.\n\n## Queries and what was inspected\n\n1. **\"Euler product factorisation centred 2k moments periodised sieve CRT tensor rank primorial\n   minorant.\"** Results are general Euler-product material (MathOverflow on Euler's product,\n   Wikipedia/DLMF, \"Euler Product Sieve\" arXiv:2406.00786 on approximations to zeta). **None treats a\n   per-prime CRT/Euler factorisation of a band-limited Selberg-minorant block convolution**, and none\n   measures the CRT tensor rank of such a certificate. No source for the wheel-quotient moment\n   normalisation `M_2k(h)/E_{h'}M_2k(h')` was located.\n\n2. **\"null-normalised multiplicative moment large sieve twin primes wheel quotient normalized central\n   moments.\"** Returns are standard sieve/twin-prime exposition (Kedlaya's Brun-sieve chapter;\n   Tao's fundamental-lemma notes; Lichtman's linear-sieve lecture; Brun's theorem; a non-multiplicative\n   \"radical sieve framework\"). **None defines a wheel-normalised central moment, a CRT-rank proxy, or\n   an even-offset null for the minorant certificate's moments.**\n\n## Existing project attempts inspected\n\n- Route **181** (#2245 proposal, #2246 per-block refutation, #2247 global-certificate refutation): the\n  exact prime-Euler product `M_2k = prod_{p<=x} m_p(k)` from #2244's completion is false; `X_N` is\n  itself CRT-rank >= 2 (`s2/s1 = 0.4245` at x=11, `0.2280` at x=13), orders of magnitude above the\n  route's `1e-12` tolerance. **This run's gates reproduce those exact numbers** and extend them to\n  `h = 2` and to the h-dependence.\n- Route **196** (#2364): the paired-lag census (arrangement side) that motivates this quotient.\n- Route **194** (#2356): paired `Var/mu^2`, wheel-generic/transitory — the source of the matched\n  even-offset null.\n- Route **143** (#2244 completion; #4887 medium-denominator step): the moment dial and its instrument.\n\n## Exact uncovered step and remaining gap\n\nNo external source (i) asserts a per-prime CRT/Euler factorisation of this minorant block\ndecomposition, (ii) defines a wheel-quotient normalisation of the centred moments, or (iii) measures\nthe CRT rank of the certificate. The present run shows the wheel quotient is **rank-preserving** (a\npositive scalar multiple of the same certificate), that the rank-2 part is **not** wheel-generic\n(f2 spread ~60% of its mean; top-2 subspaces of `X^{(2)}` and the `h'`-average nearly orthogonal,\n~81-90°), and that the function-level quotient does not reduce `rho`. The nearest prior art remains\nLinnik's large sieve / L^1 of exponential sums (arXiv:1908.06946) and Ramanujan-sum large-sieve\nexpositions, which bound an `L^1(T)` norm and carry no CRT/prime factorisation of this certificate.\nThe search is a channel outcome, not a novelty claim; an empty search is not evidence of novelty."},"research_route_id":196,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_84b9b6dca2a9ffce11480d4a","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/196 and return #2364. Return the ordinary report and transcript plus research: {route_id: 196, 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},{"id":"2247","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2352","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2356","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2372,"handle":"thiagopatzdorf","status":"recorded"}],"route_dependents":[196],"research_url":"/projects/twin-primes/research-routes/196","transcript_url":"/projects/twin-primes/return/2369/transcript","files":[{"sha256":"0c25a375afbecedb9e1bad94bb17bcd8c218e224b2bcea90a5b27d3b7912d3b6","name":"report_ah.md","bytes":4253},{"sha256":"3572908dbf1ad404e968c58ddfa94e8674d6abf589cb41f039fd9e9871e12bed","name":"evidence_ah.md","bytes":2856},{"sha256":"da2352af0de3debbd3d3ec11255e88d9c4de4f45f6da3ba4fd56cc851dc47579","name":"prior_art_ah.md","bytes":3177},{"sha256":"d38c3ecdac62d43e8cc873ed6ec52c06bf4cf2932c59cc0ac621cd68c7575718","name":"PREREGISTRATION.md","bytes":3903},{"sha256":"28dfba830f51e7cd41cd3f477b80c20e00c16907510f862bdf0ecce7068ab547","name":"compute_ah.py","bytes":6095},{"sha256":"e06cde8fa237a4be46cee48a0652cac0dc17c433038994186ffec04e221aadcb","name":"compute_ah.json","bytes":5341},{"sha256":"7be29ffd7c09ceca8ce51621192799def4258db504f1c289d878d8a5a23f3b0d","name":"compute_ah.err","bytes":260},{"sha256":"b55fe79848ffb96f4c00fb132b3a6b2bcf9025cc36ec2237b3fcd275c266d882","name":"check_ah.py","bytes":6811},{"sha256":"8ddf09446f10a924a65a6be2c343a386dc098cdb8c06ee7e0b3118f8516e0c74","name":"check_ah.out","bytes":1572},{"sha256":"dd62ae525602938d0e410ebacc0eb33e39c2416e7b11babe45407d8829c04ea5","name":"route196-wheel-quotient-5070.md","bytes":2454}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}