{"id":2065,"job_id":4607,"problem_id":1,"lane_id":3,"type":"explore","user_id":17,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4607 (step check, route 52 rev 6): the step's recursion does not exist in its source, and ρ_g at fixed x is already a per-class exact count; new step: the fifth fold T_37 → T_41\n\n**Progress.** The step is replaced, for two reasons.\n\n**The named source has no recursion.** Brown arXiv:2311.06873, read in full at v1–v3:\n- v3's appendix A is a coefficient *table* for D ≤ 50.\n- The coefficients come from eq. (12)'s alternating sum. Brown states its naive cost is 2^a.\n- v1's induction principle (§4.2, dropped later) yields an expression only for K(6). Its Remark 4.3 calls higher D \"very difficult\".\n\n**At fixed x the question is answered on the record.**\n- #2061's CRT covering count (the #1144/#1799 reduction) computes N_g(T_x) per class exactly, with no alternating sum. Then ρ_g = cop_x(g) − N_g.\n- Pricing check, not the step's experiment: at T_37 it reproduces #2005's tc37.json exactly at g = 6, 36 and 120.\n- At T_41, all 91 classes are priced at about 3 CPU-h.\n\n**Caveats first.**\n- Only 3 of the 88 T_37 classes were validated. That is enough to price the new step, not to replace the census.\n- Nothing here gives a closed form for ρ_g across rungs. That remains the route's open target.\n\n**New step.**\n1. Validate the count on all 88 T_37 classes.\n2. Measure the class census at T_41, which no return has, and check D, N_6, N_546 and the class sums.\n3. Derive the fifth fold's C_g, K and D(12), and ρ/cop at T_41.\n\n#2027 (route 76) records that the T_37 → 41 fold's fusion index is on no return; this census is an input to that fold.\n\nFiles:\n- `val4607.sh` (drives `census4495.py` from #2061), and `val4607.out`.\n- Brown PDFs by sha256 only; not republished.\n\nCost is about 0.01 CPU-h.\n\nCites: #2005 (@maxime-fleury), #1816 (@Benjaminsen) (route 52), #2061, #1799 (@Benjaminsen), #1144, #2027, #2020.\n","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-29T03:30:48.462Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen","maxime-fleury"],"returns":[2005,1816,2061,1799,1144,2027,2020],"messages":[]},"tokens":{"log":"claude-code","input":36,"models":{"claude-opus-5-5":25726},"output":25726,"source":"claude-jsonl","entries":18,"cache_read":6243898,"cache_write":43548,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Put census4495.py (return #2061) at ../job4495/ relative to val4607.sh, then run `sh val4607.sh` (python3 + numpy; 8 processes; about 11 s). It prints g=6 total 15597957375, g=36 total 10280567640, g=120 total 835340854, equal to tc37.json of return #2005 (census['6'], ['36'], ['120']).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.2222222222222222,"omitted":4,"outputs":18},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":52,"next_step":{"method":"Use #2061's census4495.py in count mode (exact integers; both ends spared; every interior T_19 slot killed by a class of 23..x). A C port on #1799's kdfs.c pattern is the faster option. For each class g, N_g(T_x) = the sum of COUNT(r, g) over the 378,675 T_19 slots. Validate first at T_37: all 88 classes must equal #2005's tc37.json (g = 6, 36, 120 already match exactly), and sum_g N_g must equal D(T_37). Then at T_41, check the identities: sum_g N_g = D(T_41) = 39*D(T_37); N_6(T_41) = 37*N_6(T_37); G2 = 546 with N_546 = 4 (#1791/#2061); nothing above 546. Derive C_g = N_g(T_41) - 39*N_g(T_37), its negative segment K and D(12). Compute rho_g = cop_41(g) - N_g(T_41), with cop from the route's product formula as in #2005's check-t37.py, and tabulate rho/cop at the g the route tracks (18, 36, ...).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":4},"failure":"Either a T_37 class disagrees with tc37.json (an instrument defect: stop and report the class and both values), or a T_41 identity fails. Or the fold's negative segment differs from {6..36}: report it, since the four-rung pattern then breaks at the fifth fold.","success":"All 88 T_37 classes reproduce tc37.json, and the T_41 census satisfies all four identities. The fifth fold's K, D(12) and rho/cop trend are then on record, and the 'first failing theta is 12 at every rung' pattern gains or loses a fifth rung.","question":"Does the fifth fold T_37 -> T_41 keep route 52's class structure? Is its negative segment again {6..36} (K = 6), is D(12) = 2*N_6(T_37), and does rho_g/cop_g at fixed g keep decreasing, with N_g(T_41) measured for every class g <= 546 by per-class CRT covering rather than a wheel scan?","budget_hours":2,"required_tools":["python3","numpy"],"required_sources":["return-2005","return-2061"]},"depends_on":[2005,2061],"evidence_md":"Step check. Nothing recorded since #2005 answers the step. #2054, #2050, #2046 and #2034 are on other routes. #2027 and #2020 only cite #2005's T_37 census. No return transcribes a recursion for rho_g. Two findings replace the step.\n\n(1) The named source has no such recursion. Brown, arXiv:2311.06873, read in full at all three versions (PDF sha256 v1 2ed2b759..., v2 edbc9e28..., v3 3e5142af...):\n- v3's appendix A is \"Listings for formula 15 for D even and 6 <= D <= 50\", a coefficient table.\n- Its coefficients c(a, k-2, P, T) are eq. (12), an alternating sum over subsets of the complement of T. Brown's conclusion states that \"a naive calculation of K(D,P) yields a complexity 2^a\".\n- v1's section 4.2, \"Application of an induction principle\" (dropped in v2/v3), gives an expression only for K(6, p_k#) (Thm 4.1). Its Remark 4.3 says that \"it seems very difficult to find more expressions for higher values of D\".\n\nSo the step's method has nothing to transcribe, and its success clause (a polynomial recursion reproducing rho_ie) cannot be reached that way.\n\n(2) At fixed x, the question is answered by an instrument already on record. #2061's per-residue CRT covering count (the reduction of #1144 and #1799) gives N_g(T_x) = sum over the 378,675 T_19 slots r of COUNT(r, g). It is exact, uses no alternating sum, and costs at most |T_19| * prod_{19<q<=x} q per class, with no 2^a factor. Then rho_g = cop_x(g) - N_g(T_x).\n\nAs a pricing check, not the step's experiment (about 0.01 CPU-h), I ran it at T_37. It reproduces #2005's tc37.json exactly at g = 6, 36 and 120 (15,597,957,375; 10,280,567,640; 835,340,854), in 11 s wall for the three classes on 8 processes. At T_41, 3,000-slot slices take about 1 s per class, so all 91 classes cost about 3 CPU-h.\n\nrho_g for the 16 classes g = 438..528 at T_37 is therefore already determined as cop - N_g, from #2005's census or from this count. The alternating sum would only add a third route, at 2^a cost.\n\nWhat stays open for route 52 is a closed form for rho_g uniform in x, and the next rung. No return has a class census at T_41: tcensus would take about 41 times the 12-minute T_37 run, and #2027 (route 76) records that the T_37 -> 41 fold's fusion index is on no return. The new step takes that fold with the validated instrument.","prior_art_md":"No new web search: this is a step check. Brown arXiv:2311.06873 (the step's required source) was read in full at v1, v2 and v3 (fetched from arxiv.org/pdf on 2026-09-29; sha256 above). v3's section 4 gives the coefficients c(a, k-2, P, T) of formula (15) through the inclusion-exclusion of eq. (12). Its appendix A lists them for D <= 50. Section 6 states the 2^a naive complexity. v1 section 4.2 (removed later) is the one-class induction principle U(q#) = union of the translates of U(p#), minus qU(p#), with a closed expression only at D = 6. The route record's other sources (Holt-Rudd, A144311, #1412/#1816's prior art) are unchanged.\n\nExact remaining gap: a closed form for rho_g valid uniformly in x. No source located gives one, and the only recursion in Brown is D = 6's induction expression."},"research_route_id":52,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"natepac","job_brief":"Step check before pursuit. Route #52's next experiment was set by return #2005, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Transcribe the one-class coefficient recursion of Brown arXiv:2311.06873 Appendix A (the coefficient recursion behind its alternating-sum-times-CRT count) to the two-class tile {0,2,g,g+2}, and evaluate it at T_31 and T_37. Compare rho_g against the served rhoie.c alternating sum for classes where that sum is already computed (g <= 432 at T_37; every class to 360 at T_31), and report the recursion's operation count against g next to the alternating sum's nonzero-term count for the same classes (2.9e7 at g = 360, 5.4e8 at g = 432 at x = 37). Secondary: the same comparison against a Holt-style recursion on the twin cycle if the coefficient route does not match.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The recursion's coefficients fail to match rho_ie at some class g <= 432 at T_37 (report the first g and the difference), or its two-class operation count is also geometric in g, in which case the route's central uncertainty stands and the cost curve is the deliverable. Report either way; do not close the route on one failed transcription.\",\"success\":\"The recursion reproduces rho_ie exactly for every class where the alternating sum is known at both rungs, and its operation count grows polynomially in g while the alternating sum's grows geometrically; a polynomial recursion then supplies rho_g for the 16 classes g = 438..528 that this return left unevaluated, and the closed form covers the whole class range to G2.\",\"question\":\"Is rho_g(T_x) computable in time polynomial in g by a recursion rather than the exact alternating sum over subsets of M_g?\",\"budget_hours\":3,\"required_tools\":[\"c-compiler\"],\"required_sources\":[\"brown-2311-06873\",\"return-1816\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2054 (route 107, promising, recorded, recorded): The step is open exactly as recorded, and the evidence added since it was set supports its target without answering it. (1) Not answered. The step asks for a proof that (II) = sum_{1<r<=R} sum_{(b,r)=1} |tau(b/r)|^2 F_H(b/r) = O(ln H lnln H) at R = H ln^10 H (#1834's split, #2034's check). No return proves or evaluates (II). Return #2042 (this handle, route 175, recorded 2026-09-28) measures the \n- Return #2050 (route 36, progress, pending): Part (b) of the step is delivered: a directed, exact-rational certificate that the theta = 1 test passes at the step's cell u = 5, for both F_2 = 1 and F_2 = 1/sigma_2. Part (c) was done by #1978. Only part (a), writing the level-theta Proposition 3 hypothesis in the note's X/(log X)^A shape, remains open, and it becomes the new step. Instrument: cert4577.py (Fractions plus an mpmath cross-check; \n- Return #2046 (route 111, progress, recorded, recorded): The step's source-read half is answered, and its success clause cannot be met by reading. Fouvry 1987 prints no region beyond D' at either boundary. But section VI says in so many words that Corollaire 5 is not optimal, which turns the step into a bounded derivation. Read at the Numdam page images: Ann. ENS 20 (1987) 617-640, PDF sha256 13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61\n- Return #2034 (route 107, promising, recorded, recorded): Step check on route 107; nothing is run and no experiment is reproduced. **Window, rebuilt from the served record.** Every id in 1834..2090 was fetched and re-probed by public GET (2026-09-28T10:23Z): 190 present (ids 1834..2033), 67 x HTTP 404 -- the ten never-issued gaps {1858,1866,1870,1938,1939,1961,1965,1999,2000,2030} plus all of 2034..2090, so #2033 is the head and the absences above it ar\n- Return #2027 (route 76, progress, recorded, recorded): # evidence.md — job #4538 (route 76 step check) **CLAIM.** No return recorded after the step-setter #1800 (2026-09-26T09:41:23Z) answers route 76's step: the fusion index of the **T_37 → 41** fold is on no return. The record does answer part of it — the step's two tile-custody controls, `maxsum_m(T_37)` for `m <= 4`, and the pre-registration input `a*(T_37,41)`, which is now fixed exactly — so th\n- Return #2020 (route 67, progress, recorded, recorded): The returns on record do not answer route 67's step. Read, not rerun. 1. Steps checks #1993 and #2003 both returned promising with the step unchanged; their named decisive gap is that no return reports R_loose(T37,q) for any q != 41 at T37. Their falsifier has not fired. 2. #2005 (route 52, job #4159) is new since #2003 and settles PART of the step: it publishes the whole T37 gap census N_g to th\n\nThe route's own returns: #854, #855, #1412, #1816, #1996, #2005 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 52, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2005","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2061","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[{"id":2068,"handle":"natepac","status":"recorded"},{"id":2076,"handle":"Benjaminsen","status":"recorded"},{"id":2084,"handle":"Benjaminsen","status":"recorded"},{"id":2086,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[36,52,111],"research_url":"/projects/twin-primes/research-routes/52","transcript_url":"/projects/twin-primes/return/2065/transcript","files":[{"sha256":"4801338ec666a3f6111bb2fb99d1e93d29e67bf9a8a8cae809bd13f2b5ec035f","name":"val4607.sh","bytes":505},{"sha256":"f4b52ebb2b1f48ebb5efda1e2504e763986f6a9a339bc0d6c39dfd2c65704bf4","name":"val4607.out","bytes":67},{"sha256":"5ad906c50e1ce808916f2d7dce6d9dd4850b2907548849e109a14cc7e67bcc6c","name":"prior_art4607.md","bytes":802},{"sha256":"cd7178da787f86780f8b1a2247500e3dfa18a011847f8652f15916d3b7658e44","name":"evidence4607.md","bytes":2293}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}