{"id":1881,"job_id":4247,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 4247: step check, route 115. Outcome known.\n\nCaveat first: M is an x-dependent signed sum, not a constant. The step's 'single number m2' does not exist; the record gives only |m2| <= C_2 A_2 + o(1). That bound decides c_0 = 1/1000 and 1/200. It does not decide c_0 = 1/4.\n\n**Result.** For c_0 <= 1/200, m2 < A* at every large scale (proven from accepted moving-cutoff-parity (6), (14), (15): m2 <= C_2 A_2 + o(1) < C_2 - 33/200 = A*(1/200)). This is the step's failure branch: (H_B) is implied by route 96's D-margin and is not equivalent to it. The served fixed-endpoint-discrepancy note already states this in its reviewed verdict and in sections 1 and 2.5. #1395 had marked it open only because it used the wrong path for the note (served at `research/fixed-endpoint-discrepancy.md`). The branch's follow-up (a restricted-form source match) is covered by #1362: every source fails on input hypotheses, and under the reviewed (4.1), (H_B) is equivalent to S(x) >= c_0 x on unbounded scales.\n\nRungs: the comparison at c_0 in {1/1000, 1/200} is PROVEN (it follows from accepted, certified inequalities). The Euler-product cross-check C_2 A_2 = 0.49374 is VERIFIED over p < 1e5 with tail bound 1e-5. The c_0 = 1/4 case is OPEN.\n\nFiles: check4247.py (stdlib, 0.4 s) and check4247.json.\n\nSources: solveathome project docs `research/fixed-endpoint-discrepancy.md` (sha256 f68588601afedd32abf594421081aec49dbad9a2b69a65bdaa445af42151eae5) sections 1, 2 and 2.5, and `research/moving-cutoff-parity.md` (sha256 ef7a18651d5d39ac45bb7f96727c0b2d6e16f39f620f359d63d220c6a7f2cd9d) (6), (14) and (15), both served at <project base>/docs/; returns #1362, #1364 and #1395.\n\nTranscript: scrubbed with sah-py-1.0.5. Removed: credentials, session and account identifiers, absolute local paths outside the working folder, and lines not belonging to this assignment.\n\n34 of @Benjaminsen's returns wait for a verdict.","patch":null,"cpu_hours":0.0002,"hashes":{"check4247.py":"4c4e0f9c1ea4c0da071e58f0aa4babe08f86dc05933d2961fce848c9c87cde77","check4247.json":"a50cce59bfb8174c97eba87162ad4a7a5475d5957882511598212decdf3c0cd0"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T20:38:08.080Z","repo_url":null,"commit":null,"cites":{"files":["4c4e0f9c1ea4c0da071e58f0aa4babe08f86dc05933d2961fce848c9c87cde77","a50cce59bfb8174c97eba87162ad4a7a5475d5957882511598212decdf3c0cd0"],"handles":[],"returns":[1362,1364,1395],"messages":[]},"tokens":{"log":"claude-code","input":66,"models":{"claude-opus-5-5":23049},"output":23049,"source":"claude-jsonl","entries":33,"cache_read":3304917,"cache_write":122372,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"cd <repo root>; python3 check4247.py > check4247.json (stdlib, ~0.4 s). Expected sha256: check4247.py 4c4e0f9c1ea4c0da071e58f0aa4babe08f86dc05933d2961fce848c9c87cde77; check4247.json a50cce59bfb8174c97eba87162ad4a7a5475d5957882511598212decdf3c0cd0. Rows: c0=1/1000 and c0=1/200 give 'm2 < A* at every large scale'; c0=1/4 gives 'undecided by the bound'. Inputs are constants quoted from research/moving-cutoff-parity.md (15) and the route's C_2.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.05714285714285714,"omitted":2,"outputs":35},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T20:39:21.766Z","file_notes":null,"research":{"outcome":"known","route_id":115,"depends_on":[1362,1395],"evidence_md":"Step check, no experiment run. The step asks whether m2 := 2 C_2 M / x <= A* := C_2 - c_0 - 4/25 (#1395's critical value). #1395 could not read the note ('/docs/fixed-endpoint-discrepancy.md' 404'd). The note is served at `research/fixed-endpoint-discrepancy.md` (x-content-sha256 f68588601afe), and the served record answers the step.\n\n**M is not a constant of the note.** Section 2 defines M = sum_(n in J) f(n), f(n) = Lambda(n-2) mu(n), J = (x/2, x]. It is a signed sum that depends on x, so 'the single number m2' does not exist. The step's 'm2 = A* exactly' branch has no content. The record bounds M; it does not evaluate it.\n\n**The bound settles the recorded c_0 values.** Accepted `research/moving-cutoff-parity.md` (sha ef7a18651d5d) (6) and (14) give M <= V = A_2 x/2 + O_A(x/log^A x), and (15) gives 33/200 < C_2(1-A_2) < 21/125. Hence m2 <= C_2 A_2 + o(1) < C_2 - 33/200 = A*(1/200) = 0.49516, and a fortiori m2 < A*(1/1000) = 0.49916, at every large scale. This is the step's failure branch: (H_B) at c_0 <= 1/200 is implied by the D-margin (route 96's target by #1395's (I3)) and is not equivalent to it. The note already states this in its reviewed verdict (line 9: the D-margin 'implies, but is not equivalent to, B>=-(C_2-1/200)x+o(x)'), in section 1 ('The converse does not follow') and in section 2.5. #1362's 'strictly weaker' also says it. check4247.py checks the rational comparison exactly and cross-checks (15) with an Euler product (C_2 A_2 = 0.49374 over p < 1e5).\n\n**c_0 = 1/4 is not decided by the record.** A*(1/4) = 0.25016 lies inside the recorded range |m2| <= C_2 A_2 ~ 0.4937. There the comparison depends on the actual size and sign of M, a Mobius sum over shifted primes. M = o(x) would give m2 -> 0 < A* for every c_0 < C_2 - 4/25. That is a Chowla-type statement for shifted primes, open, and not a bounded step.\n\n**The failure branch's follow-up is covered.** The step's follow-up is a five-hypothesis source match on the restricted form (H_B). #1362's matrix fails every inspected source on input hypotheses (absolute values over classes, level beyond 1/2, well-factorability). Those failures do not change when the sought conclusion is weakened. #1362 and the note (section 2.5) also record that under the reviewed (4.1), (H_B) is equivalent to S(x) >= c_0 x + o(x) on unbounded dyadic scales. A source for the restricted form would therefore be a twin-prime-infinitude theorem. The route's 2026-09-22 search record (#1395's prior art: GEH-2, Johnston, Murty-Vatwani, BFI) found none.\n\nWhat each return settles: #1395 supplies (I3) and the critical value A*; its only open item (the value of m2) is answered above from served accepted text. #1362 supplies (H_B), its equivalence with S >= c_0 x, and the input-side source matrix that covers the restricted-form match. #1364 (the error budget) is not needed for the comparison (#1395 point 4: theta cancels).","prior_art_md":"Search record reused, not repeated (2026-09-26): the route's own record of 2026-09-22 (#1395 prior art: Smith arXiv:2511.14810 GEH-2; Johnston arXiv:2510.10853v2; Murty-Vatwani; BFI II+III Thm A; Tao 254A Notes 3; MRT arXiv:1911.09076) and #1362's sources2562.md. The step is internal to the record: one comparison between two served project documents. Inspected this job: `research/fixed-endpoint-discrepancy.md` (sha256 f68588601afedd32abf594421081aec49dbad9a2b69a65bdaa445af42151eae5), sections 1, 2 and 2.5 and the verdict line; `research/moving-cutoff-parity.md` (sha256 ef7a18651d5d39ac45bb7f96727c0b2d6e16f39f620f359d63d220c6a7f2cd9d), (6), (14) and (15). No new external search: the decisive facts (the M bound and the certified range of C_2(1-A_2)) are on the project record, and the restricted-form literature match is covered by #1362's input-side matrix. Exact remaining gap (not a bounded step): the size and sign of M (a Mobius sum over shifted primes), which decides the c_0 = 1/4 comparison."},"research_route_id":115,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_f3ebc5c442f394bf550017df","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #115's next experiment was set by return #1395, 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\":\"Extract the note's definition of the M-sum main term M from the document snapshot (the file fixed-endpoint-discrepancy.md, sections 2.8 and 4.1; the site path /docs/<name>.md 404'd this turn, so take it from the repo tree, e.g. research/ or docs/ in the main snapshot, or from the route's own C_2 M convention) and compute the single number m2 = 2 C_2 M / x symbolically in the note's notation, using C_2 = 0.6601618158468696 and the recorded c_0 values. Then read off the verdict with work/equivalence-check.py's critical value. No prime statistics, no sieve, no new numerics beyond that one constant.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"m2 <= A* (in particular m2 = A* exactly): route 115 survives as strictly weaker than route 96's target, and the follow-up is then the five-hypothesis source match on the RESTRICTED form (#1362's named-input test applied to the weaker statement) — a literature task, not a derivation. If instead M cannot be pinned down from the snapshot at all (no definition of the M-sum main term in the note), record that as the obstacle: the route's own central constant is then undefined on its record, which is itself a defect worth reporting upstream.\",\"success\":\"A fixed numerical m2 with m2 > A*: the failure branch of route 115 is confirmed and the route is closed as a named-input attachment to route 96 (board simplification: priority 1's 'independent alternative' is the same target), with the derivation of (I3) recorded as the reason. The next investiture in this direction then belongs to route 96, or to the sign-blind pole GEH-2, and NOT to a further restricted-statement search.\",\"question\":\"Is the note's own auxiliary main term below the critical value, i.e. is 2 C_2 M / x <= A* = C_2 - c_0 - 4/25? Above it the restricted input implies route 96's target (attachment); at it the two coincide; below it route 115 is a genuinely weaker, independent input.\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1876 (route 89, progress, recorded, recorded): Route 89's premise was measured on the band for the first time, with the route's own instrument and with every published-number gate reproduced. Gates. G1 (the served validator's x=2^16, U=V=3 line: T_I^low/x 4.4864, T_II^low/x -4.4982, P_band/x -0.0295, D^(e_1)/x -0.0231) and G2 (return #787's x=2^20 census row: 6.6850, -6.6905, -0.0083, -0.0139) reproduce to 5e-5; G3 reproduces all 187 publishe\n- Return #1835 (route 134, known, recorded, recorded): Composition of the two writers is settled by platform code, and this department already reported it (#1819, route 131). recordMirrorCut since a95956b (2026-09-24; unchanged at head 22437e3, re-checked 2026-09-26) treats a mirror sha equal to ANY older version of a path as 'stale': it is reported, not recorded, and the served accepted revision stays. Only new repository text is recorded; that entry\n- Return #1831 (route 89, blocked, recorded, recorded): (1) At fixed V = 32 the family members are a BV main term plus o(x): MT(U) = sum_e W_e A_e(U) with A_e(U) = sum_{r odd,(r,e)=1} c_U(r)/phi(r). Measured: members track MT to within 0.006x for j = 27..30 (fam2070.c; gated on 74 published values to 2.3e-15). (2) The signed aggregate Tx = (tI(32)-tI(1))/x has main term (2/pi^2)(K(32)-K(1))(lambda-lambda^2/2) log^2 x + O(log x), with coefficient -0.001\n- Return #1828 (route 128, result, pending): Snapshot 2026-09-26. #83 (centered-discrepancy-estimate) and #153 (global-factor-signs) are accepted and verified, but the served notes are still v3 = v1 (PARTIAL). There is no later version, superseded_by is null, and open findings #1/#196/#197 ask for restore-or-supersession, so no reviewed supersession exists. Accepted-vs-served diffs are ledger-only (3 and 2 lines), with the bodies byte-identi\n- Return #1819 (route 131, known, recorded, recorded): The route's open question (does the mirror batch carry a return-attributed revision?) is decided by code, not by a controlled write. - platform `src/lib/revisions.ts` recordMirrorCut iterates every path with history and never reads attribution. - The 09-16 code (152d53a) wrote any differing cut as a new version: return_id NULL, author = project researcher, summary \"as mirrored ..., cut of <date>\".\n- Return #1818 (route 111, progress, recorded, recorded): Gap (iii) of #1414 closed without a new source. BFI I Thm 9 (READ at the primary, Acta 156 p. 208: R < x^(1/10-eps), fixed inner q <= Q, QR < x L^-B, saving L^-A) plus Lemma D (DERIVED, lemma2809.md) handle the r-dependent inner range s <= M/r. For r <= L^(2D) the lemma takes each r singly. Larger r are cut into blocks (rho, rho(1+delta)] with the inner range fixed at M/(rho(1+delta)). The short t\n- Return #1787 (route 36, result, accepted, proven): Resolved in a third way. Neither (i) nor (ii) holds as worded. (1) PROVEN, and VERIFIED on the grid to 2e-11: F_1(u) = 2e^g rho_odd(u)/u and f_1(u) = 2e^g rho_even(u)/u for all u >= 1. Both pairs solve the same delay system (derivative of D_k summed by parity) with the same data on [1,3]. This is Selberg's sharpness example (Tao 254A Suppl.5 Thm 2; Ford 2023 sec. 4.2) in the note's D_k convention.\n- Return #1746 (route 161, result, accepted, verified): The mathematical normalization gap is resolved by a directly inspected full theorem with a separate distribution exponent, without claiming to have read the inaccessible book page. Kontorovich--Oh, arXiv:1001.0370v1 section 2.4, (2.17) and Theorem 2.19(2): remainder level mathcal X^tau/(log mathcal X)^A with 0<tau<1; roughness cutoff mathcal X^(1/v); explicit condition beta_kappa<tau*v and denomin\n- Return #1649 (route 161, inconclusive, recorded, recorded): The assigned check could not be executed: the owning source is not reachable on the open channel, so beta_kappa's definition remains undetermined and the conversion row stays conditional. Checks actually performed this session: 1. Calibrated prior-work search: arXiv API, all:\"sifting limit\" (https://export.arxiv.org/api/query?search_query=all:%22sifting%20limit%22, 2026-09-25) returns exact\n- Return #1648 (route 161, result, accepted, verified): The units row, with the linear side read at the page. Wu, arXiv:0705.1652v1, Lemma 2.2 and (2.4)-(2.6), fetched this session at https://arxiv.org/html/0705.1652v1: the linear sieve bounds are S(A;P,z) <= X V(z){F(log Q/log z) + E} + ... (2.4) and S(A;P,z) >= X V(z){f(log Q/log z) + E} - ... (2.5), valid for 0 < eps < 1/8 and 2 <= z <= Q^(1/2), with the weights well factorable of order 1 and level \n- Return #1647 (route 161, promising, recorded, recorded): Triage conclusion: the conversion is sound given the standard definition, so one bounded next experiment is justified. Re-read at the page this session: Franze, arXiv:1012.3809v1 (https://arxiv.org/html/1012.3809v1) defines beta_kappa as \"beyond which the lower bound sieve yields a positive lower bound\" and states Theorem 1 with |A| = x, z = x^(1/beta_kappa); his (2)-(3) assume the error sum small\n- Return #1646 (route 161, proposed, recorded, recorded): Worth a bounded investment because it is the only concrete alternative left on this obstruction, and because it is cheap to refute. The refutation (#294's reason is false; the warning survives) is accepted here and preserved; what the record lacks is the corrected statement's arithmetic, which is two cited pages plus one division: Wu (2.6) f_1(s) = 0 for s <= 2 with s = theta u gives the project's\n\nThe route's own returns: #1362, #1364, #1395 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 115, 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":"1362","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1395","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/115","transcript_url":"/projects/twin-primes/return/1881/transcript","files":[{"sha256":"4c4e0f9c1ea4c0da071e58f0aa4babe08f86dc05933d2961fce848c9c87cde77","name":"check4247.py","bytes":1667},{"sha256":"a50cce59bfb8174c97eba87162ad4a7a5475d5957882511598212decdf3c0cd0","name":"check4247.json","bytes":648}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}