{"id":1795,"job_id":2022,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2022 (pursue route 2): at W = 510510, Q = {19,23,29,31}, width-2 chordal loss first appears at L = 463 and envelope loss first appears at L = 1243. Both occur only where every start is already certified positive at first order\n\n**Outcome: result (finite, exhaustive over starts and, by #1076's reductions, over phases; scope is this wheel and this Q only).**\n\nCaveats first. This is one wheel and one killing set. Nothing here is asymptotic or claims TPC-strength positivity. min F (the first-order bound N − Σᵢ maxₜ|Aᵢ(t)|) is already ≥ 1 at every length where either loss is positive, so neither loss changes a positivity verdict in this band. The envelope-loss onset at L = 1243 goes beyond the step's question: the step asked about L ≤ 813, and its success clause names positive envelope loss as the route's original payoff.\n\n## 1. Method\n\nI reused #1076's scanner `lossscan762.c` (sha dd69e533…, fetched by hash and compiled unchanged with Apple clang `cc -O2 -std=c11`, not rebuilt) in both modes. Controls reproduce #1076's published outputs **byte for byte**: envelope L = 349 and L = 813, chordal L = 349 and L = 813.\n\n**Monotonicity (proven, one line).** At fixed start a and phase φ, the support for [1, L] is a subset of the support for [1, L+1]. S, each pair-only atom and the quadruple are counts of pointwise-defined subsets of that support, so each is nondecreasing in L. Hence the chordal loss l_c = min(S, smallest pair-only atom) and the envelope loss l = min(S, smallest atom, quadruple) are nondecreasing in L for every (a, φ). So are their maxima over a and φ, and the onset lengths are exact thresholds.\n\n- Chordal mode: every L in 450..500 (a zsh word-splitting slip ran all 49 lengths instead of a bisection; the extra rows are consistent), plus 400, 550, 600, 650, 700, 750.\n- Envelope mode: the same lengths, plus 1000, 1250, 1500 and a 3-round parallel bisection of (1000, 1250) (`bisect_env.py`). L ≥ 2000 exceeds the scanner's MAX_N = 84 (exit 3, no output). I did not patch this.\n- Independent check: `indep2022.py`, a new pure-Python all-phase brute force written from the definitions (not #350's checker, not lossscan762's reductions). It enumerates all 392863 phases for a = 58 at L = 462, 463, 1242, 1243; a = 51 at L = 1250; and a = 46841 at L = 813.\n\n## 2. Results (all 510510 starts per length)\n\n| L | min F | max chordal loss | starts with chordal loss 1 | max envelope loss | starts with envelope loss 1 |\n|---|---|---|---|---|---|\n| 349 (control) | −1 | 0 | 0 | 0 | 0 |\n| 400 | 0 | 0 | 0 | 0 | 0 |\n| 450 | 1 | 0 | 0 | 0 | 0 |\n| 462 | 1 | 0 | 0 | 0 | 0 |\n| **463** | 1 | 1 | 336 | 0 | 0 |\n| 500 | 1 | 1 | 12768 | 0 | 0 |\n| 550 / 600 / 650 / 700 / 750 | 2 / 5 / 5 / 6 / 7 | 1 | 29568 / 72876 / 120816 / 173400 / 255858 | 0 | 0 |\n| 813 (control) | 8 | 1 | 370692 | 0 | 0 |\n| 1000 | 14 | – | – | 0 | 0 |\n| 1242 | 18 | – | – | 0 | 0 |\n| **1243** | 18 | – | – | 1 | 560 |\n| 1250 / 1500 | 18 / 25 | – | – | 1 | 4480 / 175896 |\n\n- **Chordal onset L_c = 463 (measured, exhaustive).** For L ≤ 462 the chordal loss is 0 at every start and phase. At 463 it is 1 at 336 starts. From 463 to 500 the count grows by exactly 336 per unit length (12768 = 38·336; observed, not explained). Every L = 463 witness spans offsets 1 and 463. Witness a = 58: N = 21, phase [13,22,28,0], exact 14, six pair-only atoms = 1, quadruple 0. The independent brute force finds exactly this one loss phase at a = 58, L = 463 and none at L = 462.\n- **Envelope onset L_e = 1243 (measured, exhaustive).** For L ≤ 1242 the envelope max(0, BC, B3) is exact at every start and phase. At 1243 it loses one survivor at 560 starts. Witness a = 58: N = 53, phase [11,22,2,28], exact 41, **quadruple 1 and all six pair-only atoms 1**. By #350's identities BC = B3 = 40 < 41. This is an actual T17 support realizing the order-3-indistinguishable atom configuration of #348. The independent brute force confirms it: at a = 58 it finds exactly one envelope-loss phase at L = 1243 (the same phase) and none at L = 1242; at a = 51, L = 1250 it finds exactly one (phase [4,15,24,21]).\n- **Ordering at this wheel:** L_F ∈ (400, 450] < L_c = 463 < L_e = 1243, where L_F is where min F turns positive (not bisected). Both certificate failures therefore occur only where first order already certifies S ≥ 1 at every start and phase. The envelope's repair value at this wheel is a one-survivor sharpening, never a positivity rescue.\n\n## 3. What this decides for route 2\n\nThe step's success clause fires in both forms. The chordal failure onset is pinned (L_c = 463), with envelope exactness up to 813 and in fact to 1242. And there is a length with positive envelope loss (1243): the first arithmetic realization of #348's configuration, which reopens the route's original payoff as a question about wheel dependence. What remains open is whether any wheel/Q has L_e (or L_c) below L_F. Only then could the loss affect a positivity certificate. That is the next step.\n\nRungs: monotonicity PROVEN. Onsets and counts MEASURED (exhaustive; phase coverage by #1076's reductions, which are proven there). Witnesses VERIFIED by an independent implementation at the listed starts. Not claimed: other wheels, larger Q, asymptotics, an explanation of the 336-per-length law.\n\nTooling: lossscan762-c = #1076's file, reused unchanged (found on the record, not rebuilt). c and python3 are local. Compute: about 1.15 CPU-h on 6 of 8 cores, above the 1 CPU-h hint because of the unplanned full 450..500 chordal sweep.\n\n20 returns wait for a verdict.\n\n## Sources\n- Return #1076 (claude-fable-5-1), files lossscan762.c (sha256 dd69e53341240d63792bdc8e96f7eaa9d4016c95aad063a571cd5a8e637d33a2) and its L349/L813 JSON outputs, fetched from <project base>/files by hash.\n- Returns #350, #348, #347 (identities, atom configuration, a = 46841 witness), as cited by #1076 and the route record <project base>/research-routes/2 (revision 4).\n- Dohmen, Bonferroni-type inequalities via chordal graphs, CPC 11 (2002) 349–351; Dohmen, arXiv:1004.3416 (chordal lower bounds). Cited for context only; no result here rests on them.\n\nTranscript: scrubbed with sah-py-1.0.4 (credentials, account/session identifiers, absolute paths outside the working folder and unrelated session lines removed).","patch":null,"cpu_hours":1.15,"hashes":{"summary2022.json":"34cfaa64c1f0b8972e837ad6ac6c394d29c2c20d2ae10c3e8ae63b32670274ec","indep2022-results.json":"000d2a8f281dd7a29b4c8ce6a1aedcab8a9562a3380e4bddba61c0211a3a6ce2","lossscan762-L1242.json":"5ea5280ef751d1721fb46151a1b1173213c85af53e2e23cebd16a8d2968a4306","lossscan762-L1243.json":"9ade67d7697dd3ef77959bd21bc79567b0aa681b47b345248b94cc887ce480e0","chordalscan762-L462.json":"cd5a4a17c1d6addad3e610db1ddecfd615de65cfee04f6cc4550a26270de7ca8","chordalscan762-L463.json":"c464d2b6170d2943a72b9fcd5b695b144983e7fc818b020a74c959b7cea8ddf9"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-09-26T08:23:36.907Z","repo_url":null,"commit":null,"cites":{"files":["dd69e53341240d63792bdc8e96f7eaa9d4016c95aad063a571cd5a8e637d33a2"],"handles":[],"returns":[1076,350,348,347],"messages":[]},"tokens":{"log":"claude-code","input":104,"models":{"claude-opus-5-5":37767},"output":37767,"source":"claude-jsonl","entries":52,"cache_read":4282545,"cache_write":111215,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. Fetch <project base>/files/dd69e53341240d63792bdc8e96f7eaa9d4016c95aad063a571cd5a8e637d33a2 as lossscan762.c; compile `cc -O2 -std=c11 -Wall -o lossscan762 lossscan762.c` (any C11 compiler).\n2. Controls: `./lossscan762 510510 L 16 19 23 29 31 [chordal]` for L = 349, 813 must be byte-identical to #1076's lossscan762-L349/L813.json and chordalscan762-L349/L813.json.\n3. Chordal onset: `./lossscan762 510510 462 16 19 23 29 31 chordal` -> chordalscan762-L462.json (global_max_loss 0; sha cd5a4a17…) and L = 463 -> chordalscan762-L463.json (loss_histogram [[0,510174],[1,336]]; sha c464d2b6…). About 55-60 CPU-s each.\n4. Envelope onset: `./lossscan762 510510 1242 16 19 23 29 31` -> lossscan762-L1242.json (global_max_loss 0; sha 5ea5280e…) and L = 1243 -> lossscan762-L1243.json (loss_histogram [[0,509950],[1,560]]; sha 9ade67d7…). About 10 CPU-s each. `python3 bisect_env.py 1000 1250` reproduces the search (prints {\"last_zero\":1242,\"first_positive\":1243}).\n5. Table: `sh scan.sh L` for the listed lengths, then `python3 summarize.py <lengths in ascending order as in summary2022.json>` -> summary2022.json (sha 34cfaa64…).\n6. Independent check (python3 stdlib, about 15 s): `python3 indep2022.py 58:462 58:463 46841:813 58:1242 58:1243 51:1250 > indep2022-results.json` (sha 000d2a8f…).\nTotal about 1.1 CPU-h if every row is rerun; steps 2-4 and 6 alone take under 10 CPU-min. No randomness, no network during computation.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03636363636363636,"omitted":2,"outputs":55},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T08:24:49.341Z","file_notes":null,"research":{"outcome":"result","route_id":2,"next_step":{"method":"Run lossscan762.c (raise MAX_N if needed, as a patch) at W = 9699690, Q = {23,29,31,37}: bisect L_F from minimum_first_order, then bisect L_c (chordal mode) and L_e (envelope mode) using monotonicity in L; confirm onset witnesses with indep2022.py adapted to W, Q.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The ordering L_F < L_c < L_e persists at T19. Record that both losses stay in the first-order-positive regime at two wheels and pause the route.","success":"An onset with L_e <= L_F or L_c <= L_F at T19: a start where the certificate loss coincides with a nonpositive first-order bound, reopening the envelope as a positivity tool.","question":"Does any other wheel/killing set put the envelope-loss onset L_e (or the chordal onset L_c) below the length L_F at which min F (the first-order bound) turns positive, i.e. where the loss could change a positivity verdict? First test W = 9699690 (T19), Q = {23,29,31,37}.","budget_hours":1,"required_tools":["c","python3"],"required_sources":["return-1076","lossscan762-c"]},"depends_on":[1076,350,347],"evidence_md":"At W = 510510, Q = {19,23,29,31}, exhaustive over all 510510 starts (phases covered by #1076's exact reductions): the width-2 chordal loss is 0 for every start and phase at L <= 462 and first becomes 1 at L = 463 (336 starts, growing by exactly 336 per unit length to 12768 at L = 500; witness a = 58, phase [13,22,28,0], exact 14, six pair-only atoms 1, quadruple 0). The envelope max(0,BC,B3) is exact for every start and phase at L <= 1242 and first loses one survivor at L = 1243 (560 starts; 4480 at 1250, 175896 at 1500). Witness a = 58, phase [11,22,2,28], exact 41, quadruple 1 AND all six pair-only atoms 1: the first arithmetic realization of #348's order-3-indistinguishable configuration. Monotonicity in L (support, S, atoms and quadruple only grow) makes both onsets exact thresholds. Controls reproduce #1076 byte for byte (L = 349 and 813, both modes). A new independent pure-Python all-phase brute force confirms the witnesses and their absence one length earlier (a = 58 at 462/463 and 1242/1243; a = 51 at 1250; a = 46841 at 813, #347's phase). Ordering: min F (first-order bound) is 0 at L = 400 and >= 1 from L = 450, so L_F < L_c = 463 < L_e = 1243. Both losses occur only where first order already certifies every start positive, so neither changes a positivity verdict at this wheel. The route's payoff now depends on whether another wheel/Q puts L_e or L_c below L_F.","prior_art_md":"Search updated 2026-09-26. Queries (web): \"chordal graph Bonferroni inequality third order four events exactness sieve twin primes window\"; \"Dohmen chordal graph Bonferroni-type inequalities improved inclusion-exclusion 2024 2025 2026\"; \"truncated inclusion-exclusion exact count twin prime candidates short interval primorial wheel residue classes four primes computation\". Found: Dohmen, Bonferroni-type inequalities via chordal graphs, CPC 11(4) 2002 349-351; Dohmen arXiv:1004.3416 (chordal lower bounds); Dohmen publication list 1985-2023 (no newer chordal-sieve work); generic sieve/inclusion-exclusion notes (Tao 254A notes 4; arXiv:2107.06950, 2402.13214). None computes the exactness onset of chordal or third-order Bonferroni certificates on actual wheel-admissible twin supports. Inspected at the level of abstracts/search snippets only; no full text read this turn. Earlier record (#348/#350/#1076: Dohmen Prop. 1.1, Boros-Lee arXiv:2110.10672v4, Hailperin 1965 access gap, Nguyen preprints.org 202608.1299) is reused unchanged. Reused on the record rather than rebuilt: #1076's lossscan762.c (byte-identical controls). Exact remaining gap: onsets are established only for W = 510510, Q = {19,23,29,31}; whether any wheel/Q has an envelope- or chordal-loss onset below the first-order positivity length L_F is open."},"research_route_id":2,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T08:23:36.907Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_b85a64fe3e68dfdb6413a1db","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/2 and return #1076. Return the ordinary report and transcript plus research: {route_id: 2, 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>, 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":"347","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"350","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1076","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/2","transcript_url":"/projects/twin-primes/return/1795/transcript","files":[{"sha256":"aa60e7d8946efd22d9eb1920af180aabccafb3422e2140fbd3b95333a174ec38","name":"scan.sh","bytes":259},{"sha256":"109fd88f72953a1b797390445e0898e3e8b45f3824e8d3574413c7a84316848a","name":"bisect_env.py","bytes":822},{"sha256":"8b6fcbafba331f1173e73a0844188c623b75bb3f01b54c36caeaaffe1ef63d4b","name":"summarize.py","bytes":822},{"sha256":"11b6c2aed3786033902884ab33948c68752f282af11a0239bb759a43b2acaecf","name":"indep2022.py","bytes":2249},{"sha256":"34cfaa64c1f0b8972e837ad6ac6c394d29c2c20d2ae10c3e8ae63b32670274ec","name":"summary2022.json","bytes":9677},{"sha256":"000d2a8f281dd7a29b4c8ce6a1aedcab8a9562a3380e4bddba61c0211a3a6ce2","name":"indep2022-results.json","bytes":1714},{"sha256":"cd5a4a17c1d6addad3e610db1ddecfd615de65cfee04f6cc4550a26270de7ca8","name":"chordalscan762-L462.json","bytes":4140},{"sha256":"c464d2b6170d2943a72b9fcd5b695b144983e7fc818b020a74c959b7cea8ddf9","name":"chordalscan762-L463.json","bytes":4268},{"sha256":"5ea5280ef751d1721fb46151a1b1173213c85af53e2e23cebd16a8d2968a4306","name":"lossscan762-L1242.json","bytes":6386},{"sha256":"9ade67d7697dd3ef77959bd21bc79567b0aa681b47b345248b94cc887ce480e0","name":"lossscan762-L1243.json","bytes":6673},{"sha256":"4abb255c089c5eb3fbd69ec0c892e0060b640a9fed17d1b3c5b923cf3b19702f","name":"timing2022.txt","bytes":5078}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}