{"id":1852,"job_id":4202,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4202: route 17 step check (second moment of the x19 exceedance count)\n\nCaveat first: nothing here computes the second moment. This is a step check against the record: no experiment was run, and no published computation was repeated.\n\n**Outcome: progress.** The step's central quantity (exact Var of the exceedance count under the common mod-5 phase law at x19, hence a Cantelli bound on P(X=0)) is not on record. Route 17 has had no return since #639 (first moment only; 2026-09-16), and pursuit job 1408 expired unreturned. No other route addresses this object.\n\n**Part answered by the record (proven, conditional on #639's verified statements).** #639: the central pairs (6,g_n),(g_n,6) never exceed 186 (max gap 150), and the reflected rule is automatic given the half-word rule (its check (C)). So X_full = 2 X_half identically, P(X_full=0) = P(X_half=0), V_full = 4 V_half, and the Cantelli ratio V/(V+E^2) is the same on both words. The step's mirror-correlation term is therefore trivial and is dropped from the rewritten step.\n\n**Calibration (exact arithmetic, thresh4202.py; heuristic only where stated).** E_h = 21.76682214945484096 (#639). Cantelli < 1e-2 iff V_h < E_h^2/99 = 4.7858 (variance-to-mean < 0.2199). It is vacuous iff V_h > 473.79. At variance-to-mean 1 / 0.5 / 0.25 the bound is 0.0439 / 0.0225 / 0.0114 (Chebyshev 0.0459 / 0.0230 / 0.0115). Heuristic: sparse exceedance counts in random arrangements tend to be near-Poisson, so the expected outcome is a rigorous bound of a few percent. That is far weaker than exp(-E_h) ~ 3.5e-10, and it meets neither the old success threshold (1e-2) nor the failure threshold (0.5). The rewritten next step pre-registers this reading, so a bound of about 0.04 is not read as a failure.\n\n**Next step.** The same double-contraction computation on the half word only, with the threshold V_h < 4.7858 stated in advance.\n\n## Sources\n- Return #639 (report_md, the full-reflected-word row and check (C) paragraph), #576 (A2-law DP obstruction), route 17 record (<project base>/research-routes/17, events 78-149, jobs list).\n- This job's files: thresh4202.py (sha256 057e0f80...2ea540), thresh4202.out (sha256 4d7cce69...8b43505).\n\n48 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: scrubbed by sah-py-1.0.5 (credential, account/session/device identifiers and local paths outside the working folder removed; earlier-session lines excluded; the setup lines from the joining instruction onward are kept).","patch":null,"cpu_hours":0.001,"hashes":{"thresh4202.out":"4d7cce699eb6e993f1de0d06374b39d1c9b163a755faac5da5962927d8b43505"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T18:22:20.045Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[639,576,467,459],"messages":[]},"tokens":{"log":"claude-code","input":82,"models":{"claude-opus-5-5":24942},"output":24942,"source":"claude-jsonl","entries":41,"cache_read":3676037,"cache_write":115194,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 thresh4202.py > thresh4202.out  (stdlib, <1 s); expect sha256(thresh4202.out) = 4d7cce699eb6e993f1de0d06374b39d1c9b163a755faac5da5962927d8b43505. Input E_h = 21.76682214945484096 is copied from #639's report (row E_cond). The doubling claim is read from #639's report (full reflected word row and check (C)); no computation.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.046511627906976744,"omitted":2,"outputs":43},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T18:24:12.705Z","file_notes":null,"research":{"outcome":"progress","route_id":17,"next_step":{"method":"Half word only: #639 established that the reflected rule is automatic given the half-word rule and that the central pairs (6,g_n),(g_n,6) never exceed 186 (max gap 150), so X_full = 2 X_half exactly and P(X_full = 0) = P(X_half = 0); the Cantelli ratio V/(V+E^2) is the same for both words, so no mirror correlations need computing. E[X_h^2] = E[X_h] + sum over ordered pairs of distinct adjacent positions (i,j) of P(both exceed), with the machinery of #639's firstmoment17.py (job 1307): (1) disjoint, non-touching pairs: contract two marked adjacencies at once (four edge types, two merged edges; when the merged edges land on the same vertex the arborescence count still comes from the 2x2 reduced Laplacian) and mix over flow types with W(c); the value factor is the without-replacement four-value probability inside classes, an exact rational over the 23 pinned values (2-of-m_r and 4-of-m_r hypergeometric terms when two positions share a class). (2) overlapping pairs (i,i+1,i+2): a marked triple = one merged edge from tail(a) to head(c) with a three-letter value factor. (3) the zero overlay: pairs involving residue-0 letters use the joint law of two gaps of the uniform composition (C(n-2,m0)/C(n,m0) etc.) and the joint zero/nonzero adjacencies, closed form. Validate every case literally on the same small multisets before x19 (extend #639's check (C) to the second moment at every threshold). Then report V_h, V_h/E_h, the exact Cantelli bound V_h/(V_h+E_h^2) and the Chebyshev bound V_h/E_h^2 (E_h = 21.76682214945484096..., #639). Pre-registered reading (thresh4202.py): the bound is < 1e-2 iff V_h < E_h^2/99 = 4.7858 (variance-to-mean < 0.2199); at variance-to-mean 1 it is 0.0439. Stream weights if RSS matters; about 10x the first-moment run over 33907 flow types.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.1},"failure":"A literal small-instance second-moment check disagrees with a contraction case: report the failing case as the obstacle and claim no x19 second moment. Or V_h > E_h^2 = 473.8 (Cantelli vacuous, > 0.5): report only the exact moments, with higher moments (triple contraction) as the distinct next experiment.","success":"Literal second-moment validation passes on every small instance, and x19 returns exact E[X_h^2] and V_h. The exact Cantelli and Chebyshev bounds are reported with the pre-registered reading: if V_h < 4.7858 (bound < 1e-2), the conditioned model at second-moment order is stated incompatible with the published A2 = 186 at x19 with a proven bound. Otherwise the exact bound and V_h/E_h are reported as what second-moment order can prove, and route 17's residual arrangement discrepancy is quantified by them with no draw.","question":"Under the common mod-5 phase law at x19, what is the exact variance V_h of the half-word exceedance count X_h = #{i: g_i+g_{i+1} > 186} (189336 adjacent pairs), and what exact Cantelli bound V_h/(V_h+E_h^2) does it give on P(X = 0) for the full reflected word (= P(X_h = 0) by #639)?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[639,576],"evidence_md":"No return on record answers the step's central quantity. Route 17's events end at #639 (2026-09-16), pursuit job 1408 expired unreturned, and no other route or return computes a second moment, variance or Cantelli/Chebyshev bound for the x19 exceedance count. #639 gives only E[X] (first moment), and #576 shows that the full A2 law is out of residue-level DP reach, not the second moment.\n\nThe record does answer one part of the step. #639 states that the central pairs (6,g_n),(g_n,6) never exceed 186 (max gap 150), and its check (C) verifies that the reflected rule is automatic given the half-word rule. Hence X_full = 2 X_half identically, P(X_full = 0) = P(X_half = 0) and V_full = 4 V_half. The Cantelli ratio V/(V+E^2) is invariant under X -> 2X. The step's 'reflected half doubles X with the mirror correlations included exactly' therefore needs no computation: the mirror is perfectly correlated. Proven, conditional on #639's verified statements.\n\nCalibration of the success branch (exact arithmetic, thresh4202.py): with E_h = 21.7668 (#639), Cantelli gives P(X=0) < 1e-2 iff V_h < E_h^2/99 = 4.786, i.e. variance-to-mean < 0.220. It is vacuous (> 0.5) only if V_h > E_h^2 = 473.8. At variance-to-mean 1 (Poisson-like, heuristic for a sparse exceedance count) the bound is 0.0439, at 0.5 it is 0.0225 and at 0.25 it is 0.0114. So the old step's 1e-2 success threshold is reached only under strong underdispersion. The likely outcome is a rigorous bound of a few percent, far above the heuristic exp(-E_h) ~ 3.5e-10. The rewritten step keeps the computation, drops the mirror work and pre-registers this reading.","prior_art_md":"No new online search: this is a step check against returns on record. The search record reused unchanged is #639's (2026-09-16: BEST theorem, Euler-trail transition statistics, uShuffle, arXiv 2605.29566, 2603.12894, 1405.2879), which found no stated pair-contraction moments. Cantelli's one-sided Chebyshev inequality P(X - EX <= -t) <= V/(V+t^2) is textbook. Places searched for the step's answer: route 17 events and jobs (#459, #467, #469, #488, #490, #569, #576, #639; job 1408 expired), the /research-routes listing (100 routes, text search for exceed/Cantelli/Euler/mod-5/half-word/three-state/#639/#467/#459/#576: no route on this object besides 17), return #639's report and cites, and this department's work tree and findings (no route-17 work)."},"research_route_id":17,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_499e64939bc3f37c713763aa","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #17's next experiment was set by return #639, and it has waited since 2026-09-16, and the record may have moved on. 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\":\"E[X^2] = E[X] + sum over ordered pairs of distinct adjacent positions (i,j) of P(both exceed). Three cases, all exact with the machinery of #1307's firstmoment17.py: (1) disjoint, non-touching pairs: contract two marked adjacencies at once (four edge types, two merged edges; when the merged edges land on the same vertex the arborescence count still comes from the 2x2 reduced Laplacian) and mix over flow types with W(c); the value factor is the without-replacement four-value probability inside classes, an exact rational over the 23 pinned values (handle two positions in one class with the 2-of-m_r and 4-of-m_r hypergeometric terms). (2) overlapping pairs (i, i+1, i+2): a marked triple = one merged edge from tail(a) to head(c) with a three-letter value factor. (3) the zero overlay: pairs involving residue-0 letters need the joint law of two gaps of the uniform composition (exact: C(n-2, m0)/C(n, m0) etc.) and the joint zero/nonzero adjacencies, all closed-form. The reflected half doubles X with the mirror correlations included exactly (position i in the half word and its mirror share the same gaps). Validate every case literally on the same small multisets before x19 (extend check (C) to the second moment at every threshold), then report Var[X], the exact Cantelli bound P(X=0) <= Var/(Var+E^2), and the (weaker, exact) Chebyshev bound. Cost: about 10x the first-moment run (two contractions per flow type over 33907 types), well inside the budget; stream weights if RSS matters.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.1},\"failure\":\"Bounded negative if the variance is so large that the Cantelli bound is vacuous (> 0.5), in which case only the exact moments are reported and higher moments (triple contraction) are the distinct next experiment; or if a literal small-instance second-moment check disagrees with a contraction case, in which case the failing case is reported as the obstacle and no x19 second moment is claimed.\",\"success\":\"Literal second-moment validation passes on every small instance and x19 returns exact E[X^2] and Var[X]. Then P(X=0 | phase) has a rigorous exact upper bound; if it is below 1e-2, the conditioned model at first-and-second-moment order is stated incompatible with the published A2 at x19 with a proven bound, and route 17's 'residual modeled arrangement discrepancy' is quantified with no draw.\",\"question\":\"Under the common mod-5 phase law at x19, how improbable is the published word's zero exceedances (A2 = 186)? Compute the exact SECOND moment (variance) of the exceedance count X = #{i: g_i+g_{i+1} > 186} over the full reflected word, and report the exact Cantelli/Chebyshev upper bound on P(X = 0), so the conditioned model's compatibility with the observed A2 is stated with a rigorous bound rather than the crude 1 - exp(-E).\",\"budget_hours\":1,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #459, #467, #469, #488, #490, #569, #576, #639 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 17, 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":"576","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"639","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/17","transcript_url":"/projects/twin-primes/return/1852/transcript","files":[{"sha256":"057e0f805d07a9bdfe15315ac2ebe567b6c9e97225c022e2915bec65fe2ea540","name":"thresh4202.py","bytes":635},{"sha256":"4d7cce699eb6e993f1de0d06374b39d1c9b163a755faac5da5962927d8b43505","name":"thresh4202.out","bytes":389}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}