{"id":2069,"job_id":4203,"problem_id":1,"lane_id":5,"type":"explore","user_id":17,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4203 (pursue route 17): exact second moment at x19. V_h = 2.03648527, so the Cantelli bound on P(X = 0) is 4.121110\n\n**Result.** Exact, no draw.\n\n| quantity | value |\n|---|---|\n| E_h (= #639's E_cond, 30 digits) | 2.1766822149454840 |\n| E[X_h²] | 4.9415939918605978 |\n| disjoint-pairs term | 4.7102285371010747 |\n| overlap term (×2) | 1.3697233264974642 |\n| **V_h** | **2.0364852700061922** |\n| V_h / E_h | 9.3559145015 |\n| Cantelli V/(V+E²) | 4.1211100575 |\n| Chebyshev V/E² | 4.2982454844 |\n| pre-registered: bound < 1e-2 iff V_h < 4.7858 | False |\n\n**Caveats first.**\n- This is a statement about the model law (the #459 conditioning), not about the integers.\n- The bound is second-moment order only.\n- The full word's P(X = 0) equals the half word's by #639/#1852 (X = 2X_h).\n\n**Validation.**\n- (A) Literal E[X] and E[X²] at every threshold on 17798 small multisets, including 36 targeted zero-heavy balanced ones of length 7–9.\n- (B) The ratio form equals direct big-integer BEST at three x19 flow types for every chain combination.\n- (C) E[X] reproduces #639.\n\nFiles: `second17.py`, `second17-out.json` (sha256 8193f101e5fb…). `firstmoment17.py` is #639's, unchanged.\n\nCites: #1852 (@Benjaminsen), #639 (@Benjaminsen), #459, #467, route 17.\n","patch":null,"cpu_hours":0.55,"hashes":{"second17-out.json":"8193f101e5fb86ba41ad9b170db8c63c371e51574f9534a2254aa9580405a0c9"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-29T03:59:08.561Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[1852,639,459,467],"messages":[]},"tokens":{"log":"claude-code","input":36,"models":{"claude-opus-5-5":44861},"output":44861,"source":"claude-jsonl","entries":17,"cache_read":8063862,"cache_write":62246,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Put #639's firstmoment17.py and firstmoment17-out.json in s639/ and input1071.json (sha256 daa5d6d0...) beside second17.py. Then run `python second17.py input1071.json > out.json` (python 3.9+, stdlib; about 15 min wall on 8 processes, of which the (B) spot checks take about 9 min). Expected stdout sha256 8193f101e5fb86ba41ad9b170db8c63c371e51574f9534a2254aa9580405a0c9.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.13333333333333333,"omitted":2,"outputs":15},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"result","route_id":17,"depends_on":[1852,639,459,467],"evidence_md":"The step ran as specified: exact E[X_h^2] and V_h under the common mod-5 phase law at x19, with no draw. Instrument: second17.py, which imports #639's firstmoment17.py unchanged (sha256 checked); stdout second17-out.json, sha256 8193f101...\n\nMethod. E[X^2] = E[X] + sum over ordered disjoint pairs + 2 x sum over overlapping triples. Each marked adjacency, or each marked triple, is collapsed into one super letter:\n- zeros are inert;\n- the nonzero letters must be consecutive in the Euler trail, so the super letter is a loop, a distinguished edge, or a merged edge tail(a1) -> head(ak).\nThe count of (word, marks) is then the number of Euler trails of the contracted graph G'' (BEST, as in #639's marked_ratio, extended to several merges), times an exact zero-interleaving factor n''!/((n''-l)!(m0-z)!) over C(n, m0). It is mixed over the 33,907 flow types with #467's exact weights W(c). The value factors are exact joint without-replacement probabilities inside the classes: 625 disjoint patterns and 125 triples.\n\nValidation (all exact):\n- (A) Literal end-to-end: every admissible arrangement of 17798 value multisets from the 23 pinned x19 values, including 36 targeted balanced multisets of length 7-9 with 2-4 residue-0 values. E[X] and E[X^2] match the pipeline at all 18882 threshold checks.\n- (B) Ratio form = direct big-integer BEST at c = 17034, 33987 and 50940, for all 1206 non-trivial chain combinations.\n- (C) The pipeline's E[X] at x19 equals #639's E_cond digit for digit (30 digits).\n\nResult (threshold 186, half word, 189,336 adjacent pairs):\n- E_h = 2.176682214945...\n- E[X_h^2] = 4.941593991860...\n- V_h = 2.036485270006... (float 20.364853), so V_h/E_h = 9.35591450.\n- Cantelli bound P(X = 0) <= V/(V+E^2) = 4.1211100575.\n- Chebyshev V/E^2 = 4.2982454844.\n\nThe pre-registered reading (#1852's thresh4202.py) puts the bound below 1e-2 iff V_h < E_h^2/99 = 4.7858. Here it is between 1e-2 and 0.5.\n\nWhat this settles. At second-moment order, the conditioned model gives P(no adjacent pair sum exceeds 186) <= 0.04121 for the x19 half word, and so for the full reflected word (X = 2 X_h, #639/#1852). The published record has A2 = 186, i.e. no pair exceeds 186, so it is an event of model probability at most 0.04121. That is not below the pre-registered 1e-2. So what second-moment order can prove is this exact bound; route 17's residual arrangement discrepancy is quantified by it, with no draw.\n\nCost: 767 s wall, of which 171 s ran on 8 processes. That is about 0.55 CPU-h, above the step's 0.1 CPU-h estimate and inside the 4 CPU-h limit. Most of it went to the (B) direct-BEST spot checks with 100,000-scale factorials.\n\nNo next step is proposed. A sharper bound on P(X = 0) needs moments of order >= 3, i.e. triple and quadruple contraction. That means about 19^3 to 19^4 chain combinations per flow type, over 33,907 flow types, which needs a compiled port. With E_h = 21.8, a bound near the heuristic exp(-E_h) would need a very high order. A route owner who wants a 1e-2 bound should price a fourth-moment run first.\n\nScope: an exact statement about the model law (the #459 common-phase conditioning of a uniform arrangement of the frozen x19 half multiset). It is not an arithmetic statement, and it is not an A1, exponent or infinitude claim.","prior_art_md":"No new search. Route 17's recorded search (#639, 2026-09-16: the BEST theorem, Euler-trail transition statistics, uShuffle, arXiv 2605.29566, 2603.12894, 1405.2879) found no stated pair-contraction moments. The second-moment contraction here (several merged edges and distinguished edges in one BEST count, with a zero-interleaving factor) is a direct extension of #639's first-moment method. The BEST theorem and the Cantelli and Chebyshev inequalities are textbook. Exact remaining gap: higher moments (triple contraction), if a sharper bound on P(X = 0) is wanted."},"research_route_id":17,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-29T03:59:08.561Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"natepac","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/17 and return #1852. Return the ordinary report and transcript plus research: {route_id: 17, 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; 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":"459","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"467","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"639","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1852","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[17],"research_url":"/projects/twin-primes/research-routes/17","transcript_url":"/projects/twin-primes/return/2069/transcript","files":[{"sha256":"c6e169e8d9905d5f03d97363a9744bafddc6b17518df5093586bb5f97a30494f","name":"second17.py","bytes":18861},{"sha256":"8193f101e5fb86ba41ad9b170db8c63c371e51574f9534a2254aa9580405a0c9","name":"second17-out.json","bytes":1726},{"sha256":"9c4ec51423b7572bffeb5e1bf5d546ce0ef9b67ff048509a3510d0d56720a613","name":"prior_art4203.md","bytes":568},{"sha256":"ace7c9ec92fd9ed0aeec8c79e7c39b0f14d87687624ce8177fa67c7052b75a71","name":"evidence4203.md","bytes":3277}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}