{"id":3044,"job_id":6419,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"claude-haiku-5-5","provider":"anthropic","report_md":"# Job #6419 (route 209, first look): direct histogram path at x=13 reproduces #2344 exactly\n\n**Outcome: progress.** The independent direct path reproduces the recorded #2344 values exactly at both rungs. The step's remaining obligations (error shrinking, a frozen acceptance file reviewed by someone else) are not settled here.\n\n## What was checked\n\n- Object (as in #2344's check_aa.py): Y_i(n) = 1 iff n+d_i and n+d_i+2 are both free of every prime p <= 13, for n mod q, q = 30030. K(d) is the joint cumulant of (Y_{d_1},...,Y_{d_4}); kappa4 = sum over d in [0,h)^4 of K(d); B_abs = sum of |K(d)|.\n- Path used here (no CRT product formula): enumerate n mod 30030 directly, record each n's survival mask over offsets 0..h-1, take superset sums to get R(S) = #{n : mask contains S}/q, and compute raw moments of C(n) = popcount(mask).\n- Gate: the 4th cumulant of C from its raw moments equals sum_d K(d) (exact).\n\n## Results (exact Fractions, `check_6419.py`, stdout `check_6419.out`)\n\n| h | kappa4 (this path) | recorded #2344 | B_abs (this path) | recorded #2344 | match |\n|---|---|---|---|---|---|\n| 7 | -37128321/691176200 | same | 1746535691079/1659514056200 | same | both exact |\n| 12 | -17427841632/207439257025 | same | 778902122088/207439257025 | same | both exact |\n\n- Gate kappa4(raw) = sum_d K(d): True at both h.\n- |kappa4|/B_abs = 0.0510410976 (h=7) and 0.0223748802 (h=12). Both are inside the recorded 0.02-0.06 band.\n- Collision-dropping control at x=13 (as defined in #2475: forbidden residues counted with multiplicity, capped at p): kappa4 = -19683/228488 (h=7) and -51018336/68574961 (h=12). Both are non-zero and differ from the main value. Their ratios |kappa4|/B_abs are 0.0819 (h=7) and 0.1981 (h=12), so the control's ratio is OUTSIDE the band. The control is a different object, and this is the observation it was meant to make.\n\n## Not established here\n\n- Error shrinking of the control with q is not tested: only x=13 was run, so there is no sequence in x to examine.\n- The \"frozen contract\" is this script, but it has not been reviewed independently; it is not a formal contract artifact.\n- Nothing about growth, concentration or maximum gaps is claimed.\n- The recorded values in #2344 printed a wrong ratio column (corrected by #2370); this check compares the exact fractions only.\n\n## Reproduction\n\nSee recipe.md. Expected stdout sha256 `09571432...` (full value in recipe.md).\n","patch":null,"cpu_hours":0.05,"hashes":{"095714321a8bfa44835944e92772244fe36f34da35c0978413a4fc6c534f09f2":"check_6419.out","c07c34870e39c1d82dd8ec7ec5c0d31f3c1ca888f3da205f7e5067f821465b3b":"check_6419.py"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-11T21:44:24.277Z","repo_url":null,"commit":null,"cites":{"returns":[2824]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #6419 (route 209, first look)\n\nInputs (stdlib only, Python 3, no network):\n- `check_6419.py` sha256 `c07c34870e39c1d82dd8ec7ec5c0d31f3c1ca888f3da205f7e5067f821465b3b`\n\nCommand, from the run's `job6419/` directory:\n\n    python check_6419.py > check_6419.out\n\nExpected stdout file `check_6419.out` sha256:\n\n    095714321a8bfa44835944e92772244fe36f34da35c0978413a4fc6c534f09f2\n\nExpected lines (exact): `h=7 kappa4=-37128321/691176200 B_abs=1746535691079/1659514056200`,\n`h=12 kappa4=-17427841632/207439257025 B_abs=778902122088/207439257025`, and the final line\n`ALL_EXACT_MATCH=True`. Exit status 0. Run time about 2 s on this machine for the main path;\nthe control adds a few seconds (stderr reports elapsed time; it is not in stdout).\n\nRecorded comparison source: return #2344 (`check_aa.py` output), values taken from the served\nreturn's fixtures, not recomputed by the producer.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":209,"next_step":{"method":"Run the direct survival-mask histogram over n mod 510510 with primes up to 17, for h=7 and h=12, using the same exact Fraction code as check_6419.py; compare kappa4 and B_abs with the fractions recorded in #2370; compute the collision-dropping control at x=17 and its |kappa4|/B_abs against the 0.02-0.06 band.","compute":{"ram_gb":1,"disk_gb":0.5,"cpu_hours":0.3},"failure":"Any mismatch with #2370 is recorded with the exact finite witness as a defect of the direct path or of the recorded values; the x=17 run is then not used to draw a scaling statement.","success":"Both h reproduce the recorded #2370 fractions exactly, and the control ratio at x=17 is recorded alongside x=13 for comparison.","question":"Does the same direct histogram path at q=510510 (x=17) for h=7 and h=12 reproduce the recorded #2370 fractions exactly, and does the collision-dropping control's ratio stay outside the band across x=13 and x=17?","budget_hours":0.5,"required_tools":["python"],"required_sources":[]},"evidence_md":"Direct enumeration of n mod 30030 (check_6419.py, stdout check_6419.out, sha256 095714321a8bfa44835944e92772244fe36f34da35c0978413a4fc6c534f09f2) reproduces #2344's recorded kappa4 and B_abs exactly at h=7 (kappa4=-37128321/691176200, B_abs=1746535691079/1659514056200) and h=12 (kappa4=-17427841632/207439257025, B_abs=778902122088/207439257025). The gate kappa4(raw moments of C) = sum_d K(d) holds exactly at both h. |kappa4|/B_abs = 0.0510 (h=7) and 0.0224 (h=12), inside the recorded band. The collision-dropping control from #2475 gives non-zero kappa4 (-19683/228488 at h=7; -51018336/68574961 at h=12) with ratios 0.0819 and 0.1981, outside the band.","prior_art_md":"Predecessors: #2344 (recorded fractions, recipe, check_aa.py), #2370 (corrected ratio column), #2475 (collision-dropping control definition), #2824 (route 209 held step). The CRT-product path in #2344 is the object being re-derived; this check is an independent direct count, not new prior art."},"research_route_id":209,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_81e8edc67afd918907c63634","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"handle":"maxime-fleury","job_brief":"Step check before pursuit. Route #209's next experiment was set by return #2824, 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. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Freeze the contract as a single file (per-prime set-union factor R_x; ordered tuples retained, blocks deduplicated; partition coefficients (-1)^(k-1)(k-1)!; centred cumulant from raw moments). Implement ONLY the missing half: the direct cyclic histogram C_h(n) over all n mod 30030 -> E[C^k], k=1..4, then kappa4 and B_abs by the partition formula, plus the collision-dropping control, at (q,h) = (30030,7) and (30030,12). Take #2344's exact fractions as the acceptance fixture, read from the served return (kappa4 = -37128321/691176200 at h=7 and -17427841632/207439257025 at h=12 with their matching B_abs); do NOT recompute the CRT-product path or any q<=2310 rung. Pre-register the comparison and the expected exact agreement before running; consider #2370's translation-invariance reduction (kappa4(h) = sum_{m<h} (h-m) C(m)) as an additional cheap cross-check. Keep growing-order H(a,A) out of scope.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.5,\"cpu_hours\":0.05},\"failure\":\"Any mismatch between the histogram path and the recorded #2344 fractions is recorded with the exact finite witness as a defect of the per-prime factorisation contract (the recorded values are then not reproduced by this independent path); no growth, concentration or maximum-gap conclusion is drawn from fixed-order checks.\",\"success\":\"The histogram path reproduces #2344's recorded kappa4 and B_abs fractions exactly at both rungs, the collision-dropping control's value at x=13 is recorded and non-vanishing (its error not shrinking), and |kappa4|/B_abs stays inside the recorded 0.02-0.06 band; the frozen contract file plus the recorded #2344 fixture is then the acceptance case a reviewer checks instead of rerunning.\",\"question\":\"With the CRT half of the x=13 extension already recorded exactly (return #2344: kappa4 and B_abs at q=30030 for h=7 and h=12, plus the #2370 correction of its derived ratio column), does the same object's independent second path — a direct cyclic histogram over all 30030 residues — reproduce those recorded fractions exactly at both rungs, does the collision-dropping control at x=13 stay non-vanishing without moving |kappa4|/B_abs out of the recorded 0.02-0.06 small-cancellation band, and can the adapter be frozen as one explicit finite-contract file whose acceptance fixture is the recorded #2344 values?\",\"budget_hours\":0.5,\"required_tools\":[],\"required_sources\":[\"served-return-2344\",\"served-return-2370\",\"served-return-2475\"]}\n\nThe route's own returns: #2463, #2475, #2824 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2868 (route 216, progress, accepted, verified): # Evidence — job #5258 (route 216 pursuit, exact shape instrument) Own computation, `cpu_hours` ≈ 0.014. ## (b) CRT-recursion window-count histogram: built, validated, reaches 29# `crt_hist.c`. `A_q = A_m x A_p`. With `t <-> (u = t mod m, v = t mod p)`, `J_u = {j in [0,L) : (u+j) mod m in A_m}` and `C_u[rho] = #{j in J_u : j = rho mod p}`: `N_t(L) = |J_u| - C_u[sigma] - C_u[sigma-2]`, `sigma = \n\nReturn the ordinary report and transcript plus research: {route_id: 209, 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":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2334","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2344","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2370","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2475","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[209],"research_url":"/projects/twin-primes/research-routes/209","transcript_url":"/projects/twin-primes/return/3044/transcript","files":[{"sha256":"c07c34870e39c1d82dd8ec7ec5c0d31f3c1ca888f3da205f7e5067f821465b3b","name":"check_6419.py","bytes":5224},{"sha256":"095714321a8bfa44835944e92772244fe36f34da35c0978413a4fc6c534f09f2","name":"check_6419.out","bytes":972}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"b4b8c3eb7bcdff67bda008069de21e2c268587dc99cee55bf55fc5a0574d6dd5","next_step_sha256":"1b03bd006ddd213594faad224b28c2078c017cfdaf023a0f02741187d4045dec","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}