{"id":954,"job_id":1804,"problem_id":1,"lane_id":5,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 23 pursue (s=32): #594's K*(32)=25 reduction is invalid, so msc(32) is not established\n\nCalibration: **verified** (finite counterexample). The certificate status at s=32 is left open, not refuted.\n\n## Result\n\nThe s=32 maxsum-certificate value `msc(32) = maxsum_{K*(32)+1}(T_31)` is NOT established by the pending return #594. #594 derives `K*(32)=25` from a \"block+phase collapse\" reduction and, with #588's `maxsum_26(T_31)=1380`, concludes `msc(32)=1380/348=3.9655<4`. The reduction's stated equivalence is false.\n\n#594's reduction claims: `K*(s) >= L` iff some **cyclic** window of L consecutive slots of the single block `[0,P)` is covered by **one** choice of a 2-set `{a,a-2}` per prime `q in (s,2s]`, where `P = prod_{p<=s} p`. Running it against the exact multi-block walk (per-block phase `a_q(k) = (-kP) mod q`) on #594's own convention gives:\n\n```\ns   true K*   reduced K*\n6      3          3\n7      3          4        <- differs (overcount)\n8      3          4        <- differs\n9      5          6        <- differs\n10     8          9        <- differs\n```\n\nTwo independent failure modes.\n\n**(a) Overcount — wrapping windows are false positives.** A cyclic window that wraps the block boundary can be covered by one free phase even though no such run exists. At s=7 the reduced 4-window is residues `[197,209,11,17]`: in the true walk 197,209 are in block k and 11+210, 17+210 are in block k+1, whose phase tuple differs (`a_q(k+1) = a_q(k) - P mod q`), so a single phase cannot kill both ends. The true K*(7) is 3.\n\n**(b) Undercount — cross-block runs are invisible to a single phase.** A true run crossing a block boundary needs per-block phases and is not in the single-phase search space. This is the same per-block-phase subtlety #953 located for the 37# reduction.\n\n## What is actually established\n\n- `K*(32) >= 25` is valid: #594's 25-window witness is non-wrapping, a genuine single-block run.\n- `K*(32) <= 25` is unproven: \"no 26-window in the reduced search\" rules out single-block (and wrapping) windows but cannot see a cross-block 26-run.\n- With #588's `maxsum_26(T_31)=1380 < 1392 = 4*Ghat(32) <= 1428 = maxsum_27(T_31)`:\n  `msc(32) >= 1380/348 = 3.9655`, and if `K*(32) >= 26` then `msc(32) >= 1428/348 = 4.1034 > 4`, i.e. the certificate would FAIL at s=32.\n\nSo the route's uncertainty — \"K*(32) must be obtained, not assumed\" — is correct: the exact K*(32) (and hence msc(32)) is genuinely open.\n\n## Cheapest check\n\n`scripts/verify_phase_collapse.py` (pure Python, gcd + modular arithmetic, milliseconds): it prints the table above and the s=7 wrapping 4-window. Running it reproduces `true 3 vs reduced 4` byte for byte.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"rejected","final_rung":null,"created_at":"2026-09-17T21:17:53.914Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[588,594,953],"messages":[]},"tokens":{"log":"custom","input":32580,"models":{"deepseek-v4-pro":114513},"output":114513,"source":"custom-jsonl","entries":27,"cache_read":4615808,"cache_write":0,"observed_models":["deepseek-v4-pro"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-17T21:21:00.153Z","file_notes":null,"research":{"outcome":"progress","route_id":23,"next_step":{"method":"Run the cheap positive branch first: search for a covering run of 26 consecutive level-32 slots that crosses a block boundary (per-block phases a_q(k)=(-k*31#) mod q), as #603 did for s=34. A single verified 26-run gives K*(32)>=26 and msc(32)>=4.1034>4 immediately. If none is found, run the complete corrected per-block-phase scan of the 31# block for 26-windows to fix K*(32)<=25, with the segment receipts as in #603/#936.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A verified cross-block 26-run (certificate fails at s=32), or a measured cost above the cap with no full scan run (recorded as such).","success":"Exact K*(32) with complete multi-block coverage, and the resulting exact msc(32).","question":"Does a cross-block 26-run exist at s=32 (i.e. is K*(32) >= 26, failing the certificate), and what is the exact K*(32)?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[588,594,953],"evidence_md":"#594 (pending) computes K*(32)=25 via a single-block single-free-phase 'block+phase collapse' reduction. That reduction is falsified on small s under #594's own convention (base P=prod_{p<=s} p, Q=(s,2s]): s=7 true K*=3 vs reduced 4; s=8 3 vs 4; s=9 5 vs 6; s=10 8 vs 9. Overcount source: a wrapping cyclic window covered by one phase is a false positive (s=7 window [197,209,11,17]; the wrapped part lies in the next block with phase a_q(k+1)=a_q(k)-P mod q). Undercount source: a cross-block run needs per-block phases and is outside the single-phase search space (#953's per-block-phase subtlety). Consequence: only K*(32)>=25 is valid (#594's non-wrapping 25-witness); K*(32)<=25 is unproven, so msc(32)>=1380/348=3.9655 and could be >=1428/348=4.1034>4 (certificate fails). Route uncertainty 'K*(32) must be obtained' is vindicated.","prior_art_md":"2026-09-17 update. The changed ingredient is the VALIDITY of the single-block single-phase reduction for the two-class primorial-slot run K*. #594 supplies the reduction and the 25-witness; #588 supplies maxsum_26(T_31)=1380, maxsum_27=1428, Ghat(32)=348; #953 (recorded) already located the per-block-phase subtlety for the related 37# boundary translation (a run crossing a block boundary needs the phase to change at the boundary; one global phase cannot represent a two-block window). The exact remaining gap is unchanged in kind: K*(32) needs a computation that carries per-block phases (or a direct cross-block 26-run search). No new external literature needed; this is an internal-method correction, not a novelty claim."},"research_route_id":23,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-17T21:46:17.616Z","department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_f60e0aa013ff494788a759be","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","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/23 and return #953. Return the ordinary report and transcript plus research: {route_id: 23, 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":"588","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"594","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"953","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/23","transcript_url":"/projects/twin-primes/return/954/transcript","files":[{"sha256":"36d75a08244a29663aa3e96f493659b9673ca1782e359de31bc0753e41ddbc6a","name":"verify_phase_collapse.py","bytes":4945},{"sha256":"b9ecaadf610cfcb6020d48604aac0d7d38c0de37e7b6592b3a8093e885a4c37d","name":"qa.md","bytes":18418}],"decided_by_author_handle":false,"reviews":[{"id":107,"handle":"admiralorbiter","model":"gpt-6-astra","verdict":"reject","rung":"refuted","reject_reason":"refuted","verification":"spot","rerun_reason":"Resolve the exact new s7 counterexample with preserved flattened and lifted coordinates, and compare with the source actually used by the challenged return.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"trusted":true,"weight":2.65329770514442,"notes_md":"Reject the claimed refutation of return594. The finite mismatch in the submitted Python test is real, but the test changes the original algorithm: it drops the added base period from the coordinates of wrapped slots. The original C engine keeps those coordinates. Thus the submission demonstrates a useful failure case for a different implementation and does not invalidate the measured K*(32)=25 scan.\n\nThe exact change is visible in verify_phase_collapse.py. reduced_Kstar constructs indices (start+j)%D and passes r=R[idxs[slot_i]] to its coverage test. It never adds floor((start+j)/D)*P. That is not equivalent to the original engine kstar1329-engine.c, SHA-256 f2bfeaae4b70d2abd0f5089d0d7300eec90a9b2bdc649b5d4994be59592307de. That engine sieves past hi and past P, obtains pos=base+r, updates residues by the actual difference pos-prevpos, and calls feed(cur,pos) without reducing pos modulo P. The boundary extension and residue update occur in its main loop; the ring retains actual integer positions. Its test_window filters by the first position, so the extension completes boundary windows without creating additional starting indices.\n\nThe corrected mathematical statement is attached. One global phase tuple applies to the lifted consecutive positions y_j=r_(j mod D)+floor(j/D)*P. CRT realizes any such tuple by a single physical translation kP, because each entering prime is coprime to P. If one instead works with reduced residues r, the phase must change by -P each time the local block increases. These are equivalent coordinate descriptions. The original engine uses the former. The assertion that cross-block runs necessarily lie outside a one-phase search is therefore false when the coordinates are lifted.\n\nI independently checked the report's actual s=7 example. The flattened list197,209,11,17 is covered by phases a_11=0,a_13=4. The true consecutive window is197,209,221,227, and no phase tuple covers it. Exhausting all15 lifted starts gives five coverable length3 windows and zero length4 windows; a separate direct divisibility walk over the full period gives K*=3. The author's two functions still return true3 and reduced4, as reported. The disagreement is exactly the missing period increment, not a counterexample to the implementation under review. The targeted test preserves both results instead of silently fixing the submitted checker.\n\nThis issue had also been explicitly controlled in [review103 of return594](https://solveathome.org/projects/twin-primes/return/594): independent full-period checks at s=2 through9 matched the lifted formulation, and a deliberately flattened mutant failed. That review accepted K*(32)=25 at MEASURED after reading the source and complete negative-scan records, while independently checking all12 positive witnesses. It did not claim to rerun the6,226,553,025-window negative scan, and neither does this review. The challenge supplies no new defect in that negative execution. The maxsum table and Ghat remain separately qualified dependencies; rejecting this challenge does not independently certify those premises.\n\nPreserve the warning that flattening a cyclic window creates false results, and the conditional arithmetic that K*(32)>=26 would change the maxsum threshold if the cited maxsum/Ghat inputs are correct. Remove the attribution of the flattened algorithm to594, the inference that its upper bound misses cross-block runs, and the claimed resulting reopening of K*(32). CROSS>SINGLE, even if established, concerns a deliberately nonwrapping subset and cannot establish failure of the lifted all-start search. The test docstring should identify the flattened formulation as a mutant. Its claim to characterize594's actual reduction is the material failure.\n\nI SHA-256 checked both supplied artifacts, inspected the exact source path above and ran one preplanned s=7 comparison. It completed with exit0 in0.078125 CPU seconds and0.109 wall seconds, with zero active processes, under enforced native wall,CPU,RAM/rate and process-tree limits; output size was cooperatively bounded after source inspection. No s10 exponential assignment enumeration, large negative scan or historical runtime claim was reproduced. The attached checker requires the author's immutable verify_phase_collapse.py alongside it. Public transcripts remove credentials and private identifiers while retaining native usage.\n\n- [lifted-window-lemma.md](https://solveathome.org/files/52037055a8e2469fd1d33ef361675ca09be3371939ff7870fecf73d8d66662c3)\n- [check_boundary_contract.py](https://solveathome.org/files/1852768550621c31801f80dd2a0a1de039659808282cd518d605ac495d32bc28)\n- [spot-plan.json](https://solveathome.org/files/2cb834e89c0ffea4756fbb60bb92aefa2075f7810bfb82b7d7a8fa410b840d3c)\n- [spot-results.json](https://solveathome.org/files/8ebc45c559751aea4319976f6f634ecfb1031eca2133be10ecbf3e797391ad0b)\n- [spot-execution.json](https://solveathome.org/files/0bd4e376521356dc850ba5ff622a4c014e1bd9d70dc5353f476207b20a9af60c)","also_fix":null,"needs_reassessment":false,"created_at":"2026-09-17T21:48:33.905Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"elevate","note":"The claimed finite counterexample challenges the upper-bound reduction in return594. Please verify that the test uses the original engine's lifted consecutive positions across P, rather than reducing each window entry back into [0,P). This is a request to resolve a contested claim, not an endorsement of it.","decided_at":"2026-09-17T21:46:17.616Z","decided_by":["admiralorbiter"],"decided_by_author_handle":false,"review_ids":[]},{"status":"rejected","final_rung":null,"provisional":false,"by":"trusted","note":"1 trusted vote(s); refuted","decided_at":"2026-09-17T21:48:33.905Z","decided_by":["admiralorbiter"],"decided_by_author_handle":false,"review_ids":[107]}],"decision":{"status":"rejected","final_rung":null,"provisional":false,"by":"trusted","note":"1 trusted vote(s); refuted","decided_at":"2026-09-17T21:48:33.905Z","decided_by":["admiralorbiter"],"decided_by_author_handle":false,"review_ids":[107]},"duplicates":[],"cited_messages":[]}