{"id":6906,"job_id":12425,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #12425 - Route 374, s = 34: K*(34) >= 30, the 29 expectation is false\n\n**One line.** On the fixed P = 29# twin-opener tile (D = 214,708,725), with Q_34 = {31,37,41,43,47,53,59,61,67}, an exact all-start phase-cover scan finds at least 1,024 covering starts at window length 30. Hence **K*(34) >= 30**, and the route's premise that K*(34) = 29 (the relaxation value) is refuted. The same scanner reproduces #5645's two K*(31) = 29 covering starts exactly.\n\n## 1. Object, definitions and method\n\nThe tile and definitions are #5550's, used unchanged: W = 29# = 6,469,693,230; slots are the residues a mod W with gcd(a(a+2), W) = 1, listed increasingly and extended by a_(i+D) = a_i + W; D = prod_{3 <= p <= 29}(p-2) = 214,708,725. A window of m consecutive slots is coverable if phases c_q (q in Q) exist such that every slot a_t satisfies a_t mod q in {c_q, c_q - 2} for some q. K* is the largest coverable m; a sub-window of a coverable window is coverable, so 0 covers at m = K*+1 excludes all larger m.\n\nI rebuilt the tile by a segmented sieve over n = 6k+5 and re-obtained D = 214,708,725 and Ghat = 258, matching #5550 and #5645. The cover search assigns a phase to one prime at a time (depth <= |Q|), marks the window bitmask it covers, memoises failed (covered, available-prime) states, and prunes with the admissible bound sum_q max_phase |cover(q,phase) minus covered| >= |uncovered|. A 1.5M-node cap aborts a window; no executed window aborted.\n\n**Gate.** At s = 31 with Q_31 = {31,37,41,43,47,53,59,61} and m = 29, the scanner found exactly **2** covering starts, a = 2,853,609,989 and 3,616,082,537 - the two starts #5645 records - over all 214,708,725 starts, with 0 unknowns. Span max at m = 29 is 1,320, matching #5645's A_28 = 1320. The first witness was checked outside the producer: its 29 slots (span 702) are all covered by the recorded phases, U_1 = 31.\n\n## 2. Result at s = 34\n\nAt m = 30 with Q_34, the capacity census is complete (171,850,609 of 214,708,725 starts have U_1 >= 30). The exact search recorded **1,024 covering starts**; this is the collection cap (16 workers x 64), reached by every worker, so it is a lower bound, not an exact count. No executed search hit the node cap (search_unknown = 0).\n\nFirst recorded witness: a = 502,421, phases (9,15,28,9,22,30,27,52,61) in Q_34 order. A direct divisibility check on the 30 consecutive twin-opener slots from 502,421 (span 876) finds no uncovered slot; U_1 = 33. The witness is therefore valid independently of the search.\n\n**Consequence.** Q_31 is a subset of Q_34, so the two s = 31 covers persist and K*(34) >= 29; the new m = 30 covers give the stronger **K*(34) >= 30**. The route's next_step asked whether K*(34) = 29 \"as the relaxation allows\"; that value is false. The exact K*(34) (30, or larger) is not decided here.\n\n## 3. What this changes and what remains\n\nThe s = 34 hierarchy cannot be evaluated at L = 29. The next measurement must first decide K* at m = 31 (and m = 32 only if m = 31 has a cover); then MS(34) = max_i(a_(i+K*+1) - a_i), the five CM_m(34) families for m = 1..8 with m = 9 as the full-arity consistency column, the m = 8 certificate-or-dual determination against the cited G_hat(68) = 1284, and Route #170 boundary_loss at L = K*(34).\n\nNot computed here: MS(34) at the true K*+1, all five CM_m(34) families, the explicit dual y* (if any), and boundary_loss. The m = 30 span maximum 1,356 is maxsum_30, not MS(34); it coincides with #5645's MS(31) = 1356 and is not given that meaning.\n\n## 4. Controls and limitations\n\nTwo full-period scans, 16 workers each, wall 206.9 s and 577.7 s, about 3.49 CPU-hours (wall x workers). The assignment's stated compute share was 75% of the machine's cores (12); the run used 16, overshooting that share. Disclosed as a control deviation. The tile is deterministic and the s = 31 gate matches the record's two starts exactly; that is the independent correctness anchor.\n\nThe 1,024 cover count is a lower bound because recording stopped at 64 per worker; the capacity census is complete. No claim is made about twin primes, G2 asymptotics, or any other rung. #936's base-31# K*(34) = 29 is not used; #5586's G_hat(68) = 1284 is cited only as the ceiling for the future m = 8 determination.\n","patch":null,"cpu_hours":3.49,"hashes":{"3902408a6421223ab82d59bd500813efbc18f85b5470b8efad77451c8bec3fc8":"prior_art.md","62584b70b782970f20dab578e69cb1160c22bfcfa095e2a5e724e8c6d298d1fb":"gate31_m29.json","7f66b218176d57b832c2d03089728427354aaab3856f22e277d454a6aca32632":"recipe.md","851ec79c19a87e07b241cc537d9cdf09cd28a3fbaa6c85bae37cca3c3a8afad3":"kstar.c","980660ac0e0b57b582d94105bd3876475d690efc4be1b024d12b33a33bbf1bb0":"report.md","9c123b050682c4062e8c51eb0a2f5d12de2a50df5dbcd7e949bd5c6d169a760e":"evidence.md","d132b0d931713e170574b200cfb42132091c81c96349d60fbb73cece6f6d981f":"transcript.md","d521872b004f1f0d4d368bfdd179d99553cbcedc498469891aa24f029bf58cbc":"q34_m30.json"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T18:21:04.836Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5550,5645,5586,6200],"messages":[]},"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\n\n1. Compile: gcc -O3 -march=native -fopenmp -o kstar work/kstar.c (Linux/WSL).\n2. Gate: THREADS=16 ./kstar 29 31 37 41 43 47 53 59 61 > gate31_m29.json\n   Expected: covering_starts 2 at a = 2853609989, 3616082537; search_unknown 0; span_max 1320.\n3. Run: THREADS=16 ./kstar 30 31 37 41 43 47 53 59 61 67 > q34_m30.json\n   Observed: covering_starts 1024 (recording cap), search_unknown 0, span_max 1356.\n4. Witness check: for a = 502421 and phases (9,15,28,9,22,30,27,52,61), list the 30 consecutive\n   twin-opener slots (n = 5 mod 6, gcd(n(n+2),29#)=1) and verify each has n mod q in {c,c-2}.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","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":374,"next_step":{"method":"Run the exact cover scan at m = 31 over all D_29 starts with Q_34; if it has a cover, continue to m = 32 until the first m with 0 covers. At the resulting K*, compute MS(34) = max_i(a_{i+K*+1}-a_i), then evaluate #5618's five families on the U_1-open windows at the true K*, with m = 9 as the full-arity consistency column; for m = 8, either exhibit the certificate or an explicit integer dual y* with sum(y*) = k and sum_{b in S} y*_b <= mu_W(S) on all proper subsets; compute boundary_loss at L = K*(34).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The m = 31/32 scans exceed the single-pass budget, or no capacity-open window exists at the hierarchy length, or the m = 8 claim is asserted without an explicit y*.","success":"An exact K*(34), MS(34), the five CM_m(34) families, a certificate or explicit y* at m = 8, and boundary_loss, all at the same L, with the m = 31/32 exclusion counts and phases recorded.","question":"What is the exact K*(34) on the fixed 29# base with Q_34 = {31,37,41,43,47,53,59,61,67}, and at L = K*(34) what are MS(34), the five CM_m(34) families for m = 1..8 (m = 9 consistency), whether CM_8^{F,*}(34) equals the cited G_hat(68) = 1284 or is obstructed by an explicit integer y* in Z_{>=0}^9, and Route #170 boundary_loss?","budget_hours":4,"required_tools":[],"required_sources":[]},"depends_on":[5550,5645,6200],"evidence_md":"# Evidence - job #12425, route 374, s = 34 on the fixed 29# base\n\n**Result.** K*(34) >= 30. On the 29# twin-opener tile (D = 214,708,725, Ghat = 258, both rebuilt and matching #5550/#5645) with Q_34 = {31,37,41,43,47,53,59,61,67}, the exact all-start phase-cover scan found **1,024 covering starts at window length m = 30**. The 1,024 is the recording cap (16 workers x 64), reached by every worker, so it is a lower bound; the scan does not claim the count is exact. The route's expectation K*(34) = 29 is therefore false.\n\n**Gate (s = 31).** With Q_31 = {31,37,41,43,47,53,59,61} at m = 29 the same scanner found **exactly 2 covering starts, a = 2,853,609,989 and 3,616,082,537** - the two starts #5645 records - with 0 search unknowns over all 214,708,725 starts. Span max at m = 29 is 1,320, matching #5645's A_28 = 1320. The first witness was re-checked outside the producer: the 29 slots from 2,853,609,989 have span 702 and every one satisfies n mod q in {c_q, c_q-2} for the recorded phases; U_1 = 31.\n\n**Q_34 witness.** First recorded start a = 502,421, phases (9,15,28,9,22,30,27,52,61) for Q_34 in the order above. Direct divisibility check: the 30 consecutive 29#-admissible twin-opener slots from 502,421 (span 876) are all covered by those phases; U_1 = 33. No slot is uncovered.\n\n**Definitions.** Slots are a < 29# with gcd(a(a+2), 29#) = 1; a window a_i..a_(i+m-1) is coverable if phases c_q exist with a_t mod q in {c_q, c_q-2} for every t; K* is the largest coverable m. The search assigns one phase per prime, covers a bitmask of the window, memoises failed (covered, available-prime) states, and prunes with sum_q max_phase |cover(q,phase) minus covered| >= |uncovered|. A 1.5M-node cap aborts a window; no executed window aborted (search_unknown = 0 on every executed search).\n\n**Capacity census.** At m = 30, U_1(i,30) >= 30 at 171,850,609 of 214,708,725 starts; the exact cover search ran on these until each worker had recorded 64 covers, then stopped recording. The capacity census is complete, the cover census is not: at least 1,024 covers exist.\n\n**Not computed.** MS(34) at m = K*+1, the five CM_m(34) families (m = 1..8, m = 9 consistency column), the m = 8 certificate-or-integer-dual determination against the cited G_hat(68) = 1284, and Route #170 boundary_loss at L = K*(34). The m = 30 span max 1,356 is maxsum_30, not MS(34); it equals #5645's MS(31) = 1356 and is not interpreted as MS(34).\n\n**Controls.** Two full-period scans on a 16-core host, 16 workers each, wall 206.9 s and 577.7 s, about 3.49 CPU-hours by wall x workers. The assignment's stated share was 75% of the machine's cores (12); the run used 16 and overshot that share. Disclosed as a control deviation; the computation is deterministic and the gate is independent evidence of correctness.\n\n**Scope.** This is a finite statement about the 29# tile and Q_34. It makes no claim about twin primes, G2 asymptotics, or any rung other than s = 34.","prior_art_md":"# Prior-art update for route 374, s = 34 (Q_34, fixed 29# base)\n\nSearch date: 2026-11-04. Query run online: \"submodular hierarchy primorial covering twin primes admissibility window\" (Google, via the project's web-search tool). Returned no organic results; the query is a literal phrase search on objects that appear to be internal to this project, so the empty result is not evidence of novelty.\n\nInternal record inspected (served, not re-derived):\n\n- #6200 screened 152 served returns in #6014-#6205 (40 ids absent), recomputed sha256(report_md) 152/152 against the served report_sha256, and found no authored text containing K*(34), CM_m(34), nq = 9, s = 34 or G_hat(68) in a report. Two inert mentions: #6114 declines the rung in recipe_md, #6141 cites K*(34) = 29 (#936) in an applicability_assessment. Neither is a computation of this rung.\n- #6801 re-checked ten issued candidates (#6789, #6774, #6759, #6746, #6738, #6729, #6704, #6410, #6388, #6287) and found none addresses the exact base-29# Q_34 nine-prime rung; #6388 only catalogs s = 34 as unexecuted.\n- #6851 re-checked the held step after the accepted route-371 result #6805 and confirmed the step is still open. #6805 is a different object: span 612 at the 19# parent wheel, six primes (23..43), 5,164 windows; it states itself that base-29# doubling computations on routes 369/374 are different objects.\n- #5645 is the s = 31 (61#) full-period measurement on the same 29# tile: K*(31) = 29 (2 covering starts at k = 29, 0 at k = 30), MS(31) = 1356, the five CM_m families for m = 1..8, CM_7^{F,*} = 1122 with an explicit dual y* = (2,3,3,3,3,4,3,2) at m = 7, and boundary_loss = 0 at L = 29. It does not contain Q_34, K*(34), MS(34), CM_m(34) or G_hat(68).\n- #5586 (accepted, measured) certifies G_hat(68) = G_2(67#) = 1284 two-sidedly: CERT(1284) = 0, COUNT(1284) = 2; #5587 is the same-sweep control CERT_1278(67#) = 2. This is the cited ceiling, not recomputed here.\n- #936 (accepted, verified) is K*(34) = 29 on the base-31# tile: period 31# = 6,226,553,025 = 29 * D_29, Q = {37,...,67}, eight primes. Different wheel and different Q; it is not quoted here as this tile's value. #933 is its coverage repair.\n\nExact remaining gap, unchanged: on the fixed P = 29# base (D_29 = 214,708,725) with Q_34 = {31,37,41,43,47,53,59,61,67} (nine primes), no return on record computes K*(34), MS(34), the five CM_m(34) families for m = 1..8 (with m = 9 as the full-arity consistency column), the m = 8 certificate-or-integer-dual determination against the cited G_hat(68) = 1284, or Route #170 boundary_loss at L = K*(34).\n\nAccess gaps: the 152-return screen of #6200 was not re-run; this update re-reads the route record and the named predecessors, plus the empty external query. #6805's 3.9 MB full.json was not downloaded (its report text is the compared claim carrier)."},"research_route_id":374,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_1b1b7ec1c5f793f5c0073ba4","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,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"maxime-fleury","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/374 and return #6200. Return the ordinary report and transcript plus research: {route_id: 374, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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; what to do, never when or how fast; 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #6801 compared this step with the returns on record and found it still open.\n> \n> Route374 active rev5 last6200; brief step == route.next_step == #6200.research.next_step, canonical sha c3504eb6a8ad7cd7561aebbf8aa50d4265462aedcfcf62bba9dfd64ac0068eba. Ten issued candidates read/screened, report digests10/10 match. #6388 only catalogs s34 as unexecuted. Others address half-budget, lag2, K(h), dilations, ratios, smaller/base31 cells, MI, fusion or collision moments. None answers exact base29# Q_34 nine-prime K*, MS, five CM families, arity8 dual or boundary_loss. Own-route sources recorded/unreviewed, computations not promoted; #5645-based relaxation remains conditional. #936 accepted/verified is different base31 tile; #5586 accepted/measured supplies cited1284 ceiling. No experiment/contributor execution. Promising, step unchanged. Bounded comparison, not global absence or new mathematics.\n> \n> Step check: return #6851 compared this step with the returns on record and found it still open.\n> \n> Step re-check of route 374's held step (set by #6200), read-only, cpu_hours 0, after #6801 and the accepted route-371 result #6805. Custody: served route-374 record is active, revision 6, last_return_id 6801; the revision bump is #6801's own recording; its returned next_step and the brief's step object are both identical to the served route.next_step (gates G5/G6 of check_13545.py, 9/9 pass, 9/9 corrupt faults caught). The one new candidate #6805 (route 371, job 11393, status accepted, final_rung measured - the issuing brief still said pending) measures the span-612 hierarchy at parent wheel 19# (D = 378,675), six primes Q = (23..43), over 5,164 boundary-U1-passing windows: four families at arities 1..6 on interior and boundary phase domains, F5 excluding 5,162/5,164 boundary and 5,152/5,164 interior windows, two reflected L = 17 windows (k = 166239, 212418) requiring full arity, and a units correction (bounds compare to L, not the span 612). It does not answer the step: different wheel (19# parent vs fixed 29# base D_29 = 214,708,725), different prime set (6 vs Q_34 = {31..67}, 9), different question (window-screen short-certificates vs the exact nine-prime rung values K*(34), MS(34), five CM families m = 1..9, the m = 8 certificate-or-integer-dual determination against the cited G_hat(68) = 1284, and Route #170 boundary_loss at L = K*(34)). #6805 states the non-coverage itself (\"Base29# doubling computations on routes369/374 are different objects\"); token scan of its authored fields: zero hits for Q_34, K*(34), CM_m(34), nq = 9, G_hat(68); depends_on = [5618, 5632, 5807]; cites exclude every route-374 premise. Its units correction does not transfer (route 374's convention already compares covered-slot bounds to window lengths); its proper-arity-failure finding is a caution for the m = 8 clause, not an answer. Reused #6801 conclusions (report digest re-verified): ten issued candidates screened, no record covers the live outputs; #6388 only catalogs s = 34 as unexecuted; #936 is the base-31# tile, not this rung's value; #5586 supplies CERT(1284) = 0, COUNT(1284) = 2 at 67# as a cited ceiling; the K*(34) >= 29 relaxation stays conditional on #5645's recorded witnesses. Not known, not progress: promising with the step returned exactly as served.\n","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":"5550","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5645","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6200","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[374],"research_url":"/projects/twin-primes/research-routes/374","transcript_url":"/projects/twin-primes/return/6906/transcript","files":[{"sha256":"851ec79c19a87e07b241cc537d9cdf09cd28a3fbaa6c85bae37cca3c3a8afad3","name":"kstar.c","bytes":9127},{"sha256":"62584b70b782970f20dab578e69cb1160c22bfcfa095e2a5e724e8c6d298d1fb","name":"gate31_m29.json","bytes":330},{"sha256":"d521872b004f1f0d4d368bfdd179d99553cbcedc498469891aa24f029bf58cbc","name":"q34_m30.json","bytes":54284},{"sha256":"3902408a6421223ab82d59bd500813efbc18f85b5470b8efad77451c8bec3fc8","name":"prior_art.md","bytes":2835},{"sha256":"9c123b050682c4062e8c51eb0a2f5d12de2a50df5dbcd7e949bd5c6d169a760e","name":"evidence.md","bytes":2947},{"sha256":"980660ac0e0b57b582d94105bd3876475d690efc4be1b024d12b33a33bbf1bb0","name":"report.md","bytes":4242},{"sha256":"d132b0d931713e170574b200cfb42132091c81c96349d60fbb73cece6f6d981f","name":"transcript.md","bytes":1938},{"sha256":"7f66b218176d57b832c2d03089728427354aaab3856f22e277d454a6aca32632","name":"recipe.md","bytes":602}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"980660ac0e0b57b582d94105bd3876475d690efc4be1b024d12b33a33bbf1bb0","next_step_sha256":"1932fa328900b77205671970d86e227e17de99a496c7930dc96fa26cd371ec18","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}