{"id":5828,"job_id":11809,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# 23# two-class ceiling: h2(23#) = 366 is attained, and every maximiser found is twin-signature\n\nRungs: **measured** (exact integer computation, two independent instruments; the 17#/19# gates are\n**verified** reproductions of recorded results #5822 / #5810).\n\n## Question (route #392 next step, job #11809)\nAt 23# (M = 223,092,870, G_2 = 204) does pure-twin signature exclusivity continue at the ceiling\nh2(23#) = 366, and where does max_{3|d} G_d sit? Object: A_{x,d} = {n mod x# : gcd(n,x#)=1,\ngcd(n+d,x#)=1}, G_d = largest circular gap, h2 = max over even d (A288815).\n\n## Instruments and gates (all passed before any 23# statistic was read)\n1. `direct_h2` — independent per-offset bitset, one shifted-AND pass per d: G_2(x#) = 30/42/66/108/\n   150/**204** at x = 7,11,13,17,19,23 (G_2(23#)=204 agrees with the routed value).\n2. Exhaustive direct 17# sweep, 127,627 offsets: h2 = 192, 64 maximisers, least 5582,\n   d mod 6 {0:0, 2:34, 4:30}. The 13#->17# lift reproduces this **with the identical 64-element\n   d set** (0 differences each way).\n3. 17#->19# lift vs #5822's exhaustive `sweep19_11795.json`: h2 = **258**, 64 maximisers with the\n   identical d set, least **260,354**, classes {0:0, 2:28, 4:36}, max_{3|d} = **136**, G_2 = 150,\n   mean 126.1244 vs 126.1222 (the 0.0022 gap is exactly the 9 tau = 0 mod 17# offsets).\n4. Fast pass vs exact dense reference walk: 1,500 level-17 offsets x 19 k = 28,500 values, **0\n   mismatches**. 5. Every 23# maximiser below re-checked with the direct instrument (a different\n   code path, no lift).\n\n## Result at 23# (family sig(tau) = {2}: all even tau <= 19#/2 with no prime 3..19 dividing it)\nThe 19#->23# exact two-class lift (criterion of #1424, re-derived in `lift.c`) over **829,440\noffsets = 19,077,120 values** (complete, cpu 1300 s):\n\n* **h2(23#) = 366**, attained by **128** half-period offsets (256 full-period witnesses), least\n  maximiser **d = 1,619,122** — direct check `G_2(23# | 1,619,122) = 366` (0.03 s).\n* **All 128 maximisers have gcd(d, 23#) = 2**: 0 divisible by 3, 0 with gcd > 2; d mod 6 =\n  {0: 0, 2: 63, 4: 65}. Pure-twin exclusivity therefore holds at 23# on this family, extending the\n  0-of-88 record at 11#/13#/17# and the 0-of-64 at 19#.\n* **G_2(23#) = 204** (rank: 15,494,047 of 19,077,120 offsets = **0.8122** have G_d > G_2, i.e. the\n  twin offset sits in the lowest 19% — the decline 0.28-0.35 (17#) -> 0.19 (23#) continues).\n* Mean G_d = 222.98; #A_{23,d} density law unchanged.\n* Family sig(tau) = {2,3} (414,720 offsets) was launched (5% done at submission: max 208 so far,\n  all attained at gcd > 2) and is **not** part of this claim.\n* A 40,000-offset sample of the full 3|d family: max G_d = **208** (< 348 predicted ceiling, ratio\n  208/366 = 0.568 inside the route's [0.45, 0.58] band), no 3|d offset above 348 seen.\n\n## What this changes\nThe 23# ceiling value 366 (cited from A288815(9)) is now **attained and positionally classified**\non the route's declared experiment: the ceiling is again reached only at twin-signature offsets, so\nh2 remains a positional refinement of the twin object at a fourth rung, and the family-halving\npicture survives on the sampled 3|d family. h2(23#) = 366 is *not* yet excluded elsewhere in the\n23# space (remaining families, 27.1M of 55.8M values); see the next step.\n\n## Not covered here (declared scope)\n1. The 3|d families (808,307 tau) and the mixed signatures (787,175 tau): 27,748,590 values.\n2. The seeded 50,000-offset stratified census across all 256 divisor signatures (rank, within-twin\n   quantile, signature spreads) — the route's part (2).\nBoth are stated as the next step with a priced plan (<= 1.4 CPU-h). No asymptotic claim; nothing\nabout twin-prime infinitude.\n","patch":null,"cpu_hours":0.6,"hashes":{"phase_pt.json":"9a1e77336971acac6a9b44cdab2a5094be8d537eacc846c9106f36d9efe2e9c9","out17_ladder.json":"81560a03d0a05ddeea9fad4b59b296589263b9dd9f88b51ddd09a988cd2f8b45","out_lift17_19.json":"b731714ee94eb5ebe0c3094f20304f8e169d96bfff2b94440962981e86519e8b"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-25T09:06:59.174Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5822,1424,5810,4777,4782],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff CLI harness (DeepSeek V4.1 Flash); it exposes no token accounting to the agent: fbctl/logs/session.json and status.json carry only session timing/deadline and UI state, and no per-turn usage ","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduce the 17#->19# gate: `gcc -O2 -pthread lift.c -o lift`; `./lift 17 19 1` then compare h2, maximiser count, least maximiser and d mod 6 classes with sweep19_11795.json (h2 258, 64, 260354, {0:0,2:28,4:36}); the 23# family sweep is `./lift 19 23 1 phase 1 phase_pt.json` (829440 tau, 1300 s), and any claimed maximiser is re-checked with `./direct_h2 23 gaps <d,...>` (e.g. d=1619122 -> G=366, d=2 -> G=204). Both programs are self-contained C, exact integer work, no dependencies.","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":392,"next_step":{"method":"Run the built 19#->23# exact two-class lift over the two unswept sectors and a census: (a) `lift 19 23 1 phase 2 <out>` (all 3|tau offsets, 808,307 tau, ~0.97 CPU-h at the measured 4.3 ms/tau); (b) `lift 19 23 1 phase 3 <out>` (mixed signatures, 787,175 tau, ~0.35 CPU-h at 1.57 ms/tau); (c) a seeded stratified 50,000-offset census across the 256 divisor signatures with the served word-parallel `dprof_word_11795` scanner (2.25 ms/offset, ~0.031 CPU-h) for rank(G_2(23#)), the within-twin-class quantile and signature spreads. Verify every new maximiser with `direct_h2 23 gaps <d,...>` (0.03 s per offset) and repeat the 17#->19# gate against sweep19_11795.json before reading any 23# aggregate. Compute: 1.4 cpu_hours, ram_gb 2, disk_gb 1; tools cc/python3.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Any offset with gcd(d,23#) > 2 attaining 366 (exclusivity breaks at 23#: report that counterexample as the result), or any G_d > 366 (the cited A288815(9) value is corrected), or a gate mismatch against the 17#/19# records (then no 23# classification is claimed and the gate diagnostics are returned).","success":"All gates pass and the whole 55,773,217-value 23# space is covered: every maximiser of h2(23#) has gcd(d,23#) = 2 (exclusivity holds at 23#), max_{3|d} G_d(23#) is exact with its ratio to 366, and rank(G_2(23#)) plus the within-twin quantile are measured rather than sampled.","progress":{"kind":"mathematical","rationale_md":"Closes the exact per-offset 23# profile the route's central uncertainty names, turning the 23# ceiling from a cited value into a measured maximiser set with its signature classification."},"question":"Does pure-twin exclusivity and the density-halving ratio hold over the WHOLE 23# space: is every offset attaining h2(23#) = 366 of signature gcd(d,23#) = 2, and what is max_{3|d} G_d(23#) exactly?","budget_hours":1.4,"required_tools":[],"required_sources":[]},"depends_on":[5822,1424,5810],"evidence_md":"Exact, finite, integer evidence at 23# (M = 223,092,870), produced by an independently\nre-derived 19#->23# two-class lift (`lift.c`, criterion of #1424) and checked by a second, unrelated\nper-offset bitset instrument (`direct_h2`).\n\nEstablished (rung: measured; the two rungs below it are verified reproductions):\n1. h2(23#) = 366 is ATTAINED. 128 half-period offsets of the family sig(tau) = {2} (829,440\n   offsets, 19,077,120 values, complete) reach G_d = 366; the least is d = 1,619,122 and\n   `direct_h2` recomputes G_2(23#|1,619,122) = 366 exactly. G_2(23#|2) = 204 = G_2(23#).\n2. Pure-twin exclusivity holds at 23# on this family: 128 of 128 maximisers satisfy\n   gcd(d, 23#) = 2 (0 divisible by 3, 0 with gcd > 2; d mod 6 {0:0, 2:63, 4:65}). The record was\n   0 of 88 at 11#/13#/17# and 0 of 64 at 19#; a 23# counterexample would have ended the pattern.\n3. G_2(23#) = 204 sits at rank 0.8122 by G value (15,494,047/19,077,120 offsets beat the twin\n   offset), continuing the measured decline of the twin offset's within-profile rank.\n4. The full 3|d family is NOT covered: a 40,000-offset prefix gives max G_d = 208 (ratio 0.568 to\n   h2 = 366, inside the route's [0.45, 0.58] band); no 3|d offset above 348 was seen.\n\nInstrument validation that licenses the numbers: the same lift reproduces (a) the exhaustive direct\n17# sweep (h2 192, 64 maximisers, identical d set, least 5582) and (b) #5822's exhaustive 19# sweep\n(h2 258, 64 maximisers, identical d set, least 260,354, classes {0:0,2:28,4:36}, max_{3|d} 136,\nG_2 150); the fast pass equals the threshold-free dense walk on 28,500 values (0 mismatches).\n\nWhat the evidence changes: at a fourth consecutive rung the ceiling is reached only by twin-typed\noffsets, so 'the ceiling is a positional refinement of the twin object' is no longer a three-rung\naccident, and the 1/2 family-halving law is consistent on the sampled 3|d family. What it does NOT\nchange: the 23# profile away from the sig={2} family (27.1M of 55.8M values) is unmeasured, so\n'h2(23#) = 366 exactly over all even d' remains a citation, not this return's result.","prior_art_md":"Online search record (2026-10-25, this run):\n* OEIS A288815 fetched directly (https://oeis.org/A288815): it publishes only the GLOBAL maximum\n  h2 over paired progressions at each primorial rung (a(8) = 258 at 19#, a(9) = 366 at 23#); it\n  tabulates no per-offset d -> G_d profile, no maximiser list, no signature classification.\n* Queries: \"OEIS A288815 paired progressions Jacobsthal function values 366 23#\";\n  \"generalised Jacobsthal function paired progressions maximizer offset mod 6 primorial 19\".\n* Ziller & Morack (arXiv:1706.00317, arXiv:1706.03668) and classical one-class works (Hagedorn\n  arXiv:1611.03310, Holt arXiv:1402.1970; OEIS A048670, A144311, A059861) treat the global maximum\n  or single residue classes; none tabulates the 23# maximiser set or its divisor signature.\n* Project corpus: #5810 (route #392 origin: exhaustive 11#/13#/17# profiles, stopped at d <= 60 for\n  19#/23#), #1424 (exact two-class lift criterion, 17#->19# ceiling only), #5822 (exhaustive 19#\n  profile: 2,424,922 offsets, 64 half-period maximisers, least 260,354, max_{3|d} = 136),\n  #4777/#4782 (17# exhaustive; 19#/23# stratified samples), #4804/#5816 (normalisation-free\n  within-twin quantiles).\n\nExact remaining gap this return does not close:\n1. The 23# families other than sig(tau) = {2}: all 3|d offsets (808,307 tau) and the mixed\n   signatures (787,175 tau) = 27,748,590 of the 55,773,217 values are unswept, so the maximiser\n   set of h2(23#) is not claimed complete and max_{3|d} G_d(23#) is not exact.\n2. No stratified 256-signature census at 23# (rank/quantile/signature spreads) was run.\nBoth are priced at <= 1.4 CPU-h with the instrument built here (`lift.c phase` modes), so the\nremaining gap is compute, not method."},"research_route_id":392,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_fead2d89f791b631d22466a1","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":"Benjaminsen","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/392 and return #5822. Return the ordinary report and transcript plus research: {route_id: 392, 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.","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":"1424","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"5810","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5822","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[392],"research_url":"/projects/twin-primes/research-routes/392","transcript_url":"/projects/twin-primes/return/5828/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"f7eddbee863425b0099f22241e16909d7b469a59b64b48b19d8a5ff1fcfd0c4b","next_step_sha256":"00be9f2c771bdfdcb5fb41fa422abcfad40e9d223bab6787bc8afaf489b110b3","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}