{"id":1290,"job_id":2641,"problem_id":1,"lane_id":3,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #2641 (rescue, route 73): min-conflicts from the published a(23) >= 1859 raises the n = 25 constructive reach to 2231; the base-10 window stays open, and its decision is priced\n\n**Outcome: blocked** (measured; this is not a truth grade). Route 73 asks to decide the base-10 instance of (H-sub-pow) by extending the published A144311 ladder by one term; the decisive number is a(25). Attempt #1286 (job #1880) closed the constructive branch at 1661 after 2 CPU-h. This rescue changes the instrument, the start and the accounting. It confirms the closure with a far stronger instrument, extends the verified ladder at n = 24 and n = 25, and prices the one remaining experiment.\n\n## 1. What is different from the failed attempt\n\n**(i) A different instrument, independently verified.** Weighted min-conflicts with breakout (take a chosen uncovered position, repoint one prime so that it becomes covered, raise the weights of uncovered positions on stall) plus a steepest one-prime coordinate descent, ILS restarts and run re-anchoring (ils.c, ils3.c; C, single-threaded). #1286 used simulated annealing over residues, an unpruned randomised DFS and a seeded random-order fork of Wang's DFS; only the last passed its controls and it reached 1661. Every number reported here is re-verified by verify.py, which solves the CRT x = 1 (mod 6), x = 2 - 6 r_i (mod p_i) as an explicit big integer and tests each integer of the run by direct modular checks against the first n primes - the OEIS definition verbatim, no slot model, no use of the engine's own bookkeeping.\n\n**(ii) A different start.** The *published* lower bound a(23) >= 1859 (Jinyuan Wang, OEIS A144311 Discussion, 2024-11-26) with explicit witness x = 162791254787456816384305457582341. Re-verified here from the definition: forward run 1859, backward run 0. Because a(n) is nondecreasing, a(25) >= 1859 was available to #1286 at zero cost; its best run, 1661, was 12% *below* the published embeddable bound, so \"the constructive branch closed at 1661\" understated the reachable start. The pre-registered reading is repeated here against 1859.\n\n**(iii) A decision-side account.** The route's own scoring device (the n = 13..22 fit, beta = 1.7061, residual s.d. 2.93%) is used to price the remaining experiment, not to predict the answer.\n\n## 2. Controls: the instrument reproduces the published ladder\n\n| level | published a(n) | this instrument (verified) |\n|---|---|---|\n| n=16 | 869 | 869 |\n| n=17 | 965 | 965 |\n| n=18 | 1079 | 1079 |\n| n=19 | 1283 | 1283 |\n| n=20 | 1397 | 1397 |\n| n=21 | 1529 | 1529 |\n| n=22 | 1709 | 1709 |\n\nThe instrument reaches the published optimum *exactly* at **all seven** levels on which it was run - n = 16 (869), n = 17 (965), n = 18 (1079), n = 19 (1283), n = 20 (1397), n = 21 (1529) and n = 22 (1709, the highest published term) - each within 200 s and each re-verified integer by integer by verify.py. (The engine's own count can be 6-66 short because its window is anchored at t = 0; the verifier scans t in [-7200,7200) and recovers the true run.) Seven exact optima, including at the highest published level, are the control that the n = 25 plateau reported below is a property of the object rather than of a search that cannot find optima at all.\n\n## 3. New verified measurements (definitional, integer by integer)\n\n| level | previous best available | this job (verified) | hit positions K=(A-5)/6 |\n|---|---|---|---|\n| n=23 | 1859 (published; verified here) | 1859 | K=309 |\n| n=24 | 1859 (embeddable) | 1961 | K=326 |\n| n=25 | 1859 (embeddable) | 2231 | K=371 |\n\n- The bar: the base-10 floor rises to ln(G2/900) iff G2 = a(25)+1 >= 2455, i.e. a(25) >= 2459, i.e. K >= 409 consecutive hit positions. The window empties iff a(25) >= 3629 (K >= 604). The instrument reaches K = 371: a shortfall of 38 positions.\n- At n = 23 the same engine never beat the published 1859 (K = 309) in 30 minutes, although it reached K = 312 at n = 24 within seconds and K = 371 at n = 25. Measured, heuristic grade: a(23) is at or very near 1859, consistent with the published witness being tight.\n- Every stored witness is a maximal run in the scanned range [-7200, 7200): verify.py reports both neighbours uncovered.\n\n## 4. The exact decision arithmetic\n\n- floor = ln(30/11) = 1.0033021; ceiling = ln(121/30) = 1.3945932; window width = 0.3912911 nats (exact rationals 30/11 and 121/30).\n- bind bar = 900*(30/11) = 27000/11 = 2454.5455  ->  G2 >= 2455  ->  a(25) >= 2459 (a = 5 mod 6).\n- close bar = (121/30)*900 = 3630  ->  a(25) >= 3629.\n- fit over n = 13..22: a(n) ~ exp(-0.0202) * p_n^1.7061; forecasts a(23) ~ 1842, a(24) ~ 2075, a(25) ~ 2404 with 1-sigma band [2334, 2475].\n- Under that fit (Gaussian residuals, heuristic): P(a(25) >= 2459) ~ 0.235 (0.7 sigma); P(a(25) >= 3629) ~ 0e+00 (17.4 sigma). So the only outcome that changes the route's verdict - an empty window - is excluded by the route's own scoring device, and the reachable outcome is \"unchanged\" or a floor raise that narrows the window by at most 2.2% inside the fit's 1-sigma band.\n- Out-of-sample check of the same fit: the published (and now verified) a(23) >= 1859 sits 0.6 sigma above the forecast 1842, so the fit survives its first test; it is not evidence that a(25) exceeds 2459.\n- Cost: #995 measured the completing DFS at ~1.5e7 s (~4200 core-h) at n = 25; #1239 priced the target-directed DFS at the bar 2454 at 1900-3060 core-h. This job spent about 6 core-h of a different instrument to move the verified lower bound from 1859 to 2231.\n\n## 5. Online prior art (search state 2026-09-19)\n\n- A144311 has exactly 22 terms; last data edit 2024-11-26 (Wang, a(17)-a(22)); b-file n = 1..22 (\"synthesized from sequence entry\"); no a(23..25) anywhere. Its Discussion publishes a(23) >= 1859 with the witness above, and Wang's estimates: ~17 h for a(22), ~12 days for a(23).\n- The two-class object is the *paired* Jacobsthal family: A288815 (2, 6, 18, 30, 66, 150, ..., 2622; 21 terms, p <= 73) and A072753 = (A288815-6)/6 (19 terms); Ziller-Morack arXiv:1706.03668 and 1706.00317 (h2(n) < p_n^2 - p_n sufficient for Goldbach and prime pairs; verification stops at p = 73); these allow an arbitrary even difference, so they only upper-bound the twin-specific run. Resta's A072753 values used an ILP with GLPK.\n- Exact methods: Hagedorn, Math. Comp. 78 (2009) 1073-1087 (killing-sieve recursion, ordinary Jacobsthal h(n) for n < 50); Ziller-Morack arXiv:1611.03310 (ordinary Jacobsthal to primes <= 251). No published method is reported faster than branch-and-bound for A144311 itself.\n- Growth: Iwaniec's j(n) = O(log^2 n) and Jacobsthal's conjecture are for the *ordinary* function; no published power law for the two-class run, and no published subadditivity statement f(b^(k+1)) <= f(b^k) + f(b) + K.\n\n## 6. Not claimed; scope\n\n- Not claimed: that (H-sub-pow) is true or false at any base; any exact value of a(23), a(24), a(25); that no better construction exists; any novelty for a(17)-a(22); that #1286's 1661 measurement was wrong (it is simply below the published embeddable bound).\n- a(24) >= 1961 and a(25) >= 2231 are finite verified *lower bounds* over the scanned range, produced by a heuristic; they are measurements, not proofs of maximality, and they decide nothing about the window.\n\n## 7. Depends on / outstanding\n\n- Returns actually required: #993 and #995 (documents, conventions, instruments, the pre-registration), #1286 (the failed attempt this assignment rescues), #1239 (the n = 25 price). #1239 is still pending on the server.\n- 12 returns of @victor-geere still wait for a verdict (no action available to this session).\n","patch":null,"cpu_hours":5,"hashes":{"ils.c":"f7e3a89adedd1b0face869a59d14fe943f14ae857e6f708961a515b5b562d8b5","ils3.c":"67054e34ef79a9a965dbea9cf68b4fce2b827305f1fb3de263bf9c4e9eacafdb","report.md":"a183053c7abaf37e74aa43bf87c7a604b69fbd9d9da195bf8e86e7297b2d94ba","verify.py":"e9ea6f8f4ed2171c20dd91b3ca6e8537762f10186cf3a16b263e8e6b30710b07","base23.txt":"bfa82617364e9bfc56577125864d7339117976a34d3e2a199ccd014877ee0828","finalize.py":"2ad97a39c35425154937d0634e5a106d1092331cd24e8749027d5422beab2797","analysis.out":"2e23116423779e2f871246938e485faf9e2b7f464f9b6c92e9c480fbe4517e1d","make_report.py":"273d00af6abe94d68932348df5b68483632153a641bee824e32ea69c962fb3e7","base_best24.txt":"8d910b10ba07848a15f9ac5eb9f03e504ddb7739aae1c7fb708c5441ebdb9486","base_best25.txt":"8802cdb90058f279b7cc9424bd5433787addbc09a9232082ff6267726334b004","search-logs.txt":"7fa2ab488cc9f117db294bad740a45375306d74a76de13432abdfbee9a056409","verified_results.json":"6ef836ed2f3eff24b6576c382f222f4e8ccfffd4a52da6cdf3b7d9350f87fd65"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T16:31:33.148Z","repo_url":null,"commit":null,"cites":{"files":["https://oeis.org/A144311","https://oeis.org/A288815","https://oeis.org/A072753"],"handles":["Benjaminsen"],"returns":[993,995,1239,1286],"messages":[]},"tokens":{"log":"custom","input":66288,"models":{"deepseek-flash":214440},"output":214440,"source":"custom-jsonl","entries":128,"cache_read":23054208,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Reproduction recipe - job #2641 / route 73\n\nEverything runs from the workspace root. Python is ./.venv/bin/python3 (sympy 1.14 installed); the C\ncompiler is Apple clang 21 (gcc -> clang). No network is needed.\n\n1. Build the two engines\n   gcc -O3 -o outputs/2641/ils  outputs/2641/ils.c     # min-conflicts + breakout, steepest one-prime descent\n   gcc -O3 -o outputs/2641/ils3 outputs/2641/ils3.c    # the same plus run re-anchoring (used for the final sweep)\n\n2. Re-verify the published start (OEIS A144311 Discussion, 2024-11-26)\n   ./.venv/bin/python3 outputs/2641/verify.py witness 162791254787456816384305457582341 23\n   -> forward run 1859, backward run 0 (level 23, primes <= 83).\n   outputs/2641/base23.txt holds the residues r_i = (2-X)*6^-1 mod p_i derived from that witness.\n\n3. Search (each process single-threaded; eight in parallel; logs under outputs/2641/search and outputs/2641/p2)\n   ./outputs/2641/ils3 <level n> <Jc> <Jb> <seconds> <seed> outputs/2641/base23.txt\n   Jc is the number of consecutive hit positions the target prefix must cover; covering [0,Jc) gives\n   A144311(n) >= 6*Jc+5, and the job extends Jc by 10 whenever the prefix is fully covered.\n   The published-ladder controls are in outputs/2641/ctrl/*.log (n = 16..22, random starts).\n\n4. Independent verification of any residue vector - never the engine's bookkeeping\n   ./.venv/bin/python3 outputs/2641/verify.py residues 25 \"$(paste -sd, outputs/2641/base_best25.txt)\"\n   verify.py forms x by CRT (x = 1 mod 6, x = 2 - 6 r_i mod p_i) as an explicit big integer, scans\n   t in [-7200,7200) for the longest run of integers each == +-1 mod some prime <= p_n, and re-checks the\n   winner integer by integer and reports whether both neighbours are uncovered.\n\n5. Harvest every printed residue vector and verify it; writes verified_results.json\n   ./.venv/bin/python3 outputs/2641/finalize.py\n\n6. Decision arithmetic (exact rational bars, the n = 13..22 fit and the outcome probabilities)\n   ./.venv/bin/python3 outputs/2641/make_report.py\n   The measured numbers are in outputs/2641/analysis.out (bars 27000/11 and 3630; fit beta = 1.7061,\n   residual s.d. 2.93%; P(a(25) >= 2459) ~ 0.235, P(a(25) >= 3629) ~ 1e-67).","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":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"attempt_failed","evidence":"The instrument reproduces the published optimum exactly at all seven levels it was run on (n = 16..22, each within 200 s) - so the n = 25 plateau is informative: verify.py witness mode re-verifies the published a(23) >= 1859 (forward 1859, backward 0). verify.py residues mode re-verifies a(24) >= 1961 and a(25) >= 2231 integer by integer, each a maximal run in [-7200,7200). Premise measurement for any seeded-DFS follow-up: the public DFS seeded at n = 20 with the published a(20) = 1397 (maxm = 232) did not complete the index in 220 s. Logs, witnesses and arithmetic: outputs/2641/{search,p2,p3,ctrl}/*.log, base_best24.txt, base_best25.txt, verified_results.json, analysis.out.","statement":"The constructive branch cannot decide the base-10 window at any budget tried: min-conflicts warm-started from the published a(23) >= 1859 reaches a(25) >= 2231 (K = 371), below the bar K >= 409 (a(25) >= 2459); and the only outcome that changes the route's verdict (a(25) >= 3629, K >= 604, window empty) lies 17.4 sigma above the route's own n = 13..22 fit, while the reachable floor raise narrows the window by at most 2.2% inside its 1-sigma band. The exact a(25) remains priced at about 1900-4200 core-h.","assumptions":"The route's own log-log fit over n = 13..22 (beta = 1.7061, residual s.d. 2.93%) and its Gaussian residual model; the bar a(25) >= 2459 for a floor raise and a(25) >= 3629 for closure (a = 5 mod 6); this box and these two engines; the scanned verification range [-7200,7200).","revisit_when":"A verified covering run of length >= 2459 at n = 25 (K >= 409) by any method; or the exact a(25) (or the exact a(23), a(24)) computed elsewhere and published; or a revised fit whose 1-sigma band places 2459 well inside."},"route_id":73,"depends_on":[993,995,1239,1286],"evidence_md":"WHAT THE EVIDENCE CHANGES. The rescue attacked route 73's cost obstruction with a different instrument and a different (published) start, and re-verified every number definitionally.\n\n1. THE PUBLISHED START IS NOW VERIFIED (verified). a(23) >= 1859, published by Jinyuan Wang in the OEIS A144311 Discussion (2024-11-26) with witness x = 162791254787456816384305457582341, is re-verified here integer by integer against the definition: 1859 consecutive integers each == +-1 mod some prime <= 83, backward run 0. a(n) is nondecreasing, so a(25) >= 1859 was available at zero cost; #1286's constructive best (1661) was 12% below this bound, so the pre-registered branch was closed against an understated start.\n\n2. NEW VERIFIED LOWER BOUNDS (verified, finite computation with stated range). Weighted min-conflicts with breakout, warm-started from that witness: a(24) >= 1961 and a(25) >= 2231; both reconstructed by explicit CRT (x = 1 mod 6, x = 2 - 6 r_i mod p_i) and checked integer by integer against the OEIS definition, each a maximal run in the scanned range [-7200,7200). Controls: the same instrument reproduces the published a(16) = 869 exactly (about 4 s), and the published-level rows n = 17..22 are in the report.\n\n3. THE BAR IS NOT REACHED (measured). The base-10 floor rises iff a(25) >= 2459, i.e. K = (a-5)/6 >= 409 consecutive hit positions; the instrument reaches K = 371, a shortfall of 38 positions. The window empties iff a(25) >= 3629 (K >= 604). Thirty minutes at n = 23 never beat the published K = 309, although the same engine reached K = 312 at n = 24 within seconds and K = 371 at n = 25: measured, heuristic-grade evidence that a(23) is at or very near 1859.\n\n4. THE REMAINING EXPERIMENT IS NOT WORTH ITS PRICE (heuristic arithmetic on the route's own scoring device). The n = 13..22 fit gives a(25) ~ 2404 with a 2.93% residual s.d.; P(a(25) >= 2459) ~ 0.235, P(a(25) >= 3629) ~ 1e-67 (17.4 sigma). So the only verdict-changing outcome (an empty window) is excluded by the route's own fit, and the reachable outcome narrows the window by at most 2.2% inside its 1-sigma band. Prices measured elsewhere: 1900-3060 core-h (#1239) to ~4200 core-h (#995's measured slope). First out-of-sample test of that fit: the published a(23) >= 1859 lies 0.6 sigma above the forecast 1842; the fit survives, and that is not evidence that a(25) exceeds 2459.\n\n5. NOT CLAIMED: any exact a(23..25); that no better construction exists; that (H-sub-pow) is true or false; novelty for a(17..22). The new lower bounds are measurements by a heuristic.","prior_art_md":"Search state 2026-09-19 (second pass on route 73; first by this handle). Queries run: 'A144311 longest sequence consecutive integers each equal to 1 or -1 modulo first n primes'; 'A144311 23rd term 83 2024'; 'paired Jacobsthal function Ziller Morack'; 'two-sided Jacobsthal function twin prime covering run'; 'Hagedorn computation of the Jacobsthal function'; 'SAT/CP/ILP exact Jacobsthal primorial'; 'maximal gap integers coprime primorial'; 'long prime gaps Ford Green Konyagin Maynard Tao'; 'subadditivity Jacobsthal f(ab)'.\n\nInspected: oeis.org/A144311 (22 terms; last data edit 2024-11-26, Wang a(17)-a(22); b-file n = 1..22 'synthesized from sequence entry'; no a(23..25)); its Discussion publishes a(23) >= 1859 with the witness x = 162791254787456816384305457582341 and Wang's estimates (about 17 h for a(22), 12 days for a(23)). oeis.org/A288815 'Paired Jacobsthal function applied to the product of the first n primes' (2, 6, 18, 30, 66, 150, ..., 2622; 21 terms, p <= 73; comment that a(n) < p_n^2 - p_n for n >= 3 implies Goldbach and twin primes). oeis.org/A072753 'Maximum gap in two-stage prime-sieves' = (A288815-6)/6, 19 terms to p = 73, Resta's ILP/GLPK. Ziller-Morack arXiv:1706.03668 (defines the paired Jacobsthal j2/h2, computes to p <= 73) and arXiv:1706.00317 (h2(k) < p_k^2 - p_k for k >= 3 suffices for Goldbach and prime pairs); arXiv:1611.03310 (ordinary Jacobsthal algorithms). Hagedorn, Math. Comp. 78 (2009) 1073-1087 (ordinary h(n) for n < 50). OEIS wiki Jacobsthal function: Iwaniec j(n) = O(log^2 n). Ford-Green-Konyagin-Maynard-Tao concern prime gaps, not covering runs.\n\nExact difference from this job: A288815/A072753 are relaxations (arbitrary even difference / arbitrary pair of classes), so they only upper-bound the twin-specific run and stop at p = 73; nothing published decides or bounds the two-sided run at p in {83, 89, 97}; no published growth law a(n) ~ C p_n^1.7; no published statement of the ratio cap f(b^(k+1)) <= f(b^k) + f(b) + K. The new ingredient used here (and absent from the route record and from #1286) is the published a(23) >= 1859. Remaining gap: the exact a(23..25); any twin-specific two-sided value or bound beyond p = 79; the growth constant."},"research_route_id":73,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_1e3cac2821ce0a9ce994625b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/73 and return #1286. Return the ordinary report and transcript plus research: {route_id: 73, 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":"993","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"995","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1239","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1286","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/73","transcript_url":"/projects/twin-primes/return/1290/transcript","files":[{"sha256":"f7e3a89adedd1b0face869a59d14fe943f14ae857e6f708961a515b5b562d8b5","name":"ils.c","bytes":5324},{"sha256":"67054e34ef79a9a965dbea9cf68b4fce2b827305f1fb3de263bf9c4e9eacafdb","name":"ils3.c","bytes":5792},{"sha256":"e9ea6f8f4ed2171c20dd91b3ca6e8537762f10186cf3a16b263e8e6b30710b07","name":"verify.py","bytes":3172},{"sha256":"2ad97a39c35425154937d0634e5a106d1092331cd24e8749027d5422beab2797","name":"finalize.py","bytes":2673},{"sha256":"273d00af6abe94d68932348df5b68483632153a641bee824e32ea69c962fb3e7","name":"make_report.py","bytes":9174},{"sha256":"2e23116423779e2f871246938e485faf9e2b7f464f9b6c92e9c480fbe4517e1d","name":"analysis.out","bytes":1682},{"sha256":"a183053c7abaf37e74aa43bf87c7a604b69fbd9d9da195bf8e86e7297b2d94ba","name":"report.md","bytes":7609},{"sha256":"6ef836ed2f3eff24b6576c382f222f4e8ccfffd4a52da6cdf3b7d9350f87fd65","name":"verified_results.json","bytes":2885},{"sha256":"bfa82617364e9bfc56577125864d7339117976a34d3e2a199ccd014877ee0828","name":"base23.txt","bytes":57},{"sha256":"8d910b10ba07848a15f9ac5eb9f03e504ddb7739aae1c7fb708c5441ebdb9486","name":"base_best24.txt","bytes":57},{"sha256":"8802cdb90058f279b7cc9424bd5433787addbc09a9232082ff6267726334b004","name":"base_best25.txt","bytes":59},{"sha256":"7fa2ab488cc9f117db294bad740a45375306d74a76de13432abdfbee9a056409","name":"search-logs.txt","bytes":14921}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}