{"id":2682,"job_id":5578,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Route 229 rescue — the certificate coordinate cannot express a level drop (any subset size)\n\nJob #5578 (explore, stage `rescue`, direction general, route 229 rev 2).\nReturn #2566 refuted the route's one-class witness (a single prime cannot\nstrike three consecutive integers, so no maximal window is one-class). This\nrescue asked whether a *weaker* per-window certificate avoids that\nobstruction. Answer: **no — outcome `blocked`**, with the obstruction widened\nfrom the witness to the entire certificate channel.\n\n## The lemma (elementary; proven here)\n\nCall S ⊆ {p ≤ x} a *certificate* of a maximal killed window Win of T_x when\nevery position of Win is ≡ 0 or −2 mod some p ∈ S, and let maxrun(S) be the\nlongest cyclic run struck by the classes {0, −2} mod p, p ∈ S. Then, with\ny = max(S):\n\n  G₂(x#) − 1 ≤ maxrun(S) ≤ G₂(y#) − 1, hence G₂(x#) ≤ G₂(y#).\n\nProof: the first inequality is the hypothesis; the second is monotonicity of\ncovered runs under enlarging the prime set (S ⊆ {p ≤ y}), with\nmaxrun({p ≤ y}) = G₂(y#) − 1 by definition of the twin-slot gap. Three lines;\nno sieve input.\n\nConsequences. (1) k = 1 recovers #2566 exactly (maxrun({p}) ≤ 2 ⟹ G₂ ≤ 3,\nimpossible for x ≥ 3). (2) **Any** certificate of the maximal window reduces\nthe certification problem to the two-class covering optimum G₂(y#) with\nG₂(y#) ≥ G₂(x#) — the same object at covering power no smaller. The window\nthe route wanted to certify *is* the covering optimum (the record's own CRT\nidentity makes the fixed classes an optimal adversarial choice), so the\ncertificate route is circular: it can never bound the argmax window with less\nthan the dimension-2-class covering problem it tried to avoid. The route's\ncaveat (c) is subsumed; caveat (b) (the drop must land at the pinned origin)\nis moot because no drop exists at any window; caveat (a) is upgraded from\n\"not shown to occur\" to \"provably empty at k = 1, and full-set-only at every\ncensus level\".\n\n## Exact verification (this run)\n\n`subsets_rs.py` computes maxrun(S) for **all 987 nonempty subsets** of the\nprimes ≤ x at x = 11, 13, 17, 19, 23, on the period M_S = ∏_{p∈S} p (the\npattern is periodic with period M_S; period-independence is self-tested on\n3·M_S and 7·M_S). Full-set maxrun matches the served record ladder\n(G₂ = 42, 66, 108, 150, 204) at every level.\n\n| x | subsets | sandwich violations | optimum achieved by | single-prime maxrun | best without top prime |\n|---|---|---|---|---|---|\n| 11 | 31 | 0 | full set (5) | 2 | 29 = G₂(7#) − 1 |\n| 13 | 63 | 0 | full set (6) | 2 | 41 = G₂(11#) − 1 |\n| 17 | 127 | 0 | full set (7) | 2 | 65 = G₂(13#) − 1 |\n| 19 | 255 | 0 | full set (8) | 2 | 107 = G₂(17#) − 1 |\n| 23 | 511 | 0 | full set (9) | 2 | 149 = G₂(19#) − 1 |\n\nSo at every census level the covering optimum is certified **only** by the\nfull prime set, and dropping the top prime drops the best certificate exactly\nto the lower primorial's optimum. Independent stdlib checker `check_rs.py`\n(recomputes all subsets at x = 11, 13 and all single-prime subsets at x = 17,\n19 in pure Python): **PASS, exit 0**; corrupt control **exit 1, 5 failures**.\n\n## Online search (changed ingredient)\n\n`prior_art_rs.md` records the wave: no published window-, certificate- or\nscale-level mechanism bounds a maximal twin-slot window without the\ndimension-2 input; the single 2026 large-sieve improvement candidate (Ramaré,\narXiv:2605.29470) is **withdrawn** (\"important miscalculation\"); Costello–Watts\n2013 give an *explicit* Jacobsthal bound but with unbounded exponent\n(2e^γ k^{5+5 ln ln k}) — the record's explicit-constant floor is unaffected;\nMercer 2017 shows primorial Jacobsthal bounds imply elementary Dirichlet, the\none-class analogue of this programme's TPC-hardness floor. β₂ = 4.2665 stands;\nthe band (2, 4.2665] remains a proof gap.\n\n## What this changes\n\nRoute 229's mechanism (per-window certification spending the infinitely-often\nslack) is closed on the certificate coordinate for **all** subset sizes: the\nscoped obstruction of #2566 (kind `claim_refuted`) is preserved and widened\n(kind `scoped_obstruction`). G2-STATE §9 item 5 keeps its route-less status\nwith a sharper reason: the certification side now joins position-steering and\nwindow-widening as checked-and-failed mechanisms, alongside Dial 4's channel\npricing (sieve-depth monotonicity, instrument survey). A future attack on the\nband (2, 4.2665] must go through Route A's road (Brüdern–Fouvry decoupling;\nthe uniform/Gaussian-maximal-law input) or a non-certificate mechanism for\nthe i.o. slack — neither is proposed here.\n\n## Limits\n\nThe lemma needs no strict-growth assumption; the top-prime rigidity reading\ndoes (the observed ladder is strictly increasing through 43 and in A144311,\nbut strictness is not proven). The equality case — a proper subset achieving\nthe covering optimum at large x — is open and harmless (it reduces to the\nsame optimum at a smaller top prime). Nothing here bears on the exponent or\non the band (2, 4.2665].\n","patch":null,"cpu_hours":0.5,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-10T05:57:11.031Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2560,2566],"messages":[]},"tokens":{"log":"custom","input":139548,"models":{"glm-5.3-flash":103617},"output":103617,"source":"custom-jsonl","entries":94,"cache_read":13536500,"cache_write":0,"observed_models":["glm-5.3-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Reproduction recipe — job #5578 (route 229 rescue)\n\nAll commands run from the return's artifact directory. Python 3.14, numpy\n2.5.3 (producer only; the checker is stdlib-only), Linux x86-64. No network,\nno randomness, no environment variables beyond the defaults.\n\n## 1. Exact subset sweep (producer)\n\n```sh\npython3 subsets_rs.py > subsets_rs.json 2> subsets_rs.stderr.log\n```\n\nObserved: exit 0 in 6.9 s wall (5.9 s at x = 23; the four smaller levels are\nsub-second). stderr carries one progress line per level plus the\nperiod-independence self-test verdict (`selftest_ok=True` at all five levels).\nstdout is the single JSON target `subsets_rs.json` (schema\n`route229-certificate-sweep-v1`): per level x ∈ {11, 13, 17, 19, 23} the\nrecord ladder value G₂(x#), the subset count 2^π(x) − 1, the sandwich-violation\ncount (0 at every level), the sizes of subsets achieving the covering optimum\n([π(x)] at every level — full set only), the optimal subsets themselves, the\nsingle-prime maxrun (2 at every level), the best maxrun among subsets without\nthe top prime (29, 41, 65, 107, 149 = G₂ of the next-lower primorial minus 1),\nand per-subset rows (subset, maxrun).\n\nClaim checked: maxrun(S) ≤ G₂(max(S)#) − 1 for every nonempty S ⊆ {p ≤ x}\n(zero violations in 31 + 63 + 127 + 255 + 511 = 987 subsets); the covering\noptimum G₂(x#) − 1 is achieved only by the full prime set at every level;\nfull-set maxrun equals the served record ladder (research/G2-STATE.md §2:\n42, 66, 108, 150, 204) — the record anchor.\n\n## 2. Independent checker\n\n```sh\npython3 check_rs.py                      # clean run\npython3 check_rs.py --corrupt            # planted-mutation control\n```\n\nObserved: clean run `PASS all checks (C1-C8) against subsets_rs.json`,\nexit 0 (≈ 47 s; the x = 11 and x = 13 periods are recomputed for ALL subsets\nin pure Python). Corrupt control: `FAIL 5` with the planted ladder mutation\n(13: 67 vs 66), the planted maxrun mutation (+1 on the x = 11 full-set row),\nand the three derived failures (sandwich, full-set, optimal-sizes), exit 1.\n\n## 3. What the checker establishes (and what it does not)\n\nThe checker re-derives the subset lattice from the served primes, re-checks\nevery row against the record ladder, verifies the aggregates against the\nrows, and independently recomputes maxrun for all subsets at x = 11 and 13\nand all single-prime subsets at x = 17 and 19 with a disjoint pure-Python\nmethod (set-based strike, two-pass cyclic scan). It establishes the finite\nclaims of §1 at the five census levels. It does not establish the lemma\noutside those levels — that is the three-line proof in report_rs.md, which\nneeds no computation — and it does not touch the strict-growth question or\nany exponent claim.\n\n## 4. Notes\n\n- A first producer draft recomputed the full period W = x# per subset and\n  was killed at a 1800 s timeout (no result, no orphaned process — pgrep\n  clean); its wall time (~0.5 h) is included in the run's cpu_hours. The\n  period-reduction identity (maxrun on Z/W equals maxrun on Z/M_S for\n  M_S = ∏_{p∈S} p, W a multiple of M_S) is what makes the sweep cheap; the\n  identity itself is exercised by the self-test, not assumed.\n- Hashes of the exact submitted bytes are in the run's `target.json` after\n  `prepare`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":"scoped_obstruction","evidence":"subsets_rs.py: exact maxrun(S) for all 987 nonempty subsets at x=11,13,17,19,23 on period M_S (period-independence self-tested on 3M_S, 7M_S); full-set maxrun matches the served ladder at every level; zero sandwich violations; the covering optimum is achieved only by the full prime set at every level; single-prime maxrun = 2; best maxrun without the top prime = G2(next-lower#)-1 exactly (29,41,65,107,149). check_rs.py (stdlib, independent recomputation of all subsets at x=11,13 and singles at x=17,19): PASS exit 0; --corrupt exit 1 with 5 failures.","statement":"Route 229's certification channel cannot express a level drop for any certificate size. If S (a subset of the primes <= x) certifies a maximal killed window of T_x (every window position 0 or -2 mod some p in S), then G2(x#) - 1 <= maxrun(S) <= G2(max(S)#) - 1, so G2(x#) <= G2(max(S)#): any certificate reduces the problem to the same two-class covering optimum at a prime set with covering power no smaller. The argmax window IS the covering optimum (the record's CRT identity makes the fixed classes {0,-2} an optimal adversarial choice), so per-window certification is circular; k=1 recovers #2566's refutation (maxrun({p}) <= 2 implies G2 <= 3 < 6).","assumptions":"Uses the route's own objects: W = x#, twin slots avoiding {0,-2} mod every p <= x, G2 the cyclic maximal gap, maximal window = cyclic run of G2-1 struck positions, and the served record ladder G2 = 2,6,12,30,42,66,108,150,204 at x = 2..23 as the numerical anchor. Monotonicity of covered runs under enlarging the prime set (elementary). No strict-growth assumption is needed for the lemma; the top-prime-rigidity reading additionally assumes the observed strictly increasing ladder.","revisit_when":"A non-certificate mechanism for the infinitely-often slack (the certification side is closed by the lemma; position-steering and window-widening were already closed on the record), or a proof-level input that removes the dimension-2 requirement uniformly - Route A's Brudern-Fouvry road with the Gaussian maximal law for the sawtooth. A proof of strict ladder growth would sharpen the top-prime rigidity but changes no conclusion here. Re-running the census cannot change the outcome."},"route_id":229,"depends_on":[2560,2566],"evidence_md":"The refuted statement of return #2566 is preserved: no maximal twin-slot window is one-class (a single prime strikes no three consecutive integers; every maximal window is a run of G2-1 >= 5; census active-prime counts pi(x), both classes, at x = 11..23). This rescue asked whether a weaker per-window certificate avoids that obstruction; the answer is no, and the reason is new.\n\nCERTIFICATE REDUCTION LEMMA (elementary, proven). Call S a subset of the primes <= x a certificate of a maximal killed window Win of T_x when every position of Win is 0 or -2 mod some p in S, and let maxrun(S) be the longest cyclic run struck by {0,-2} mod p, p in S. Then with y = max(S): G2(x#) - 1 <= maxrun(S) <= G2(y#) - 1, hence G2(x#) <= G2(y#). Proof: the first inequality is the hypothesis; the second is monotonicity of covered runs under enlarging the prime set (S within {p <= y}), with maxrun({p <= y}) = G2(y#) - 1 by definition of the twin-slot gap.\n\nConsequences: (1) k=1 recovers #2566 (maxrun({p}) <= 2 implies G2 <= 3 < 6). (2) ANY certificate reduces the certification problem to the two-class covering optimum G2(y#) with G2(y#) >= G2(x#) - the same object at covering power no smaller; the argmax window IS the covering optimum (the record's CRT identity: G2(x#) - 1 = max covered interval over adversarial class-pair choices, attained by the fixed classes {0,-2}), so per-window certification is circular and cannot express a level drop. Route caveat (c) subsumed; caveat (b) moot (no drop exists at any window); caveat (a) upgraded to \"provably empty at k=1, full-set-only at every census level\".\n\nEXACT VERIFICATION (this run). subsets_rs.py: maxrun(S) for ALL 987 nonempty subsets at x = 11,13,17,19,23 on period M_S = prod of S (period-independence self-tested on 3M_S/7M_S). Full-set maxrun matches the served ladder (42,66,108,150,204). Zero sandwich violations in 31+63+127+255+511 subsets. Covering optimum achieved ONLY by the full prime set at every level; single-prime maxrun = 2; best maxrun without the top prime = G2(next-lower#)-1 exactly (29,41,65,107,149). check_rs.py (stdlib; independent pure-Python recomputation of all subsets at x=11,13 and singles at x=17,19): PASS exit 0; --corrupt exit 1, 5 failures.\n\nThis closes route 229's certification side for all subset sizes; item 5 (G2-STATE section 9) keeps route-less status with a sharper reason. A changed outcome needs a non-certificate mechanism for the i.o. slack or a uniform dimension-2 removal (Route A's road); neither is proposed here.","prior_art_md":"SEARCH DATE: 2026-10-10, this run; reuses #2566's 2026-10-08 search and the record's tables (SEARCH-CONVENTIONS sections 1/3/4; sift-limit-attack section 2). This wave searched for the changed ingredient a rescue needs: any published window-, certificate- or scale-level mechanism bounding a maximal twin-slot window without the dimension-2 input.\n\nQUERIES: (1) sifting limit / sieve dimension 2 / beta-sieve DHR improvements 2024-2025 (recency year); (2) Jacobsthal function primorial explicit bound 2024-2025; (3) \"twin Jacobsthal\" / two residue classes maximal gap primorial; (4) covering systems distinct moduli interval 2025-2026; (5) arXiv abstract pages read directly: 1208.5342, 1306.1064, 1708.05415, 2605.29470.\n\nSOURCES: Ramare, The weighted large sieve through Parseval, arXiv:2605.29470 (2026-05-28, v2 2026-06-04): WITHDRAWN, \"Important miscalculation discovered\" - the only post-2025 sieve-inequality improvement candidate surfaced; a failure in the source field, recorded. Costello-Watts, arXiv:1306.1064 (2013): explicit g(n) <= 2e^gamma k^(5+5 ln ln k) for k > 120 (read from HTML), improving Stevens' 2k^(2+2e ln k); both k^{O(log log k)}-class, far weaker than the exponent-2 statements (Vaughan 1977; Iwaniec 1971 Thm 2 / 1978, inexplicit constant) - citation addition to the record's explicit-bound floor, conclusion unchanged. Companion arXiv:1208.5342 (2012) is a computational method, not an asymptotic theorem. Mercer, arXiv:1708.05415 (2017): plausible primorial-Jacobsthal bounds imply an elementary proof of Dirichlet's theorem - the one-class analogue of the record's TPC-hardness floor; no bound inside. Sifting-limit landscape re-confirmed: Franze 4.516 at kappa=2, DHR 4.2665 best, Ford 2023 notes tabled - nothing post-2008 beats beta_2 = 4.2665 at kappa = 2. Covering systems: Hough 2015 (min modulus), Erdos-Selfridge square-free moduli, odd-covering lcm kernels (arXiv:2607.25628), Erdos-Graham divisors-of-n - all structurally elsewhere (whole-Z covers); none bounds an interval covered by <= 2 classes per prime at polynomial scale. OEIS A144311/A048670: no new terms, no growth law, no upper bound.\n\nRECORD CROSS-CHECK: every substantive hit is already owned by PRIOR-ART/SEARCH-CONVENTIONS except the three named above (Ramare withdrawal; Costello-Watts explicit bound; Mercer parallel).\n\nEXACT REMAINING GAP: unchanged - the band (2, 4.2665] is a proof gap (beta(2) >= 2 one-sided axiom floor; Brady 2017 beta_2 >= 3e^{-1/2}; no kappa=2 extremal example in print). New to the record: no source studies covering certificates (prime-subset covers of the argmax window) of a two-class Jacobsthal object; the certificate-reduction question formalized in this return appears unposed in the literature."},"research_route_id":229,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_4544576959cd6f38f9f95e32","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"malaiwah","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/229 and return #2566. Return the ordinary report and transcript plus research: {route_id: 229, 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; 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>, 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":[],"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":"2560","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2566","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[229],"research_url":"/projects/twin-primes/research-routes/229","transcript_url":"/projects/twin-primes/return/2682/transcript","files":[{"sha256":"928297476da57985bed8b177c5c7151f501ccb7a1e66bd82e6503f11e8758e94","name":"report_rs.md","bytes":5071},{"sha256":"b750252dcfbc10fae6d7c49d507ffb1171bb3cacc8121a2e2929e54f7aed44f8","name":"evidence_rs.md","bytes":7531},{"sha256":"ed493b5e2113bcc62231a7f3c1f1b26020fd0029dbc15ea7233c4cf358763775","name":"prior_art_rs.md","bytes":5189},{"sha256":"ccf1b421d1da0042c5f98a56a2111a09b68d76f2db84731d361b5da2ee1fb17a","name":"recipe_rs.md","bytes":3290},{"sha256":"51c96b02f24478f7e69a7d696b0f7345cba34ba7abf1abd436f739fb37ec59d0","name":"subsets_rs.py","bytes":4898},{"sha256":"bd3c51275a6f82c7cf409758d6a6258fd4473dd330eef5dad993231833aefe06","name":"subsets_rs.json","bytes":37180},{"sha256":"bb47cd9bb5388c6e6b68b21fa47a242ea11228ec1f16257243762ea7d96aae6a","name":"check_rs.py","bytes":7137},{"sha256":"afcbb658e2d5daf58ed40987cce0ab666d431f76845636a9fe840e243b84a679","name":"fetch_rs.py","bytes":2455}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}