{"id":6538,"job_id":12835,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"Improvement claim: none. Route 424 stays blocked. The refutation in #6431 of the delta = 0 premise is kept. This rescue looked for a changed ingredient that reaches (H-sub-pow) without that premise. None exists in the literature or the corpus. The bounded negative is recorded below.\n\nRungs:\n- Classification of #6431's findings: the delta = 0 claim, the k-independent ratio and K = ln 5 are refuted statements. The R0 delta-meter is a failed instrument. The uniform ratio cap (*) G2hat(b^{k+1}) <= e^K G2hat(b) G2hat(b^k) is an unresolved task: a proof gap at a fixed base and a possible truth gap across all bases. heuristic (reading of #6431 and the K-pass).\n- Weaker requirements: (a) a fixed base instead of all bases, (b) a stepped law (#156/#61). Both still need (*) uniformly in k, or the unproven delta >= 0. Neither avoids the obstruction. heuristic.\n- Per-prime insertion lemma: if the maximal twin-survivor gap g at modulus P# satisfies g < p/2, then G2(P# p) <= 3 g. Three consecutive survivors span < p but only two classes are removed mod p. proven (elementary, r424_obstacle.md section 2.4). On the chain g >> p (G2hat(82) = 1710), so the lemma never applies. Where it does apply, 3 per prime compounds over pi(b^{k+1}) - pi(b^k) primes. It is a dead end.\n- Online search for a uniform ratio or doubling bound for Jacobsthal-type maximal gaps found only absolute single-level bounds (Iwaniec, arXiv:1209.3464, Kanold, Stevens, arXiv:2211.13255, 1611.03310, 1706.03668, 1708.05415, OEIS A048670/A144311). By T3, absolute bounds never yield a uniform ratio. heuristic (search record).\n\nWhat changed. #6431 closed the route's path to (H-sub-pow) but not (H-sub-pow) itself. This return pins the exact remaining gap: a uniform-in-level cap on the ratio of two-class maximal gaps at consecutive chain levels. That cap is the two-class Jacobsthal window problem in its window coordinate. No weaker requirement, published ingredient, elementary per-prime bound or reachable instrument avoids it. The route's own uncertainty (K_eff > 1.80 at n <= 2^16) and its experiment (2^20 to 2^30) are far beyond the exact range (n <= 82). No published computation was rerun.\n\nNo next step: no distinct experiment avoids the obstruction.","patch":null,"cpu_hours":0,"hashes":{"r424_obstacle.md":"adb9e7b7a947322d5b78776336c0229c39e5cbd49c45bde2abbd5c2acdcd1670"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-29T07:31:07.732Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6431,6387,6358],"messages":[]},"tokens":{"log":"summary","input":168,"models":{"claude-opus-5-5":47059},"output":47059,"source":"reported","entries":0,"cache_read":3953668,"cache_write":234435,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"File: r424_obstacle.md (text only; no code, no computation). Section 1 classifies the #6431 findings. Section 2 examines five candidate repairs; 2.4 contains the per-prime insertion lemma and its proof. Section 3 is the search record. Section 4 states the exact remaining gap and the revisit conditions. To check the lemma: take survivors s1<s2<s3 with consecutive gaps <= g < p/2, so s3-s1 < p; they occupy three distinct residues mod p, while sieving by p removes only two classes.","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":"scoped_obstruction","evidence":"r424_obstacle.md (classification, five repairs examined, per-prime lemma with proof, search record); #6431 refutation; K-pass sections 2-4 (cited, not rerun).","statement":"After #6431 refuted delta = 0, every path from route 424 to (H-sub-pow) needs (*) G2hat(b^{k+1}) <= e^K G2hat(b) G2hat(b^k) uniformly in k: a ratio cap on two-class maximal gaps at consecutive chain levels. No weaker requirement (fixed base, stepped law), published bound (all absolute; T3), elementary per-prime insertion bound (factor 3 only when g < p/2, void on the chain) or reachable instrument (R0 meter fails control; exact range n <= 82) supplies it.","assumptions":"G2hat(n) = G2(P(n)#), maximal distance between consecutive twin survivors; f = ln G2hat; trusted zone [1.3946, 11.3568) and reachable data as in the K-pass of the record.","revisit_when":"A uniform-in-level ratio or doubling bound for G2 or the two-class Jacobsthal function at primorials is proved or published; or delta >= 0 is proved on the stepped ladder with c derived from the ladder (#6431 conditions); or a delta instrument passes the one-class control; or an all-s K* bound arrives from Q-kstar-prereg."},"route_id":424,"depends_on":[6431],"evidence_md":"#6431 refuted the route's premise (delta = 0 via Selberg-Delange/#6358, k-independent ratio, K = ln 5). Its refutation stands. This rescue tested whether any changed ingredient reaches (H-sub-pow) without that premise. Outcome: none does. The decisive obstruction is the inequality (*) G2hat(b^{k+1}) <= e^K G2hat(b) G2hat(b^k), uniform in k, for some K in the trusted zone [1.3946, 11.3568).\n\n- Fixed base instead of all bases. The sign lemma is finite at fixed b for every delta, so delta < 0 is harmless there. (*) is still needed uniformly in k, and for K >= 3 the base must be >= 72 to buy beta < beta_2. The obstruction is unchanged.\n- Stepped law. Already priced finite for delta >= 0 (#156/#61). It needs the same unproven sign.\n- Per-prime insertion. Proven: if the maximal survivor gap g at P# is < p/2, then the gap at P# p is <= 3g (three consecutive survivors span < p; only two classes are removed mod p). On the chain g >> p (G2hat(82) = 1710 against next primes <= 101), so the lemma is void. Even where valid, 3 per prime compounds over pi(b^{k+1}) - pi(b^k) primes. A constant cap needs phi(p) = 1 + O(1/p), and bounding long removed runs (about 2g/p) is the two-class Jacobsthal window problem.\n- Instruments. The R0 diagonal meter fails the one-class control (+0.5157 +/- 0.1969 vs 2). The route's uncertainty sits at n <= 2^16 and its experiment at 2^20-2^30, against an exact range of n <= 82 (47# priced at 1.49 days; first new rung (2,6) at n = 100).\n\nSo the record keeps (H-sub-pow) OPEN, with the gap narrowed to one named missing method. Data: 15 reachable pairs, sup D = 1.0033 at (16,4), so no K >= 1.0033 is falsified.","prior_art_md":"Online search, 2026-10-29. Changed ingredient: a ratio, doubling or submultiplicativity bound for Jacobsthal-type maximal gaps (one- or two-class) that is uniform in the level, i.e. h(P# Q)/h(P#) <= C(Q) or G2(P(by)#)/G2(P(y)#) <= C(b).\n\nSources found and their content:\n- arXiv:1209.3464: explicit h(k) <= k(k+1)/2 (e^gamma ln(k ln k + k/2) + 0.0289). Absolute.\n- Iwaniec 1978: h(k) << (k log k)^2. Kanold: 2^k. Stevens: 2k^{2+e log k}. All absolute.\n- arXiv:2211.13255: monotonicity (non-strict ordering) for Jacobsthal-type functions. No multiplicative cap.\n- arXiv:1611.03310 (algorithms; Hagedorn n < 50) and arXiv:1708.05415: computation and lower-bound constructions.\n- arXiv:1706.03668 (Ziller-Morack): two-class h2 values for up to 21 primes. Data only.\n- OEIS A048670, A144311: data.\n- Lower bounds (Rankin; FGKMT, as cited in two-class-lower-bounds.md) are single-level.\n- Prior search inherited from #6431 and the route: Selberg-Delange (de la Breteche-Tenenbaum arXiv:2010.12929; Tenenbaum III.5) gives (log x)^{rho-1} for averages, not a delta for a maximal-gap ladder. No source gives delta for G2hat or K = ln 5.\n\nFailures in the source field: no published or corpus inequality bounds such a ratio uniformly. T3: two absolute bounds never give a uniform ratio. #1947: bounded-factor comparison is load-bearing, so it cannot be assumed.\n\nExact remaining gap: (*) at one base b, uniform in k, with K in [1.3946, 11.3568). For all bases, also the sign delta >= 0 of the second-order term, unknown for G2."},"research_route_id":424,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_aed3ae67369b160aaa22abed","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"r424-uniform-ratio-cap-hsubpow","kind":"negative","negative":{"kind":"scoped_obstruction","evidence_md":"r424_obstacle.md; #6431; T3; #1947.","revisit_when_md":"A uniform-in-level ratio or doubling bound for G2 or the two-class Jacobsthal function at primorials is proved or published; or delta >= 0 is proved on the stepped ladder with c derived from the ladder (#6431 conditions); or a delta instrument passes the one-class control; or an all-s K* bound arrives from Q-kstar-prereg."},"domain_md":"Twin-survivor maximal gaps at primorial moduli P(n)#; reachable exact range n <= 82; K in [1.3946, 11.3568).","statement_md":"(H-sub-pow) for G2hat cannot be reached from route 424 without a uniform-in-k cap G2hat(b^{k+1}) <= e^K G2hat(b) G2hat(b^k). Absolute single-level bounds, the stepped-law pricing and the per-prime factor-3 insertion lemma (valid only when g < p/2) do not supply it.","assumptions_md":"Gap = max distance between consecutive twin survivors; delta sign unknown.","artifact_sha256":["adb9e7b7a947322d5b78776336c0229c39e5cbd49c45bde2abbd5c2acdcd1670"],"transfer_conditions_md":"Applies to any (H-sub-pow) or Fekete-type route that needs a constant bounding the maximal-gap ratio between chain levels, one or two class, unless it brings a level-uniform inequality."}],"topic_ids":[]},"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":"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/424 and return #6431. Return the ordinary report and transcript plus research: {route_id: 424, 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":"6431","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[424],"research_url":"/projects/twin-primes/research-routes/424","transcript_url":"/projects/twin-primes/return/6538/transcript","files":[{"sha256":"adb9e7b7a947322d5b78776336c0229c39e5cbd49c45bde2abbd5c2acdcd1670","name":"r424_obstacle.md","bytes":5585}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"05c6aa54cdb6e37f40a2720df437bf75c83c8040834becadb27ee6023f47a8db","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[{"key":"r424-uniform-ratio-cap-hsubpow","kind":"negative","negative":{"kind":"scoped_obstruction","evidence_md":"r424_obstacle.md; #6431; T3; #1947.","revisit_when_md":"A uniform-in-level ratio or doubling bound for G2 or the two-class Jacobsthal function at primorials is proved or published; or delta >= 0 is proved on the stepped ladder with c derived from the ladder (#6431 conditions); or a delta instrument passes the one-class control; or an all-s K* bound arrives from Q-kstar-prereg."},"domain_md":"Twin-survivor maximal gaps at primorial moduli P(n)#; reachable exact range n <= 82; K in [1.3946, 11.3568).","statement_md":"(H-sub-pow) for G2hat cannot be reached from route 424 without a uniform-in-k cap G2hat(b^{k+1}) <= e^K G2hat(b) G2hat(b^k). Absolute single-level bounds, the stepped-law pricing and the per-prime factor-3 insertion lemma (valid only when g < p/2) do not supply it.","assumptions_md":"Gap = max distance between consecutive twin survivors; delta sign unknown.","artifact_sha256":["adb9e7b7a947322d5b78776336c0229c39e5cbd49c45bde2abbd5c2acdcd1670"],"transfer_conditions_md":"Applies to any (H-sub-pow) or Fekete-type route that needs a constant bounding the maximal-gap ratio between chain levels, one or two class, unless it brings a level-uniform inequality.","scope_sha256":"e115bb0ee13e2e76cf90f95327fa4848254a4a654dab707e1fec65c5b6e957e2","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}