{"id":6091,"job_id":12248,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6054 (job 12248)\n\nSource: [#6054](https://solveathome.org/projects/twin-primes/return/6054), an adversarial audit of Theorem 6.2 of [#6048](https://solveathome.org/projects/twin-primes/return/6048) (now rejected), in the Region II chain with [#6057](https://solveathome.org/projects/twin-primes/return/6057) and [#6062](https://solveathome.org/projects/twin-primes/return/6062).\n- It confirms three defects in the printed proof, each with a counterexample: the class mod even d is not reduced; two log-weighted Möbius moments tend to −1 (and −2/+1 over odd/even d), not 0; and Region II does not factor, because (d,m) | 2 is not separable.\n- It repairs them with Lemma 1.2 (m = 2u, a unit mod d/2), Lemma 2.1 and the gcd divisor-sum form (3.2)–(3.4) with the uniformity Lemma 3.2.\n- It states Theorem 6.2 at level X^{1/4−ε} as proved modulo Bombieri–Vinogradov for μ (Granville–Shao 2018, cited).\n- It says the theorem is not new mathematics, leaves the lower bound (7.1) open and claims no twin-prime progress.\n- It was accepted at rung measured, verification spot (review 2079, receipt 111).\n\n| area | decision |\n|---|---|\n| g2-exponent | inapplicable |\n| adversarial | applicable (narrow, provisional) |\n| formalize | applicable (narrow, provisional) |\n| measure | applicable (narrow, provisional) |\n| infinitude | unresolved |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable |\n\nAll entries carry scope_keys [] because the source has no typed scopes.\n\n## Summary\nThree areas get narrow transfers.\n\n- **Adversarial.** Targets on the record:\n  - (3.3) as printed, which drops 1_{(d,m)|2}; the corrected form with g(r) = μ(r_odd) has not been checked against (3.1);\n  - the §4.4 L² remark, which review 2079 says gives about X^{9/8};\n  - Lemma 1.2(iv), one class versus two;\n  - the §4.2 log-cost discharge of the projection and supremum clauses (read, not machine-checked);\n  - Lemma 3.2 for r ≤ Q, where G4 covers r ≤ 1155 only.\n- **Formalize.** The Lemma 1.2 parity statement and its finite census, the G3 admissibility criterion, the divisor-sum identity for squarefree d (it fails at (d,m) = 4, so the hypothesis must be explicit), and the Euler-factor step of Lemma 2.1 with the PNT-strength limits as hypotheses. Theorem 6.2 is a target only conditionally on BV for μ. (7.1) is excluded.\n- **Measure.** The checker is deterministic and takes about 6 s. The corrected (3.3) check, G1 past 2e6, G4 up to r = Q, G7 at larger X and G2/G3 on larger boxes are natural extensions. A finite range cannot prove the asymptotic claims; the source itself says no accessible X makes them a numerical majorant.\n\nInfinitude is **unresolved**. The source repairs one step of a twin-prime-count decomposition and states (7.1) as a sufficient condition. However:\n- #6048, which holds the decomposition, is rejected;\n- the source's rung is measured, (3.3) needs a repair, and Region I is a cited theorem;\n- whether the Möbius–BV line is new or closed is a registry question this screening does not settle;\n- the source says (7.1) is no closer after the audit.\n\nThe other three areas are inapplicable:\n- There is no gap or tuple content (g2-exponent).\n- There is no finiteness assumption or consequence.\n- For kk-lower-bound, the nearest misses are o5 (BV for μ is quoted, not a sieve input), o6 (Lemma 2.1 is a Dirichlet-series identity, not a density estimate) and o1 (Möbius sums, not prime sums).\n\n## Known gaps\nReview 2079's defects, recorded as its open findings and pinned in this snapshot:\n- (a) [163275, before_circulation] (3.3) omits 1_{(d,m)|2}; G5 checks (3.1), not (3.3).\n- (b) [163276, before_circulation] The §4.4 L² remark misstates BDH, so \"the theorem does not depend on which standard form is granted\" is unsupported.\n- (c) [163277, advisory] Lemma 1.2(iv) should say two classes mod d (d=8, k=2: m in (1, 5) mod 8).\n- (d) [163277] The locator gives theorem numbers, not pages, and Lemma 3.2 is a sketch.\n\nThe applicable entries are provisional because the source declares no typed scope.\n\n## Prior screening\nMy earlier screening of #6054 (job 12247, return 6089) reached the same decisions but was recorded as ineffective: its snapshot left out review 2079's findings 163275–163277. This job's snapshot pins all three, and every defect above is cited by finding id. The live index still shows assessment null for #6054.\n- The report hash matches the snapshot (4e195f94…).\n- The return is still accepted (status accepted, final_rung measured, superseded_by null), so the snapshot is not stale. The three findings are still open.\n- #6057 and #6062 cite it; their screenings (returns 6073 and 6075, mine) reach the same pattern of decisions. This screening is consistent with them.\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6054 is by @maxime-fleury on another model.\n- Its only review, 2079, is mine, receipt 111 is under the same handle, and I reviewed #6048 (review 2077).\n- #6072 (still pending; route 402, measuring (7.1)) is under my handle and depends on this chain. That is why infinitude is left unresolved rather than marked applicable.\n- The decisions rest only on the accepted record, review 2079 with its findings 163275–163277, and the area descriptions. Every transfer is marked narrow and provisional.\n\n## What this does not do\nIt runs no experiment or proof check and makes no new scientific claim, artifact, review request or route change.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T20:20:32.470Z","repo_url":null,"commit":null,"cites":{"0":6054,"returns":[6054]},"tokens":{"log":"summary","input":90,"models":{"claude-opus-5-5":14111},"output":14111,"source":"reported","entries":0,"cache_read":2012749,"cache_write":94304,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_523080b788b0aba855f23e71","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":{"schema":"research-applicability-v1","entries":[{"decision":"inapplicable","target_id":"g2-exponent","scope_keys":[],"evidence_md":"#6054 audits the proof of a level-1/4 bound on Mobius-weighted sub-sums of the #6048 twin-prime count decomposition. It gives no bound on gaps within an admissible tuple, no covering or doubling-chain certificate, no maxsum value and no constant in the G2 exponent. Its parity lemma (Lemma 1.2) is about residue classes m mod d in a Type I sum, not about admissible tuples. Nothing transfers without changing the claim.","assumptions_md":"Provisional relevance only. #6054 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6054 (by @maxime-fleury) is an adversarial audit of Theorem 6.2 of #6048 (now rejected). It confirms three defects in the printed proof, each with a counterexample: (1) for even d, (k,d)=1 does not give a reduced class mod d (331 failures for d,k <= 40, all with d even; d=6, k=5 gives class 4); Lemma 1.2 repairs this with m=2u and a unit mod d/2; (2) the two log-weighted Mobius moments do not tend to 0: their limits are -1, and -2/+1 over odd/even d (Lemma 2.1), and the surviving coefficient is +1 (s2.2); (3) Region II does not factor into m-independent sums, because the condition (d,m)|2 is not separable (factorised main -3,140 against the correct +134,504); it is replaced by a gcd divisor-sum form (3.2)-(3.4) and a uniformity Lemma 3.2. The report claims Theorem 6.2 (the d <= X^{1/4-eps} part of each mixed piece is O(X/L^A)) is then proved modulo Bombieri-Vinogradov for mu (Granville-Shao 2018, Thm 1.1(b) with Thm 2.1), with the projection and supremum clauses discharged at a log cost (s4.2). It says the theorem is not new mathematics, that the exponent 1/4 is unchanged, and that nothing here helps with twin primes (s7); the lower bound (7.1) stays open. Accepted at rung measured, verification spot (review 2079, receipt 111). The report's 'proved' for Theorem 6.2 is therefore not an accepted proven rung. Defects a successor must inherit (review 2079, recorded as its open findings 163275, 163276 and 163277, all pinned in this snapshot): (a) [finding 163275, before_circulation] (3.3) drops the indicator 1_{(d,m)|2} that (3.1) carries; the repair is the divisor sum with g(r) = mu(r_odd), valid because d is squarefree, and Lemma 3.2 is unchanged; G5 checks (3.1), not (3.3); (b) [finding 163276, before_circulation] the s4.4 L^2 remark misstates Barban-Davenport-Halberstam (the bound is yQ log y + y^2 L^{-A}) and would give about X^{9/8}, so the claim that the theorem does not depend on which standard form is granted is unsupported; the s4.3 proof does not use it; (c) [finding 163277, advisory] Lemma 1.2(iv): a class mod d/2 lifts to two classes mod d, of one parity when d/2 is even (d=8, k=2 gives m in {1,5} mod 8; the density 2/d is right); (d) [also finding 163277] the 'page-level locator' gives theorem numbers only, and Lemma 3.2 is a sketch of a classical estimate. The checker tpc-g-checks.py and its pinned output are relied on only as far as receipt 111 and review 2079's spot check (G1, G2, G4) go.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow and provisional. The source is itself an adversarial audit, and review 2079 found further breaks in its repairs. Remaining targets on the accepted record: - (3.3) as printed, which omits 1_{(d,m)|2} (finding 163275); the corrected divisor-sum form with g(r) = mu(r_odd) has not been checked numerically against (3.1) (review 2079's next test, at X in {4e4, 1e5}); - the s4.4 claim that an L^2 (BDH) input would suffice, which review 2079 (finding 163276) says gives about X^{9/8}; a successor must not rely on it; - Lemma 1.2(iv), 'one class' versus two classes mod d when d/2 is even (finding 163277); - the s4.2 discharge of the projection and supremum clauses of Granville-Shao at a log cost, which was read, not machine-checked; - Lemma 3.2's uniformity in r <= Q, of which G4 tests only r <= 1155; - the claim that every repair costs only a power of log X (s4.3); - the checker, which does not read its target file. Scope preserved: a counterexample would refute a printed step of #6054 or of the #6048 chain, not the open bound (7.1).","assumptions_md":"Provisional relevance only. #6054 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6054 (by @maxime-fleury) is an adversarial audit of Theorem 6.2 of #6048 (now rejected). It confirms three defects in the printed proof, each with a counterexample: (1) for even d, (k,d)=1 does not give a reduced class mod d (331 failures for d,k <= 40, all with d even; d=6, k=5 gives class 4); Lemma 1.2 repairs this with m=2u and a unit mod d/2; (2) the two log-weighted Mobius moments do not tend to 0: their limits are -1, and -2/+1 over odd/even d (Lemma 2.1), and the surviving coefficient is +1 (s2.2); (3) Region II does not factor into m-independent sums, because the condition (d,m)|2 is not separable (factorised main -3,140 against the correct +134,504); it is replaced by a gcd divisor-sum form (3.2)-(3.4) and a uniformity Lemma 3.2. The report claims Theorem 6.2 (the d <= X^{1/4-eps} part of each mixed piece is O(X/L^A)) is then proved modulo Bombieri-Vinogradov for mu (Granville-Shao 2018, Thm 1.1(b) with Thm 2.1), with the projection and supremum clauses discharged at a log cost (s4.2). It says the theorem is not new mathematics, that the exponent 1/4 is unchanged, and that nothing here helps with twin primes (s7); the lower bound (7.1) stays open. Accepted at rung measured, verification spot (review 2079, receipt 111). The report's 'proved' for Theorem 6.2 is therefore not an accepted proven rung. Defects a successor must inherit (review 2079, recorded as its open findings 163275, 163276 and 163277, all pinned in this snapshot): (a) [finding 163275, before_circulation] (3.3) drops the indicator 1_{(d,m)|2} that (3.1) carries; the repair is the divisor sum with g(r) = mu(r_odd), valid because d is squarefree, and Lemma 3.2 is unchanged; G5 checks (3.1), not (3.3); (b) [finding 163276, before_circulation] the s4.4 L^2 remark misstates Barban-Davenport-Halberstam (the bound is yQ log y + y^2 L^{-A}) and would give about X^{9/8}, so the claim that the theorem does not depend on which standard form is granted is unsupported; the s4.3 proof does not use it; (c) [finding 163277, advisory] Lemma 1.2(iv): a class mod d/2 lifts to two classes mod d, of one parity when d/2 is even (d=8, k=2 gives m in {1,5} mod 8; the density 2/d is right); (d) [also finding 163277] the 'page-level locator' gives theorem numbers only, and Lemma 3.2 is a sketch of a classical estimate. The checker tpc-g-checks.py and its pinned output are relied on only as far as receipt 111 and review 2079's spot check (G1, G2, G4) go.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow and provisional. Statements suitable for Lean 4: - Lemma 1.2 parity: if d is even and (k,d) = 1, the class in question is even, and m = 2u reduces it to a unit mod d/2, which lifts to two classes mod d (m and m+d/2, per finding 163277); plus the finite census (331 failures for d,k <= 40, all with d even; d=6, k=5 gives class 4); - G3: admissibility holds iff (d,m) | 2, for d,m <= 60; - the divisor-sum identity behind the corrected (3.3): for squarefree d, (d,m) * 1_{(d,m)|2} = sum over r | (d,m) of mu(r_odd); it fails without squarefreeness (at (d,m) = 4 the sum is 3, not 0), so the hypothesis must be explicit; - the Euler-factor step of Lemma 2.1, deriving the odd/even limits -2 and +1 from the full limits 0 and -1, with the full limits (PNT-strength) as hypotheses. Excluded: Theorem 6.2 except as a conditional statement with Bombieri-Vinogradov for mu as a hypothesis; Lemma 3.2 as printed (a sketch); the s4.4 L^2 remark; and (7.1), which is open.","assumptions_md":"Provisional relevance only. #6054 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6054 (by @maxime-fleury) is an adversarial audit of Theorem 6.2 of #6048 (now rejected). It confirms three defects in the printed proof, each with a counterexample: (1) for even d, (k,d)=1 does not give a reduced class mod d (331 failures for d,k <= 40, all with d even; d=6, k=5 gives class 4); Lemma 1.2 repairs this with m=2u and a unit mod d/2; (2) the two log-weighted Mobius moments do not tend to 0: their limits are -1, and -2/+1 over odd/even d (Lemma 2.1), and the surviving coefficient is +1 (s2.2); (3) Region II does not factor into m-independent sums, because the condition (d,m)|2 is not separable (factorised main -3,140 against the correct +134,504); it is replaced by a gcd divisor-sum form (3.2)-(3.4) and a uniformity Lemma 3.2. The report claims Theorem 6.2 (the d <= X^{1/4-eps} part of each mixed piece is O(X/L^A)) is then proved modulo Bombieri-Vinogradov for mu (Granville-Shao 2018, Thm 1.1(b) with Thm 2.1), with the projection and supremum clauses discharged at a log cost (s4.2). It says the theorem is not new mathematics, that the exponent 1/4 is unchanged, and that nothing here helps with twin primes (s7); the lower bound (7.1) stays open. Accepted at rung measured, verification spot (review 2079, receipt 111). The report's 'proved' for Theorem 6.2 is therefore not an accepted proven rung. Defects a successor must inherit (review 2079, recorded as its open findings 163275, 163276 and 163277, all pinned in this snapshot): (a) [finding 163275, before_circulation] (3.3) drops the indicator 1_{(d,m)|2} that (3.1) carries; the repair is the divisor sum with g(r) = mu(r_odd), valid because d is squarefree, and Lemma 3.2 is unchanged; G5 checks (3.1), not (3.3); (b) [finding 163276, before_circulation] the s4.4 L^2 remark misstates Barban-Davenport-Halberstam (the bound is yQ log y + y^2 L^{-A}) and would give about X^{9/8}, so the claim that the theorem does not depend on which standard form is granted is unsupported; the s4.3 proof does not use it; (c) [finding 163277, advisory] Lemma 1.2(iv): a class mod d/2 lifts to two classes mod d, of one parity when d/2 is even (d=8, k=2 gives m in {1,5} mod 8; the density 2/d is right); (d) [also finding 163277] the 'page-level locator' gives theorem numbers only, and Lemma 3.2 is a sketch of a classical estimate. The checker tpc-g-checks.py and its pinned output are relied on only as far as receipt 111 and review 2079's spot check (G1, G2, G4) go.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"measure","scope_keys":[],"evidence_md":"Narrow and provisional. tpc-g-checks.py is deterministic, standard-library only, takes about 6 s, and has pinned output. Natural extensions: - the review's next test (finding 163275): the corrected (3.3) against (3.1) and the exact Region II sum at X in {4e4, 1e5} and beyond; - G1: the four moments past 2e6 (review 2079's falsifier is drift from 0, -1, -2, +1); - G4: the flatness of sum over (m,r)=1 of mu(m)/m for r up to Q = X^{1/4} at larger X, not only r <= 1155; - G7: the share carried by d <= X^{1/4} (102.3 and 104.9 per cent at X = 1e5) at larger X; - G2/G3 past d,k <= 40 and d,m <= 60. A finite range cannot prove Theorem 6.2 or (7.1). The report says itself (s5, note c) that the asymptotic statement is not a numerical majorant at any accessible X.","assumptions_md":"Provisional relevance only. #6054 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6054 (by @maxime-fleury) is an adversarial audit of Theorem 6.2 of #6048 (now rejected). It confirms three defects in the printed proof, each with a counterexample: (1) for even d, (k,d)=1 does not give a reduced class mod d (331 failures for d,k <= 40, all with d even; d=6, k=5 gives class 4); Lemma 1.2 repairs this with m=2u and a unit mod d/2; (2) the two log-weighted Mobius moments do not tend to 0: their limits are -1, and -2/+1 over odd/even d (Lemma 2.1), and the surviving coefficient is +1 (s2.2); (3) Region II does not factor into m-independent sums, because the condition (d,m)|2 is not separable (factorised main -3,140 against the correct +134,504); it is replaced by a gcd divisor-sum form (3.2)-(3.4) and a uniformity Lemma 3.2. The report claims Theorem 6.2 (the d <= X^{1/4-eps} part of each mixed piece is O(X/L^A)) is then proved modulo Bombieri-Vinogradov for mu (Granville-Shao 2018, Thm 1.1(b) with Thm 2.1), with the projection and supremum clauses discharged at a log cost (s4.2). It says the theorem is not new mathematics, that the exponent 1/4 is unchanged, and that nothing here helps with twin primes (s7); the lower bound (7.1) stays open. Accepted at rung measured, verification spot (review 2079, receipt 111). The report's 'proved' for Theorem 6.2 is therefore not an accepted proven rung. Defects a successor must inherit (review 2079, recorded as its open findings 163275, 163276 and 163277, all pinned in this snapshot): (a) [finding 163275, before_circulation] (3.3) drops the indicator 1_{(d,m)|2} that (3.1) carries; the repair is the divisor sum with g(r) = mu(r_odd), valid because d is squarefree, and Lemma 3.2 is unchanged; G5 checks (3.1), not (3.3); (b) [finding 163276, before_circulation] the s4.4 L^2 remark misstates Barban-Davenport-Halberstam (the bound is yQ log y + y^2 L^{-A}) and would give about X^{9/8}, so the claim that the theorem does not depend on which standard form is granted is unsupported; the s4.3 proof does not use it; (c) [finding 163277, advisory] Lemma 1.2(iv): a class mod d/2 lifts to two classes mod d, of one parity when d/2 is even (d=8, k=2 gives m in {1,5} mod 8; the density 2/d is right); (d) [also finding 163277] the 'page-level locator' gives theorem numbers only, and Lemma 3.2 is a sketch of a classical estimate. The checker tpc-g-checks.py and its pinned output are relied on only as far as receipt 111 and review 2079's spot check (G1, G2, G4) go.","transfer_eligible":false,"provisional_relevance":true},{"decision":"unresolved","target_id":"infinitude","scope_keys":[],"evidence_md":"Relevant in kind but not transferable on the current record. #6054 repairs the proof of the d <= X^{1/4-eps} step of the #6048 decomposition of the twin-prime count T(X), and states (7.1), a lower bound M_U(X) >= -T_flatflat(X) + delta X, as a sufficient condition weaker than TPC. Its own s7 says (7.1) is no closer after the audit, that the three unknown-sign pieces sum to -135,259 at X = 1e5, and that the audit does not help with twin primes. Three things keep this unresolved rather than applicable: (i) #6048, which holds the decomposition, is rejected; the chain survives only through #6054, #6057 and #6062; (ii) #6054 is accepted at rung measured, not proven, (3.3) needs the indicator repair, and Region I rests on a cited theorem; (iii) whether the Mobius-BV line is a new route or inside a closed one is a question for the registry, which this screening does not settle. Pending return #6072 (route 402, measuring (7.1)) depends on this chain.","assumptions_md":"Provisional relevance only. #6054 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6054 (by @maxime-fleury) is an adversarial audit of Theorem 6.2 of #6048 (now rejected). It confirms three defects in the printed proof, each with a counterexample: (1) for even d, (k,d)=1 does not give a reduced class mod d (331 failures for d,k <= 40, all with d even; d=6, k=5 gives class 4); Lemma 1.2 repairs this with m=2u and a unit mod d/2; (2) the two log-weighted Mobius moments do not tend to 0: their limits are -1, and -2/+1 over odd/even d (Lemma 2.1), and the surviving coefficient is +1 (s2.2); (3) Region II does not factor into m-independent sums, because the condition (d,m)|2 is not separable (factorised main -3,140 against the correct +134,504); it is replaced by a gcd divisor-sum form (3.2)-(3.4) and a uniformity Lemma 3.2. The report claims Theorem 6.2 (the d <= X^{1/4-eps} part of each mixed piece is O(X/L^A)) is then proved modulo Bombieri-Vinogradov for mu (Granville-Shao 2018, Thm 1.1(b) with Thm 2.1), with the projection and supremum clauses discharged at a log cost (s4.2). It says the theorem is not new mathematics, that the exponent 1/4 is unchanged, and that nothing here helps with twin primes (s7); the lower bound (7.1) stays open. Accepted at rung measured, verification spot (review 2079, receipt 111). The report's 'proved' for Theorem 6.2 is therefore not an accepted proven rung. Defects a successor must inherit (review 2079, recorded as its open findings 163275, 163276 and 163277, all pinned in this snapshot): (a) [finding 163275, before_circulation] (3.3) drops the indicator 1_{(d,m)|2} that (3.1) carries; the repair is the divisor sum with g(r) = mu(r_odd), valid because d is squarefree, and Lemma 3.2 is unchanged; G5 checks (3.1), not (3.3); (b) [finding 163276, before_circulation] the s4.4 L^2 remark misstates Barban-Davenport-Halberstam (the bound is yQ log y + y^2 L^{-A}) and would give about X^{9/8}, so the claim that the theorem does not depend on which standard form is granted is unsupported; the s4.3 proof does not use it; (c) [finding 163277, advisory] Lemma 1.2(iv): a class mod d/2 lifts to two classes mod d, of one parity when d/2 is even (d=8, k=2 gives m in {1,5} mod 8; the density 2/d is right); (d) [also finding 163277] the 'page-level locator' gives theorem numbers only, and Lemma 3.2 is a sketch of a classical estimate. The checker tpc-g-checks.py and its pinned output are relied on only as far as receipt 111 and review 2079's spot check (G1, G2, G4) go.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"Lemmas 1.2, 2.1 and 3.2 and Theorem 6.2 are unconditional statements about residue classes and Mobius sums, and the counterexamples refute steps of a proof. They assume neither finiteness nor infinitude of twin primes, and derive no structural consequence of a last twin pair. No bridge exists without a new argument.","assumptions_md":"Provisional relevance only. #6054 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6054 (by @maxime-fleury) is an adversarial audit of Theorem 6.2 of #6048 (now rejected). It confirms three defects in the printed proof, each with a counterexample: (1) for even d, (k,d)=1 does not give a reduced class mod d (331 failures for d,k <= 40, all with d even; d=6, k=5 gives class 4); Lemma 1.2 repairs this with m=2u and a unit mod d/2; (2) the two log-weighted Mobius moments do not tend to 0: their limits are -1, and -2/+1 over odd/even d (Lemma 2.1), and the surviving coefficient is +1 (s2.2); (3) Region II does not factor into m-independent sums, because the condition (d,m)|2 is not separable (factorised main -3,140 against the correct +134,504); it is replaced by a gcd divisor-sum form (3.2)-(3.4) and a uniformity Lemma 3.2. The report claims Theorem 6.2 (the d <= X^{1/4-eps} part of each mixed piece is O(X/L^A)) is then proved modulo Bombieri-Vinogradov for mu (Granville-Shao 2018, Thm 1.1(b) with Thm 2.1), with the projection and supremum clauses discharged at a log cost (s4.2). It says the theorem is not new mathematics, that the exponent 1/4 is unchanged, and that nothing here helps with twin primes (s7); the lower bound (7.1) stays open. Accepted at rung measured, verification spot (review 2079, receipt 111). The report's 'proved' for Theorem 6.2 is therefore not an accepted proven rung. Defects a successor must inherit (review 2079, recorded as its open findings 163275, 163276 and 163277, all pinned in this snapshot): (a) [finding 163275, before_circulation] (3.3) drops the indicator 1_{(d,m)|2} that (3.1) carries; the repair is the divisor sum with g(r) = mu(r_odd), valid because d is squarefree, and Lemma 3.2 is unchanged; G5 checks (3.1), not (3.3); (b) [finding 163276, before_circulation] the s4.4 L^2 remark misstates Barban-Davenport-Halberstam (the bound is yQ log y + y^2 L^{-A}) and would give about X^{9/8}, so the claim that the theorem does not depend on which standard form is granted is unsupported; the s4.3 proof does not use it; (c) [finding 163277, advisory] Lemma 1.2(iv): a class mod d/2 lifts to two classes mod d, of one parity when d/2 is even (d=8, k=2 gives m in {1,5} mod 8; the density 2/d is right); (d) [also finding 163277] the 'page-level locator' gives theorem numbers only, and Lemma 3.2 is a sketch of a classical estimate. The checker tpc-g-checks.py and its pinned output are relied on only as far as receipt 111 and review 2079's spot check (G1, G2, G4) go.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"None of main or o1-o7 is addressed. main is a survivor-gap lower bound, not this object. Nearest misses: - o5 (the quoted general sieve input). #6054 quotes Bombieri-Vinogradov for mu (Granville-Shao 2018, Theorem 1.1(b) with Theorem 2.1) as a published input; it is a level-of-distribution theorem for mu, not a sieve input in the paper's sense, and it is quoted, not proved; - o6 (two-sided density and Euler estimates). Lemma 2.1 computes limits of Mobius log-moments through an Euler factor at 2, which is a Dirichlet-series identity, not a density estimate for sieve survivors; - o1 (prime harmonic asymptotic). The moments in Lemma 2.1 are Mobius sums, not sums over primes, and the source gives no constant for sum 1/p. o2-o4 and o7 have no counterpart.","assumptions_md":"Provisional relevance only. #6054 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6054 (by @maxime-fleury) is an adversarial audit of Theorem 6.2 of #6048 (now rejected). It confirms three defects in the printed proof, each with a counterexample: (1) for even d, (k,d)=1 does not give a reduced class mod d (331 failures for d,k <= 40, all with d even; d=6, k=5 gives class 4); Lemma 1.2 repairs this with m=2u and a unit mod d/2; (2) the two log-weighted Mobius moments do not tend to 0: their limits are -1, and -2/+1 over odd/even d (Lemma 2.1), and the surviving coefficient is +1 (s2.2); (3) Region II does not factor into m-independent sums, because the condition (d,m)|2 is not separable (factorised main -3,140 against the correct +134,504); it is replaced by a gcd divisor-sum form (3.2)-(3.4) and a uniformity Lemma 3.2. The report claims Theorem 6.2 (the d <= X^{1/4-eps} part of each mixed piece is O(X/L^A)) is then proved modulo Bombieri-Vinogradov for mu (Granville-Shao 2018, Thm 1.1(b) with Thm 2.1), with the projection and supremum clauses discharged at a log cost (s4.2). It says the theorem is not new mathematics, that the exponent 1/4 is unchanged, and that nothing here helps with twin primes (s7); the lower bound (7.1) stays open. Accepted at rung measured, verification spot (review 2079, receipt 111). The report's 'proved' for Theorem 6.2 is therefore not an accepted proven rung. Defects a successor must inherit (review 2079, recorded as its open findings 163275, 163276 and 163277, all pinned in this snapshot): (a) [finding 163275, before_circulation] (3.3) drops the indicator 1_{(d,m)|2} that (3.1) carries; the repair is the divisor sum with g(r) = mu(r_odd), valid because d is squarefree, and Lemma 3.2 is unchanged; G5 checks (3.1), not (3.3); (b) [finding 163276, before_circulation] the s4.4 L^2 remark misstates Barban-Davenport-Halberstam (the bound is yQ log y + y^2 L^{-A}) and would give about X^{9/8}, so the claim that the theorem does not depend on which standard form is granted is unsupported; the s4.3 proof does not use it; (c) [finding 163277, advisory] Lemma 1.2(iv): a class mod d/2 lifts to two classes mod d, of one parity when d/2 is even (d=8, k=2 gives m in {1,5} mod 8; the density 2/d is right); (d) [also finding 163277] the 'page-level locator' gives theorem numbers only, and Lemma 3.2 is a sketch of a classical estimate. The checker tpc-g-checks.py and its pinned output are relied on only as far as receipt 111 and review 2079's spot check (G1, G2, G4) go.","transfer_eligible":false,"provisional_relevance":true}],"targets":[{"id":"g2-exponent","lane":"g2-exponent","title":"Lower the G₂ upper-bound exponent (4.27 toward 2)","domain_md":"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"adversarial","lane":"adversarial","title":"Break accepted lemmas: counterexample search against validators","domain_md":"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"formalize","lane":"formalize","title":"Lean 4 formalization of accepted lemmas","domain_md":"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"measure","lane":"measure","title":"Extend numbered measurement scripts to larger ranges","domain_md":"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"infinitude","lane":"infinitude","title":"Routes toward infinitude not in the refuted registry","domain_md":"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"finiteness-structure","lane":"finiteness-structure","title":"What structure a finite twin count would force (disproof-shaped lane)","domain_md":"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"kk-lower-bound","paper":"kk-lower-bound","title":"Mapped paper claims and separate obligations","domain_md":"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.","topic_ids":["kk-lower-bound.main","kk-lower-bound.o1","kk-lower-bound.o2","kk-lower-bound.o3","kk-lower-bound.o4","kk-lower-bound.o5","kk-lower-bound.o6","kk-lower-bound.o7"]}],"reviewer":{"model":"claude-opus-5-5","effort":"high","handle":"Benjaminsen","trusted":true,"user_id":1,"authority_basis":"project_grant","same_contributor":false},"effective":true,"check_job_id":12248,"input_sha256":"a35601bb67c9b629f732f29bcdcfb02bf2d307a3a7796eed1beb5538af29b719","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"4e195f94ad06b00370ffcdbc634e1ab5ed69e7f96bb9407177cdab5a5d5a61d2","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6054,"material_evidence":{"sha256":"0d419cbadcfc3f91fd0c8728d11b3fc8df1701e991688a0be1e71ef134f02d19","entries":{"return:6054":"ae5821eb5330046251b554c6887225eb9bd56fbe4a52b25a5f6956a41525ca87","finding:163275":"5e167fb3e78960cebad79ffba3b3c744cbf2217641ddf2025f1ab9e79fd51fa4","finding:163276":"5f1ea951b007518f34b4b994dd2a35edda479fe7aeeca4d2322bfcc97824bf16","finding:163277":"f14a5310e0a4f14e9bbd88ddb41b4c08e6e52cd1cd53c51fdba053e366295406","review:6054:1:claude-opus-5-5":"139c822b5208b5be51f705d732ada01cc8fca2b4874026f84602d987939b000d","artifact:6054:4e195f94ad06b00370ffcdbc634e1ab5ed69e7f96bb9407177cdab5a5d5a61d2":"4e195f94ad06b00370ffcdbc634e1ab5ed69e7f96bb9407177cdab5a5d5a61d2","artifact:6054:57b2fb3028040d451ad58bb1a62bbe5b9960de865c6045698389489063fd90b6":"57b2fb3028040d451ad58bb1a62bbe5b9960de865c6045698389489063fd90b6","artifact:6054:79c09339fb04638c466b0550207b8233aae61473d664b51a42b99a4470927d45":"79c09339fb04638c466b0550207b8233aae61473d664b51a42b99a4470927d45"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6054,"consideration_only":true,"scientific_authority":false,"provisional_relevance":true,"base_assessment_return_id":null},"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Bounded applicability screening of accepted scientific return #6054: provisional consideration before validated transfer. This is relevance judgment, not another acceptance of the source or endorsement of unvalidated scopes. Do not run experiments, reproduce a computation or re-prove the source. No prior screening. Read the source and its current scope assessments once.\n\nSource snapshot: {\"version\":\"research-applicability-v1\",\"consideration_only\":true,\"source_return_id\":6054,\"report_sha256\":\"4e195f94ad06b00370ffcdbc634e1ab5ed69e7f96bb9407177cdab5a5d5a61d2\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6054\":\"ae5821eb5330046251b554c6887225eb9bd56fbe4a52b25a5f6956a41525ca87\",\"review:6054:1:claude-opus-5-5\":\"139c822b5208b5be51f705d732ada01cc8fca2b4874026f84602d987939b000d\",\"finding:163275\":\"5e167fb3e78960cebad79ffba3b3c744cbf2217641ddf2025f1ab9e79fd51fa4\",\"finding:163276\":\"5f1ea951b007518f34b4b994dd2a35edda479fe7aeeca4d2322bfcc97824bf16\",\"finding:163277\":\"f14a5310e0a4f14e9bbd88ddb41b4c08e6e52cd1cd53c51fdba053e366295406\",\"artifact:6054:4e195f94ad06b00370ffcdbc634e1ab5ed69e7f96bb9407177cdab5a5d5a61d2\":\"4e195f94ad06b00370ffcdbc634e1ab5ed69e7f96bb9407177cdab5a5d5a61d2\",\"artifact:6054:57b2fb3028040d451ad58bb1a62bbe5b9960de865c6045698389489063fd90b6\":\"57b2fb3028040d451ad58bb1a62bbe5b9960de865c6045698389489063fd90b6\",\"artifact:6054:79c09339fb04638c466b0550207b8233aae61473d664b51a42b99a4470927d45\":\"79c09339fb04638c466b0550207b8233aae61473d664b51a42b99a4470927d45\"},\"sha256\":\"0d419cbadcfc3f91fd0c8728d11b3fc8df1701e991688a0be1e71ef134f02d19\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: []\nConfigured areas: [{\"id\":\"g2-exponent\",\"title\":\"Lower the G₂ upper-bound exponent (4.27 toward 2)\",\"lane\":\"g2-exponent\",\"topic_ids\":[],\"domain_md\":\"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"adversarial\",\"title\":\"Break accepted lemmas: counterexample search against validators\",\"lane\":\"adversarial\",\"topic_ids\":[],\"domain_md\":\"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"formalize\",\"title\":\"Lean 4 formalization of accepted lemmas\",\"lane\":\"formalize\",\"topic_ids\":[],\"domain_md\":\"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"measure\",\"title\":\"Extend numbered measurement scripts to larger ranges\",\"lane\":\"measure\",\"topic_ids\":[],\"domain_md\":\"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"infinitude\",\"title\":\"Routes toward infinitude not in the refuted registry\",\"lane\":\"infinitude\",\"topic_ids\":[],\"domain_md\":\"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"finiteness-structure\",\"title\":\"What structure a finite twin count would force (disproof-shaped lane)\",\"lane\":\"finiteness-structure\",\"topic_ids\":[],\"domain_md\":\"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"kk-lower-bound\",\"title\":\"Mapped paper claims and separate obligations\",\"paper\":\"kk-lower-bound\",\"topic_ids\":[\"kk-lower-bound.main\",\"kk-lower-bound.o1\",\"kk-lower-bound.o2\",\"kk-lower-bound.o3\",\"kk-lower-bound.o4\",\"kk-lower-bound.o5\",\"kk-lower-bound.o6\",\"kk-lower-bound.o7\"],\"domain_md\":\"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.\"}]\n\nFor each area return applicability_assessment:{source_return_id:6054,input_sha256:\"a35601bb67c9b629f732f29bcdcfb02bf2d307a3a7796eed1beb5538af29b719\",entries:[{target_id,decision:\"applicable|inapplicable|unresolved\",scope_keys:[<declared source keys; [] when the accepted report has no typed scope>],evidence_md,assumptions_md}]}. Cite source evidence and explain the possible bridge or mismatch with each area's exact domain. Missing scope authority is an unresolved validation condition, not a demand for another full review or experiment. Applicability remains provisional unless every referenced scope satisfies the existing exact scientific authority rules; only validated entries can be transferred automatically. Preserve the claim's original scope; an extension is unresolved until ordinary scientific review. An unresolved entry is a recorded gap, not an instruction for a new experiment. Submit no new scientific claims, artifacts, review request or route changes in this administrative screening. Record existing evidence even when every area is inapplicable. GET <project>/research-applicability is the live index; OUTCOMES document integration is separate.","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":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6091/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"6a58e52e2b48b9dea9da07d93c74ccd756c633cbaff5abfb5bbce85e7fc204cd","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}