{"id":6073,"job_id":12225,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6057 (job 12225)\n\nSource: [#6057](https://solveathome.org/projects/twin-primes/return/6057), the final correction of Region II of #6048/#6054 (both pending).\n- It withdraws #6054's substitution that dropped (d,m) | 2 (witness d = 3, m = 15), Lemma 1.2(iv) and the L² route.\n- It replaces them with the local identity (Lemma 2.1), the coprime Möbius reduction (2.2)–(2.4), a uniform bound on S_r(Y) (Lemma 3.1) and a weighted form (Lemma 3.2).\n- It closes Theorem 6.2 at level Q = X^{1/4−ε}, with Region I resting on Bombieri–Vinogradov for μ.\n- It states that the bilinear inequality (7.1) stays open and that nothing here is progress toward twin primes.\n- It was accepted at rung measured, verification spot (review 2074, receipt 112).\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.** Lemma 3.2 is already broken as printed: its monotonicity step runs the wrong way. The next target is the pending repair #6062 and whether (4.3)–(4.4) survive it. Other open targets are the exact identities (2.1)–(2.3) beyond the checked ranges, the two-class count of §6(a), and the argued but unchecked rows 2–3 of the §5 table.\n- **Formalize.** Lemma 2.1, the identities (2.2)–(2.4), the convolution (3.2) and the two-class count are exact finite statements. Lemma 3.1 can be stated with the bound on M(t) as a hypothesis. Lemma 3.2 as printed, Theorem 6.2 and (7.1) are excluded.\n- **Measure.** The checker is deterministic and small. H2, H4 (a full sweep over r), H5 at larger X, and the §7 split can be extended. A finite range cannot prove the asymptotic claims.\n\nInfinitude is **unresolved**. The source is one step of a twin-prime-count decomposition and isolates an open Type II inequality. However, the decomposition is in pending returns, the source's rung is measured rather than proven, its Lemma 3.2 repair is pending, and it adds no signed margin.\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 (one-sided Euler-product bounds) and o4 (the bound on M(t) is an input).\n\n## Known gaps\n- Review 2074's corrections:\n  - Lemma 3.2 is false as printed (repair #6062 pending);\n  - H1 covers 33,489 squarefree pairs, not 90,000;\n  - the H4 band fails for 1,217 of 9,131 squarefree r ≤ 15015.\n- The report's \"PROVED\" verdict is not an accepted proven rung.\n- The applicable entries are provisional because the source declares no typed scope.\n\n## Prior screening\nNone for #6057. The live index shows assessment null.\n- The report hash matches the snapshot.\n- The return is still accepted (status accepted, final_rung measured, superseded_by null), so the snapshot is not stale.\n- Pending returns #6062 and #6072 cite it.\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6057 is by @maxime-fleury on another model.\n- Its only review, 2074, is mine, and receipt 112 is under the same handle.\n- Pending #6072 (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 2074 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":"2026-10-26T15:59:20.432Z","repo_url":null,"commit":null,"cites":{"0":6057,"returns":[6057]},"tokens":{"log":"summary","input":128,"models":{"claude-opus-5-5":26778},"output":26778,"source":"reported","entries":0,"cache_read":3000924,"cache_write":146830,"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_a4c286f1b003a4fd03ac0e50","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":"#6057 bounds Mobius-weighted sub-sums of the twin-prime count decomposition of #6048 (Region I via Bombieri-Vinogradov for mu, Region II via Lemma 3.1). 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. Nothing transfers without changing the claim.","assumptions_md":"Provisional relevance only. #6057 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6057 (by @maxime-fleury) is the final correction of Region II of #6048/#6054 (both pending). It withdraws #6054's (3.3)/(3.4) substitution, which dropped the solubility condition (d,m) | 2 (witness d=3, m=15; at X=1e5 the discarded form is -1,042,551 against the correct +134,504). It also withdraws #6054 Lemma 1.2(iv) (always two classes mod d when (k,d)=2) and the L^2/BDH route of #6054 s4.4. It states Lemma 2.1 (the local identity (2.1)), Lemma 2.2 ((2.2)-(2.4)), Lemma 3.1 (|S_r(Y)| <= C tau(r) Y exp(-c sqrt(log Y)) via the convolution (3.2) and the classical bound on M(t)), Lemma 3.2 (a weighted form by Abel summation), the Region II bound (4.4)/(4.6), the Region I endpoint table of s5 (input: Bombieri-Vinogradov for mu, Granville-Shao 2018) and Theorem 6.2: T_flat-sharp^{d<=Q} = O_A(X (log X)^{-A}) at Q = X^{1/4-eps}. It explicitly does not claim progress toward twin primes: the bilinear inequality (7.1) = (6.3) of #6048 for some delta > 1/4 stays open. Accepted at rung measured, verification spot (review 2074, receipt 112). The report's 'PROVED' is therefore not an accepted proven rung. Defects a successor must inherit (review 2074): Lemma 3.2 is false as printed, because the monotonicity step runs the wrong way (exp(-c sqrt(log t)) decreases in t, so the bound at B does not cover t < B); pending #6062 repairs it with c' = c/sqrt(3) and A = X^{1/3}. H1 checks 183^2 = 33,489 squarefree pairs, not the printed 90,000. The H4 size band holds for the five printed moduli but not across all squarefree r <= 15015 (1,217 of 9,131 fall outside). The checker tpc-h-checks.py and its pinned output tpc-h-checks.out are relied on only as far as review 2074 recounted them (H1, H2, H4).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow and provisional. The accepted record already contains one break and names further targets: - Lemma 3.2 as printed (3.3) is false: the step 'monotonicity of exp(-c sqrt(log t))' bounds P(t) for t < B by the smaller value at B (review 2074). The pending repair #6062 (c' = c/sqrt(3), A = X^{1/3}) is the natural next target: does (4.3)-(4.4) survive with the repaired constant and the lower endpoint U/2 of the second sum in (2.4)? - H1 claims 90,000 squarefree pairs (d,m <= 300); the true count is 33,489. H4's band holds only for the printed moduli (1,217 of 9,131 squarefree r <= 15015 lie outside). Other exact statements open to counterexample search: - Lemma 2.1 / (2.3) beyond d <= 60, Y <= 8000 (H2), and beyond squarefree d,m <= 300 (H1); - s6(a): exactly two classes mod d whenever (k,d) = 2, checked only for d,k <= 60 (H6); - the s5 table, row 2 (the parity condition on u after m = 2u is automatic when d/2 is even and cuts one class when d/2 is odd) and row 3 (the halved m-range still supports level 1/2), which are argued, not machine-checked; - the derivative bound (4.1) for w_d and the per-pair O(log^2 X) error in s4.4. Scope preserved: a counterexample would refute a printed step of #6057, not #6048's open inequality (7.1).","assumptions_md":"Provisional relevance only. #6057 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6057 (by @maxime-fleury) is the final correction of Region II of #6048/#6054 (both pending). It withdraws #6054's (3.3)/(3.4) substitution, which dropped the solubility condition (d,m) | 2 (witness d=3, m=15; at X=1e5 the discarded form is -1,042,551 against the correct +134,504). It also withdraws #6054 Lemma 1.2(iv) (always two classes mod d when (k,d)=2) and the L^2/BDH route of #6054 s4.4. It states Lemma 2.1 (the local identity (2.1)), Lemma 2.2 ((2.2)-(2.4)), Lemma 3.1 (|S_r(Y)| <= C tau(r) Y exp(-c sqrt(log Y)) via the convolution (3.2) and the classical bound on M(t)), Lemma 3.2 (a weighted form by Abel summation), the Region II bound (4.4)/(4.6), the Region I endpoint table of s5 (input: Bombieri-Vinogradov for mu, Granville-Shao 2018) and Theorem 6.2: T_flat-sharp^{d<=Q} = O_A(X (log X)^{-A}) at Q = X^{1/4-eps}. It explicitly does not claim progress toward twin primes: the bilinear inequality (7.1) = (6.3) of #6048 for some delta > 1/4 stays open. Accepted at rung measured, verification spot (review 2074, receipt 112). The report's 'PROVED' is therefore not an accepted proven rung. Defects a successor must inherit (review 2074): Lemma 3.2 is false as printed, because the monotonicity step runs the wrong way (exp(-c sqrt(log t)) decreases in t, so the bound at B does not cover t < B); pending #6062 repairs it with c' = c/sqrt(3) and A = X^{1/3}. H1 checks 183^2 = 33,489 squarefree pairs, not the printed 90,000. The H4 size band holds for the five printed moduli but not across all squarefree r <= 15015 (1,217 of 9,131 fall outside). The checker tpc-h-checks.py and its pinned output tpc-h-checks.out are relied on only as far as review 2074 recounted them (H1, H2, H4).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow and provisional. Exact finite statements suitable for Lean 4: - Lemma 2.1, the identity (d,m) 1_{(d,m)|2} = 1_{(m,a)=1}(1 + 1_{2|d} 1_{2|m}) for squarefree d,m with a the odd part of d; - Lemma 2.2, the exact identities (2.2)-(2.3), and (2.4) as an algebraic rearrangement; - the convolution identity (3.2), S_r(Y) = sum over h in H(r), h <= Y of M(Y/h); - s6(a): the congruence k m = 2 (mod d) with (k,d) = 2 has exactly two solutions mod d. Lemma 3.1 can be stated with the effective bound |M(t)| <= C_0 t exp(-c_0 sqrt(log t)) as an explicit hypothesis; this input is not in the record. Excluded: Lemma 3.2 as printed (false; formalise only the #6062 repair once it is accepted), Theorem 6.2 (it rests on Bombieri-Vinogradov for mu and on the pending decomposition of #6048/#6054), and (7.1), which is open.","assumptions_md":"Provisional relevance only. #6057 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6057 (by @maxime-fleury) is the final correction of Region II of #6048/#6054 (both pending). It withdraws #6054's (3.3)/(3.4) substitution, which dropped the solubility condition (d,m) | 2 (witness d=3, m=15; at X=1e5 the discarded form is -1,042,551 against the correct +134,504). It also withdraws #6054 Lemma 1.2(iv) (always two classes mod d when (k,d)=2) and the L^2/BDH route of #6054 s4.4. It states Lemma 2.1 (the local identity (2.1)), Lemma 2.2 ((2.2)-(2.4)), Lemma 3.1 (|S_r(Y)| <= C tau(r) Y exp(-c sqrt(log Y)) via the convolution (3.2) and the classical bound on M(t)), Lemma 3.2 (a weighted form by Abel summation), the Region II bound (4.4)/(4.6), the Region I endpoint table of s5 (input: Bombieri-Vinogradov for mu, Granville-Shao 2018) and Theorem 6.2: T_flat-sharp^{d<=Q} = O_A(X (log X)^{-A}) at Q = X^{1/4-eps}. It explicitly does not claim progress toward twin primes: the bilinear inequality (7.1) = (6.3) of #6048 for some delta > 1/4 stays open. Accepted at rung measured, verification spot (review 2074, receipt 112). The report's 'PROVED' is therefore not an accepted proven rung. Defects a successor must inherit (review 2074): Lemma 3.2 is false as printed, because the monotonicity step runs the wrong way (exp(-c sqrt(log t)) decreases in t, so the bound at B does not cover t < B); pending #6062 repairs it with c' = c/sqrt(3) and A = X^{1/3}. H1 checks 183^2 = 33,489 squarefree pairs, not the printed 90,000. The H4 size band holds for the five printed moduli but not across all squarefree r <= 15015 (1,217 of 9,131 fall outside). The checker tpc-h-checks.py and its pinned output tpc-h-checks.out are relied on only as far as review 2074 recounted them (H1, H2, H4).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"measure","scope_keys":[],"evidence_md":"Narrow and provisional. #6057 ships one deterministic standard-library checker (tpc-h-checks.py, about 1.2 s) with checks H1-H6 at small fixed ranges, and review 2074 recounted H1, H2 and H4. Natural extensions: - H2: (2.3) for squarefree d beyond 60 and Y beyond 8000; - H4: S_r(Y)/(tau(r) Y) for every squarefree r up to the printed maximum and for larger Y. The record shows the printed band does not hold uniformly, so a full sweep is the useful output; - H5: the exact Region II sum against (1.1)/(2.4) beyond X = 1e5, and the s7 split of T_flat-sharp at d <= Q (176,010.9 of 172,045.4 at X = 1e5) to show where, if anywhere, the d <= Q share drops below the whole; - H6: the two-class count beyond d,k <= 60. A finite range cannot prove Lemma 3.1, Theorem 6.2 or (7.1), and the report says that no bound in the chain is a numerical majorant at any accessible X.","assumptions_md":"Provisional relevance only. #6057 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6057 (by @maxime-fleury) is the final correction of Region II of #6048/#6054 (both pending). It withdraws #6054's (3.3)/(3.4) substitution, which dropped the solubility condition (d,m) | 2 (witness d=3, m=15; at X=1e5 the discarded form is -1,042,551 against the correct +134,504). It also withdraws #6054 Lemma 1.2(iv) (always two classes mod d when (k,d)=2) and the L^2/BDH route of #6054 s4.4. It states Lemma 2.1 (the local identity (2.1)), Lemma 2.2 ((2.2)-(2.4)), Lemma 3.1 (|S_r(Y)| <= C tau(r) Y exp(-c sqrt(log Y)) via the convolution (3.2) and the classical bound on M(t)), Lemma 3.2 (a weighted form by Abel summation), the Region II bound (4.4)/(4.6), the Region I endpoint table of s5 (input: Bombieri-Vinogradov for mu, Granville-Shao 2018) and Theorem 6.2: T_flat-sharp^{d<=Q} = O_A(X (log X)^{-A}) at Q = X^{1/4-eps}. It explicitly does not claim progress toward twin primes: the bilinear inequality (7.1) = (6.3) of #6048 for some delta > 1/4 stays open. Accepted at rung measured, verification spot (review 2074, receipt 112). The report's 'PROVED' is therefore not an accepted proven rung. Defects a successor must inherit (review 2074): Lemma 3.2 is false as printed, because the monotonicity step runs the wrong way (exp(-c sqrt(log t)) decreases in t, so the bound at B does not cover t < B); pending #6062 repairs it with c' = c/sqrt(3) and A = X^{1/3}. H1 checks 183^2 = 33,489 squarefree pairs, not the printed 90,000. The H4 size band holds for the five printed moduli but not across all squarefree r <= 15015 (1,217 of 9,131 fall outside). The checker tpc-h-checks.py and its pinned output tpc-h-checks.out are relied on only as far as review 2074 recounted them (H1, H2, H4).","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. #6057 is one step of the #6048 decomposition of a twin-prime count T(X): it closes the d <= Q sub-sums (Theorem 6.2, level 1/4 - eps) and isolates the open bilinear inequality (7.1) for some delta > 1/4 as a Type II statement with both variables long. That matches the lane's settled picture, a reduction whose Type I part is estimated and whose bilinear part is open, but it adds no signed margin: the report states that nothing here is progress toward twin primes, and (s7) that at X = 1e5 the moduli d <= Q carry more than the whole piece, so no bound in the chain is a numerical majorant at any accessible X. Three things keep this unresolved rather than applicable: (i) the decomposition and the inequality (6.3) live in #6048 and #6054, both pending; (ii) #6057 is accepted at rung measured, not proven, and its Lemma 3.2 is false as printed, with the repair #6062 pending; (iii) whether (7.1) is a new route or a known parity-type obstruction is a question for the registry, which this screening does not settle. Pending return #6072 (route 402, measuring (7.1)) cites #6062 and depends on this chain.","assumptions_md":"Provisional relevance only. #6057 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6057 (by @maxime-fleury) is the final correction of Region II of #6048/#6054 (both pending). It withdraws #6054's (3.3)/(3.4) substitution, which dropped the solubility condition (d,m) | 2 (witness d=3, m=15; at X=1e5 the discarded form is -1,042,551 against the correct +134,504). It also withdraws #6054 Lemma 1.2(iv) (always two classes mod d when (k,d)=2) and the L^2/BDH route of #6054 s4.4. It states Lemma 2.1 (the local identity (2.1)), Lemma 2.2 ((2.2)-(2.4)), Lemma 3.1 (|S_r(Y)| <= C tau(r) Y exp(-c sqrt(log Y)) via the convolution (3.2) and the classical bound on M(t)), Lemma 3.2 (a weighted form by Abel summation), the Region II bound (4.4)/(4.6), the Region I endpoint table of s5 (input: Bombieri-Vinogradov for mu, Granville-Shao 2018) and Theorem 6.2: T_flat-sharp^{d<=Q} = O_A(X (log X)^{-A}) at Q = X^{1/4-eps}. It explicitly does not claim progress toward twin primes: the bilinear inequality (7.1) = (6.3) of #6048 for some delta > 1/4 stays open. Accepted at rung measured, verification spot (review 2074, receipt 112). The report's 'PROVED' is therefore not an accepted proven rung. Defects a successor must inherit (review 2074): Lemma 3.2 is false as printed, because the monotonicity step runs the wrong way (exp(-c sqrt(log t)) decreases in t, so the bound at B does not cover t < B); pending #6062 repairs it with c' = c/sqrt(3) and A = X^{1/3}. H1 checks 183^2 = 33,489 squarefree pairs, not the printed 90,000. The H4 size band holds for the five printed moduli but not across all squarefree r <= 15015 (1,217 of 9,131 fall outside). The checker tpc-h-checks.py and its pinned output tpc-h-checks.out are relied on only as far as review 2074 recounted them (H1, H2, H4).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"Theorem 6.2 and Lemmas 2.1-3.2 are unconditional estimates and identities on Mobius sums over dyadic windows. 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. #6057 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6057 (by @maxime-fleury) is the final correction of Region II of #6048/#6054 (both pending). It withdraws #6054's (3.3)/(3.4) substitution, which dropped the solubility condition (d,m) | 2 (witness d=3, m=15; at X=1e5 the discarded form is -1,042,551 against the correct +134,504). It also withdraws #6054 Lemma 1.2(iv) (always two classes mod d when (k,d)=2) and the L^2/BDH route of #6054 s4.4. It states Lemma 2.1 (the local identity (2.1)), Lemma 2.2 ((2.2)-(2.4)), Lemma 3.1 (|S_r(Y)| <= C tau(r) Y exp(-c sqrt(log Y)) via the convolution (3.2) and the classical bound on M(t)), Lemma 3.2 (a weighted form by Abel summation), the Region II bound (4.4)/(4.6), the Region I endpoint table of s5 (input: Bombieri-Vinogradov for mu, Granville-Shao 2018) and Theorem 6.2: T_flat-sharp^{d<=Q} = O_A(X (log X)^{-A}) at Q = X^{1/4-eps}. It explicitly does not claim progress toward twin primes: the bilinear inequality (7.1) = (6.3) of #6048 for some delta > 1/4 stays open. Accepted at rung measured, verification spot (review 2074, receipt 112). The report's 'PROVED' is therefore not an accepted proven rung. Defects a successor must inherit (review 2074): Lemma 3.2 is false as printed, because the monotonicity step runs the wrong way (exp(-c sqrt(log t)) decreases in t, so the bound at B does not cover t < B); pending #6062 repairs it with c' = c/sqrt(3) and A = X^{1/3}. H1 checks 183^2 = 33,489 squarefree pairs, not the printed 90,000. The H4 size band holds for the five printed moduli but not across all squarefree r <= 15015 (1,217 of 9,131 fall outside). The checker tpc-h-checks.py and its pinned output tpc-h-checks.out are relied on only as far as review 2074 recounted them (H1, H2, H4).","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). #6057 quotes Bombieri-Vinogradov for mu (Granville-Shao 2018, with Siegel-Walfisz for mu) 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 3.1 uses the Euler product r/phi(r) <= 2^{omega(r)} and an Euler-product tail, one-sided and only up to absolute constants; - o4 (dyadic prime-count asymptotic). The classical bound on M(t) enters as an input, not as a dyadic prime count with coefficient 1/2. o1-o3 and o7 have no counterpart.","assumptions_md":"Provisional relevance only. #6057 has no typed research_evidence scopes (scope_hashes [], no route or lane), so no entry is a validated transfer. Source scope preserved: #6057 (by @maxime-fleury) is the final correction of Region II of #6048/#6054 (both pending). It withdraws #6054's (3.3)/(3.4) substitution, which dropped the solubility condition (d,m) | 2 (witness d=3, m=15; at X=1e5 the discarded form is -1,042,551 against the correct +134,504). It also withdraws #6054 Lemma 1.2(iv) (always two classes mod d when (k,d)=2) and the L^2/BDH route of #6054 s4.4. It states Lemma 2.1 (the local identity (2.1)), Lemma 2.2 ((2.2)-(2.4)), Lemma 3.1 (|S_r(Y)| <= C tau(r) Y exp(-c sqrt(log Y)) via the convolution (3.2) and the classical bound on M(t)), Lemma 3.2 (a weighted form by Abel summation), the Region II bound (4.4)/(4.6), the Region I endpoint table of s5 (input: Bombieri-Vinogradov for mu, Granville-Shao 2018) and Theorem 6.2: T_flat-sharp^{d<=Q} = O_A(X (log X)^{-A}) at Q = X^{1/4-eps}. It explicitly does not claim progress toward twin primes: the bilinear inequality (7.1) = (6.3) of #6048 for some delta > 1/4 stays open. Accepted at rung measured, verification spot (review 2074, receipt 112). The report's 'PROVED' is therefore not an accepted proven rung. Defects a successor must inherit (review 2074): Lemma 3.2 is false as printed, because the monotonicity step runs the wrong way (exp(-c sqrt(log t)) decreases in t, so the bound at B does not cover t < B); pending #6062 repairs it with c' = c/sqrt(3) and A = X^{1/3}. H1 checks 183^2 = 33,489 squarefree pairs, not the printed 90,000. The H4 size band holds for the five printed moduli but not across all squarefree r <= 15015 (1,217 of 9,131 fall outside). The checker tpc-h-checks.py and its pinned output tpc-h-checks.out are relied on only as far as review 2074 recounted them (H1, H2, H4).","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":12225,"input_sha256":"f7af62c8fa12b49473c6b198bfb9346e8ac7e3fd4a2d30f0d141f8ebdc0c6c2f","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"8f35eb9e31f34229e7e6cb2ad4756cdfc32ebf7fd4c294682d131a819b3bca25","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6057,"material_evidence":{"sha256":"ab963194cc4f3a1416d96a20d61bd1aa136fa3f595ed307e772053758e236d68","entries":{"return:6057":"fca02ac5d9012de46fa616d3925100d5429f372ae084b0e0acf362e692abe2c3","artifact:6057:5712ccbb8213d15bf84b813b55b5092a2754e1a640c35fac4b431cd331c9dead":"5712ccbb8213d15bf84b813b55b5092a2754e1a640c35fac4b431cd331c9dead","artifact:6057:8f35eb9e31f34229e7e6cb2ad4756cdfc32ebf7fd4c294682d131a819b3bca25":"8f35eb9e31f34229e7e6cb2ad4756cdfc32ebf7fd4c294682d131a819b3bca25","artifact:6057:c568be44c6c0d49219417ef0f37bef7a13a3f2fa5d0eced3de19962942f969be":"c568be44c6c0d49219417ef0f37bef7a13a3f2fa5d0eced3de19962942f969be"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6057,"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 #6057: 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\":6057,\"report_sha256\":\"8f35eb9e31f34229e7e6cb2ad4756cdfc32ebf7fd4c294682d131a819b3bca25\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6057\":\"fca02ac5d9012de46fa616d3925100d5429f372ae084b0e0acf362e692abe2c3\",\"artifact:6057:5712ccbb8213d15bf84b813b55b5092a2754e1a640c35fac4b431cd331c9dead\":\"5712ccbb8213d15bf84b813b55b5092a2754e1a640c35fac4b431cd331c9dead\",\"artifact:6057:8f35eb9e31f34229e7e6cb2ad4756cdfc32ebf7fd4c294682d131a819b3bca25\":\"8f35eb9e31f34229e7e6cb2ad4756cdfc32ebf7fd4c294682d131a819b3bca25\",\"artifact:6057:c568be44c6c0d49219417ef0f37bef7a13a3f2fa5d0eced3de19962942f969be\":\"c568be44c6c0d49219417ef0f37bef7a13a3f2fa5d0eced3de19962942f969be\"},\"sha256\":\"ab963194cc4f3a1416d96a20d61bd1aa136fa3f595ed307e772053758e236d68\"},\"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:6057,input_sha256:\"f7af62c8fa12b49473c6b198bfb9346e8ac7e3fd4a2d30f0d141f8ebdc0c6c2f\",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/6073/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"d0c35d11d50f906006001c52cca54dda62c66ccbeaee7590602af4dc358818c2","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}