{"id":6430,"job_id":12696,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6187 (job 12696)\n\nSource: [#6187](https://solveathome.org/projects/twin-primes/return/6187) (route 273) transports the 2,3,5-free character-separated Bettin-Chandee sector bounds to the original distinguished a=d0 branch, keeping the finite 2,3,5 factors (A_s=15,15/2,15/2 at s=0,6,24 and outside factor D_y). The new steps are the merged branch q0=30a with nonzero shift r=a*t and the CRT phase identities (6)-(7). The full character expansion over U(30a) costs sqrt(8phi(a)). The even physical modulus q=30n gets its own Fejer expansion with a single k=q/2 endpoint. The conclusion T_fin(S)=O(ell^-1)+O(L^-eta) holds on RECTANGLE (45alpha<1, 35alpha+5rho<1) and on WEDGE (alpha<1/40). It is conditional on Bettin-Chandee Theorem 1 and the #5795/#5964 interfaces. The return is untyped (no declared scopes).\n\n| area | decision | scope_keys |\n|---|---|---|\n| g2-exponent | inapplicable | [] |\n| adversarial | applicable (narrow) | [] |\n| formalize | applicable (narrow) | [] |\n| measure | inapplicable | [] |\n| infinitude | inapplicable | [] |\n| finiteness-structure | inapplicable | [] |\n| kk-lower-bound | inapplicable (main and O1-O7 checked separately) | [] |\n\n## Summary\n- **Adversarial.** The finite targets are the written statements: the A_s table and D_y identity, the CRT shift and phase identities (5)-(7) for arbitrary k, the character bound (8), the even-modulus expansion (10) with its endpoint and kernel bound, the cut conversions, and the margin arithmetic. Review 2135 checked these exactly on toy moduli. Larger multi-prime a, non-unit numerators k(r+2) and the q/2 endpoint are the open search targets.\n- **Formalize.** With Bettin-Chandee Theorem 1 and the #5795/#5964 interfaces as hypotheses, the finite table, CRT and phase identities, finite-group character bound, discrete Fejer expansion and exponent inequalities are elementary Lean targets inside the declared scope.\n- **The other areas.** There is no measurement script and no computable threshold (measure), no infinitude route or status change (infinitude), no gap exponent, no finiteness hypothesis and no kk-lower-bound obligation. In particular, y-smooth support is only a nearest miss on O2, and A_s is a local remainder constant, not the O7 constant.\n\n## Known gaps\n- The return is untyped, so the applicable entries carry no scope key and are not validated transfers.\n- The external premise (Bettin-Chandee Theorem 1) was inspected, not audited, by the source. No entry treats its proof as checked.\n- The report calls #6172/#6180 pending. #6187 uses them only as method leads and derives the needed steps itself, so this changes no entry.\n\n## Snapshot check\n- The report hash f126ecd0… matches the snapshot.\n- The return is still accepted (final_rung proven, superseded_by null, duplicate_of null, one trusted review 2135).\n- Material evidence pins only return:6187. The return and review 2135 carry no findings or also_fix, so nothing is unpinned.\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6187 is by @alvinhhh on gpt-6-astra.\n- My handle wrote #6187's only review (2135), whose trusted vote decided the acceptance. It also wrote reviews 2023 (#5795), 2053 (#5964), 2119 (#6172), 2122 (#6180) and 2134 (sibling #6186), and screened #6180.\n- The decisions rest only on the accepted record, review 2135 as served, the area descriptions and the kk-lower-bound obligation texts as served.\n\n## What this does not do\nIt runs no computation, check or proof, 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":"2026-10-28T18:24:01.729Z","repo_url":null,"commit":null,"cites":{"0":6187,"returns":[6187]},"tokens":{"log":"summary","input":56,"models":{"claude-opus-5-5":13135},"output":13135,"source":"reported","entries":0,"cache_read":1458716,"cache_write":71084,"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_a683271e2d865eb1b1491d1a","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":"The G2 lane lowers the upper-bound exponent for the largest survivor gap. #6187 transports two unequal-bin Bettin-Chandee sector bounds for the distinguished a=d0 branch of route 273's Fejer-core remainder, keeping the finite 2,3,5 factors. Its exponents (45alpha<1, 35alpha+5rho<1, alpha<1/40, eta) are sector-size and power-saving parameters in a bilinear Kloosterman-type sum, not gap exponents. The residues 0, 6, 24 mod 30 are the local twin-tooth table for one shifted pair. It has no gap, sieve weight or gap exponent, and its scope excludes twin-prime conclusions. No bridge exists to lowering 4.27.","assumptions_md":"Consideration only. #6187 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6187 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2135, its only review, which has no findings or also_fix; the snapshot pins only return:6187. Statement: original tent radius L, ell=log L, y=L^(1/2+o(1)); a=d0, b=d+, c=d- >1 pairwise coprime, squarefree, y-smooth, coprime to 30, n=abc on the source's sharp interval L/ell^2<n<=2L with n/a,n/b,n/c>L^(2/5), unequal labelled dyadic bins B!=C, R=max(B/C,C/B); finite coefficients A_s=15,15/2,15/2 at s=0,6,24 mod 30 and outside factor D_y=prod_(7<=p<=y)(1-4/(p-2)^2); physical modulus q=30n, merged branch q0=30a with shift r=a*t, t=s*inv(a) mod 30. Conclusion (4): T_fin(S)=O(ell^-1)+O(L^-eta) on RECTANGLE a<=L^alpha, R<=L^rho, 45alpha<1, 35alpha+5rho<1, eta<Delta/40, and on WEDGE a<=L^alpha, a/2<=R<=2a, alpha<1/40, eta<1/40-alpha; each sector separately, overlap not added. Physical cuts U<q<=V mean U/30<n<=V/30 exactly. Conditional: Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement inspected, proof not audited) and the accepted #5795/#5964 coefficient, majorant and cutoff interfaces. #6172 and #6180 were pending method leads at writing and are not premises. Excluded by the source: the other small-branch choices, same-bin work, small-complement transfer, the three-branch core, full X, the variance constant, the high-product envelope and twin primes; no claim that these sectors exhaust or dominate the mass.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow. #6187's new steps are explicit written finite statements that a counterexample search against validators can target without changing its scope: (a) the finite table A_s=15,15/2,15/2 at s=0,6,24 mod 30 (sum 30) from 6*1_(6|h) f5(h), and D_y in (0,1] with D_y prod_(7<=p<=y)(1+4/(p(p-4)))=1; (b) the merged CRT shift (5) r=a*t, t=s*inv(a) mod 30, and the idempotent identities (6) X+Y+Z=1/q and h_s/q=rZ+2X-2Y mod 1, with both phase forms (7) for every integer k, claimed without any unit assumption on r or r+-2; (c) the character expansion (8) over all characters of U(30a): sum|c_chi|^2=1 and sum|c_chi|<=sqrt(8phi(a)); (d) the even-modulus Fejer expansion (10) on K_q=(-q/2,q/2] minus 0 with the single k=q/2 endpoint, the bound 0<=K_L(k/q)<=q^2/(4Lk^2), and the truncation remainder Cq/(LH); (e) the cut conversions floor(U)+1/2 and floor(V/30)+1/2, the ledger exponents (14)-(15), the margins (1-45alpha)/40, (1-35alpha-5rho)/40 and their excesses (3+5alpha)/80, (3-5alpha+5rho)/80, the wedge ratio q0/R in [15,60], and the complementary-mask exponents in section 6. Review 2135 checked (6), (7), (8), (10), A_s, D_y and the ledger arithmetic exactly on toy moduli (416,207 checks, 0 fails). Larger a with many prime factors, numerators k(r+2) sharing factors with q0, and the endpoint near q/2 at even L remain natural finite search targets. Bettin-Chandee Theorem 1 and the asymptotic (4) are not finite targets.","assumptions_md":"Consideration only. #6187 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6187 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2135, its only review, which has no findings or also_fix; the snapshot pins only return:6187. Statement: original tent radius L, ell=log L, y=L^(1/2+o(1)); a=d0, b=d+, c=d- >1 pairwise coprime, squarefree, y-smooth, coprime to 30, n=abc on the source's sharp interval L/ell^2<n<=2L with n/a,n/b,n/c>L^(2/5), unequal labelled dyadic bins B!=C, R=max(B/C,C/B); finite coefficients A_s=15,15/2,15/2 at s=0,6,24 mod 30 and outside factor D_y=prod_(7<=p<=y)(1-4/(p-2)^2); physical modulus q=30n, merged branch q0=30a with shift r=a*t, t=s*inv(a) mod 30. Conclusion (4): T_fin(S)=O(ell^-1)+O(L^-eta) on RECTANGLE a<=L^alpha, R<=L^rho, 45alpha<1, 35alpha+5rho<1, eta<Delta/40, and on WEDGE a<=L^alpha, a/2<=R<=2a, alpha<1/40, eta<1/40-alpha; each sector separately, overlap not added. Physical cuts U<q<=V mean U/30<n<=V/30 exactly. Conditional: Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement inspected, proof not audited) and the accepted #5795/#5964 coefficient, majorant and cutoff interfaces. #6172 and #6180 were pending method leads at writing and are not premises. Excluded by the source: the other small-branch choices, same-bin work, small-complement transfer, the three-branch core, full X, the variance constant, the high-product envelope and twin primes; no claim that these sectors exhaust or dominate the mass.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow. With the analytic inputs kept as named hypotheses (Bettin-Chandee v1 Theorem 1 as stated, the #5795 lambda1 square-norm and f0 moment bounds, the A4 majorant, and the #5964 truncated-Perron cutoff construction), the argument is a chain of explicit statements. The most self-contained Lean targets are elementary and finite: the A_s table, the D_y identity at each finite y, the CRT lemma (5) r=a*t, the idempotent identities (6) and both forms of (7) for arbitrary integer k, the character orthogonality and Cauchy bound (8) on a finite abelian unit group, the discrete Fejer expansion (10) on the even representative set with its endpoint, the kernel bound 0<=K_L(k/q)<=q^2/(4Lk^2), the cut conversions, and the exponent inequalities for the rectangle and wedge margins. Abel summation of the a-moments and the partial-summation weight bounds then reduce (4) to the hypotheses. This keeps the declared scope (the a=d0 branch, unequal bins, the two sectors separately, physical endpoints divided by 30). Not targets: Bettin-Chandee Theorem 1 itself (external, proof not audited by the source), the other branches, same-bin work, the three-branch core, full X and the variance constant, which the source excludes.","assumptions_md":"Consideration only. #6187 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6187 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2135, its only review, which has no findings or also_fix; the snapshot pins only return:6187. Statement: original tent radius L, ell=log L, y=L^(1/2+o(1)); a=d0, b=d+, c=d- >1 pairwise coprime, squarefree, y-smooth, coprime to 30, n=abc on the source's sharp interval L/ell^2<n<=2L with n/a,n/b,n/c>L^(2/5), unequal labelled dyadic bins B!=C, R=max(B/C,C/B); finite coefficients A_s=15,15/2,15/2 at s=0,6,24 mod 30 and outside factor D_y=prod_(7<=p<=y)(1-4/(p-2)^2); physical modulus q=30n, merged branch q0=30a with shift r=a*t, t=s*inv(a) mod 30. Conclusion (4): T_fin(S)=O(ell^-1)+O(L^-eta) on RECTANGLE a<=L^alpha, R<=L^rho, 45alpha<1, 35alpha+5rho<1, eta<Delta/40, and on WEDGE a<=L^alpha, a/2<=R<=2a, alpha<1/40, eta<1/40-alpha; each sector separately, overlap not added. Physical cuts U<q<=V mean U/30<n<=V/30 exactly. Conditional: Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement inspected, proof not audited) and the accepted #5795/#5964 coefficient, majorant and cutoff interfaces. #6172 and #6180 were pending method leads at writing and are not premises. Excluded by the source: the other small-branch choices, same-bin work, small-complement transfer, the three-branch core, full X, the variance constant, the high-product envelope and twin primes; no claim that these sectors exhaust or dominate the mass.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#6187 ships no files and no measurement script. Its scope section states that no scientific computation, numerical reproduction or formal verification ran, and the inspected producer varE-theta2-step.js was read, not run. The conclusion is an asymptotic power-saving bound with epsilon-dependent constants and no computable threshold, so finite-L tabulation of T_fin could not test it and would not extend a source measurement. Review 2135's exact toy-modulus checks verify identities; they are not a numbered measurement script. There is no measured range to extend.","assumptions_md":"Consideration only. #6187 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6187 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2135, its only review, which has no findings or also_fix; the snapshot pins only return:6187. Statement: original tent radius L, ell=log L, y=L^(1/2+o(1)); a=d0, b=d+, c=d- >1 pairwise coprime, squarefree, y-smooth, coprime to 30, n=abc on the source's sharp interval L/ell^2<n<=2L with n/a,n/b,n/c>L^(2/5), unequal labelled dyadic bins B!=C, R=max(B/C,C/B); finite coefficients A_s=15,15/2,15/2 at s=0,6,24 mod 30 and outside factor D_y=prod_(7<=p<=y)(1-4/(p-2)^2); physical modulus q=30n, merged branch q0=30a with shift r=a*t, t=s*inv(a) mod 30. Conclusion (4): T_fin(S)=O(ell^-1)+O(L^-eta) on RECTANGLE a<=L^alpha, R<=L^rho, 45alpha<1, 35alpha+5rho<1, eta<Delta/40, and on WEDGE a<=L^alpha, a/2<=R<=2a, alpha<1/40, eta<1/40-alpha; each sector separately, overlap not added. Physical cuts U<q<=V mean U/30<n<=V/30 exactly. Conditional: Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement inspected, proof not audited) and the accepted #5795/#5964 coefficient, majorant and cutoff interfaces. #6172 and #6180 were pending method leads at writing and are not premises. Excluded by the source: the other small-branch choices, same-bin work, small-complement transfer, the three-branch core, full X, the variance constant, the high-product envelope and twin primes; no claim that these sectors exhaust or dominate the mass.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#6187 bounds the distinguished a=d0 branch of route 273's Fejer-core remainder in two unequal-bin sectors with finite factors retained. Its scope excludes the other branches, same-bin work, the three-branch core, full X, the variance constant and twin primes, and the source requests no route closure or status change. It is a conditional estimate on one piece of a decomposition, not a step that changes any infinitude route's status.","assumptions_md":"Consideration only. #6187 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6187 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2135, its only review, which has no findings or also_fix; the snapshot pins only return:6187. Statement: original tent radius L, ell=log L, y=L^(1/2+o(1)); a=d0, b=d+, c=d- >1 pairwise coprime, squarefree, y-smooth, coprime to 30, n=abc on the source's sharp interval L/ell^2<n<=2L with n/a,n/b,n/c>L^(2/5), unequal labelled dyadic bins B!=C, R=max(B/C,C/B); finite coefficients A_s=15,15/2,15/2 at s=0,6,24 mod 30 and outside factor D_y=prod_(7<=p<=y)(1-4/(p-2)^2); physical modulus q=30n, merged branch q0=30a with shift r=a*t, t=s*inv(a) mod 30. Conclusion (4): T_fin(S)=O(ell^-1)+O(L^-eta) on RECTANGLE a<=L^alpha, R<=L^rho, 45alpha<1, 35alpha+5rho<1, eta<Delta/40, and on WEDGE a<=L^alpha, a/2<=R<=2a, alpha<1/40, eta<1/40-alpha; each sector separately, overlap not added. Physical cuts U<q<=V mean U/30<n<=V/30 exactly. Conditional: Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement inspected, proof not audited) and the accepted #5795/#5964 coefficient, majorant and cutoff interfaces. #6172 and #6180 were pending method leads at writing and are not premises. Excluded by the source: the other small-branch choices, same-bin work, small-complement transfer, the three-branch core, full X, the variance constant, the high-product envelope and twin primes; no claim that these sectors exhaust or dominate the mass.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"#6187 is a conditional bound on sums of multiplicative weights against Fejer tent remainders over smooth moduli, with fixed local factors at 2, 3 and 5, via a Kloosterman-type bilinear estimate. These objects are defined independently of whether there are finitely or infinitely many twin primes. It uses no finiteness hypothesis and derives no structure a finite twin count would force, and no disproof-shaped consequence.","assumptions_md":"Consideration only. #6187 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6187 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2135, its only review, which has no findings or also_fix; the snapshot pins only return:6187. Statement: original tent radius L, ell=log L, y=L^(1/2+o(1)); a=d0, b=d+, c=d- >1 pairwise coprime, squarefree, y-smooth, coprime to 30, n=abc on the source's sharp interval L/ell^2<n<=2L with n/a,n/b,n/c>L^(2/5), unequal labelled dyadic bins B!=C, R=max(B/C,C/B); finite coefficients A_s=15,15/2,15/2 at s=0,6,24 mod 30 and outside factor D_y=prod_(7<=p<=y)(1-4/(p-2)^2); physical modulus q=30n, merged branch q0=30a with shift r=a*t, t=s*inv(a) mod 30. Conclusion (4): T_fin(S)=O(ell^-1)+O(L^-eta) on RECTANGLE a<=L^alpha, R<=L^rho, 45alpha<1, 35alpha+5rho<1, eta<Delta/40, and on WEDGE a<=L^alpha, a/2<=R<=2a, alpha<1/40, eta<1/40-alpha; each sector separately, overlap not added. Physical cuts U<q<=V mean U/30<n<=V/30 exactly. Conditional: Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement inspected, proof not audited) and the accepted #5795/#5964 coefficient, majorant and cutoff interfaces. #6172 and #6180 were pending method leads at writing and are not premises. Excluded by the source: the other small-branch choices, same-bin work, small-complement transfer, the three-branch core, full X, the variance constant, the high-product envelope and twin primes; no claim that these sectors exhaust or dominate the mass.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Checked against main and O1-O7 separately. main (survivor-gap lower bound, sufficient A=12): #6187 has no survivor sieve, gap or crossing-gap statement. O1 (prime harmonic asymptotic with constant and o(1) remainder): none appears; sum lambda0(a)<<ell^2 and F(t)<<t log(2t) are one-sided upper bounds. O2 (two-variable uniform smooth-number asymptotic, u<=Z^(1-eps)): nearest miss on support only. The weights live on y-smooth integers, but smoothness enters only as a support restriction carried through the #5795 bounds, which hold uniformly in smoothness; no smooth-number count or asymptotic in u appears. O3 (every fixed A>4): alpha, rho, eta and epsilon are sector and saving parameters, and the ell-powers (ell^24, H=ceil(ell^4)) are loss bookkeeping, unrelated to kk's A. O4 (dyadic prime-count coefficient 1/2): no prime-count asymptotic; the dyadic decomposition labels the bins B, C. O5 (quoted general sieve input and external one-class statement): the external input is Bettin-Chandee Theorem 1, a bilinear Kloosterman-fraction bound, not the sieve input O5 names. O6 (two-sided density and Euler estimates): D_y is a convergent product in (0,1] used only as a bounded factor, and the A4 majorant sum<<x log^3(2x) is one-sided. O7 (exact eq. (28) constant): A_s is an exact local constant for this remainder, not kk's eq. (28) constant; the analytic constants depend on epsilon. No bridge exists.","assumptions_md":"Consideration only. #6187 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6187 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2135, its only review, which has no findings or also_fix; the snapshot pins only return:6187. Statement: original tent radius L, ell=log L, y=L^(1/2+o(1)); a=d0, b=d+, c=d- >1 pairwise coprime, squarefree, y-smooth, coprime to 30, n=abc on the source's sharp interval L/ell^2<n<=2L with n/a,n/b,n/c>L^(2/5), unequal labelled dyadic bins B!=C, R=max(B/C,C/B); finite coefficients A_s=15,15/2,15/2 at s=0,6,24 mod 30 and outside factor D_y=prod_(7<=p<=y)(1-4/(p-2)^2); physical modulus q=30n, merged branch q0=30a with shift r=a*t, t=s*inv(a) mod 30. Conclusion (4): T_fin(S)=O(ell^-1)+O(L^-eta) on RECTANGLE a<=L^alpha, R<=L^rho, 45alpha<1, 35alpha+5rho<1, eta<Delta/40, and on WEDGE a<=L^alpha, a/2<=R<=2a, alpha<1/40, eta<1/40-alpha; each sector separately, overlap not added. Physical cuts U<q<=V mean U/30<n<=V/30 exactly. Conditional: Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement inspected, proof not audited) and the accepted #5795/#5964 coefficient, majorant and cutoff interfaces. #6172 and #6180 were pending method leads at writing and are not premises. Excluded by the source: the other small-branch choices, same-bin work, small-complement transfer, the three-branch core, full X, the variance constant, the high-product envelope and twin primes; no claim that these sectors exhaust or dominate the mass.","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":12696,"input_sha256":"ac0d4dae848e835d5d190ac255f7b166040f5736e05f1d95cee4f48dd4537974","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"f126ecd0b829c3d29214d15e50de237841efebc075fc9831d392d2c8d1a87115","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6187,"material_evidence":{"sha256":"4104d53bd1ca8a8de77c3f7345de853f2e93b12f56917055900e201c0586cc25","entries":{"return:6187":"65e52cfbba553e0666004ef369eb4da2eebf0ff9dc10b8bd83467e4d2e87abe5"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6187,"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 #6187: 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\":6187,\"report_sha256\":\"f126ecd0b829c3d29214d15e50de237841efebc075fc9831d392d2c8d1a87115\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6187\":\"65e52cfbba553e0666004ef369eb4da2eebf0ff9dc10b8bd83467e4d2e87abe5\"},\"sha256\":\"4104d53bd1ca8a8de77c3f7345de853f2e93b12f56917055900e201c0586cc25\"},\"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:6187,input_sha256:\"ac0d4dae848e835d5d190ac255f7b166040f5736e05f1d95cee4f48dd4537974\",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/6430/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"80328a07af208fee2b5f7648e2213c77fe08fa08db34eb9052128f8c85357cf9","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}