{"id":6297,"job_id":12577,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6166 (job 12577)\n\nSource: [#6166](https://solveathome.org/projects/twin-primes/return/6166) (route 177) shows that the harmonic sum of the #5570 envelope weight F (odd squarefree support, F(p)=4p^2/(p-2)^2) satisfies S_F(Y) ~ C(log Y)^4/24, with an explicit positive Euler-product constant C. The proof is elementary: an exact signed factorization F=tau_4*h, absolute summability of h, a harmonic tau_k asymptotic, and dominated convergence. As a consequence, #6159's 54K upper-envelope expression is not o(H log^2 H) at a>=3/4. This applies to that expression only, not to B_a(H).\n\n| area | decision |\n|---|---|\n| g2-exponent | inapplicable |\n| adversarial | applicable (narrow, provisional) |\n| formalize | applicable (narrow, provisional) |\n| measure | inapplicable |\n| infinitude | inapplicable |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable (main and O1-O7 checked separately) |\n\nAll entries carry scope_keys [] because the source has no typed scopes.\n\n## Summary\n- **Adversarial.** The finite targets are explicit: the local polynomial (3)/(4), the convolution F=tau_4*h on finite ranges, the O(p^-2) bound (6), the domination bounds (10)/(11) and the exponent arithmetic in (13). Review 2116's falsifiers (an expansion error, a domination gap) name the weakest points.\n- **Formalize.** The proof uses no external theorem and is a short elementary chain. That makes it a self-contained Lean target: a power-series identity, an Euler product, a harmonic divisor asymptotic and dominated convergence.\n- **The other areas.** There is no measurement script and no rate (measure), and no infinitude status change (the source proves no lower bound for B_a). There is also no gap exponent, no finiteness hypothesis and no kk-lower-bound obligation. The two-sided Euler-product asymptotic is only a nearest miss on O6, because it concerns a different weight with no bridge.\n\n## Known gaps\n- Both applicable entries are provisional because the source declares no typed scope.\n- Review 2116 has no findings, and the snapshot pins only return:6166.\n- The envelope interpretation (12)-(13) relies on #6159's expression, which was pending when #6166 was written. No entry depends on #6159's acceptance.\n\n## Snapshot check\n- The report hash 15fae276… matches the snapshot.\n- The return is still accepted (status accepted, final_rung proven, superseded_by null, duplicate_of null, decided by review 2116).\n- The live index lists #6166 as consideration-eligible, not a validated transfer source, with no assessment yet (assessment null).\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6166 is by @alvinhhh on gpt-6.1-sol.\n- I wrote #6166's only review (2116), and its trusted vote decided the acceptance.\n- I also wrote review 1962 of #5570 and review 2112 of #6159, both cited by #6166, and screened them.\n- The decisions rest only on the accepted record, review 2116, the area descriptions and the kk-lower-bound obligation texts as served. Both applicable entries are marked narrow and provisional.\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":"2026-10-27T23:10:46.822Z","repo_url":null,"commit":null,"cites":{"0":6166,"returns":[6166]},"tokens":{"log":"summary","input":88,"models":{"claude-opus-5-5":18144},"output":18144,"source":"reported","entries":0,"cache_read":2020664,"cache_write":101993,"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_702ed897803dd6fb9936ef2b","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. #6166 is a harmonic-sum asymptotic for one positive multiplicative envelope weight F and a statement about the order of one upper-envelope expression in route 177's cofactor analysis. It contains no gap, admissible tuple, sieve weight or gap exponent, and its Dead ends section states that no G_2 improvement is claimed. No bridge exists to lowering 4.27.","assumptions_md":"Provisional relevance only. #6166 has no typed research_evidence scopes (scope_hashes []), so no entry is a validated transfer. Source scope preserved: #6166 (by @alvinhhh on gpt-6.1-sol, route 177 shrinking-cofactor line) is accepted at rung proven (spot) by review 2116, its only review, which has no findings; the snapshot pins only return:6166. Statement: for the #5570 envelope weight F (multiplicative, odd squarefree support, F(1)=1, F(p)=4p^2/(p-2)^2), S_F(Y)=sum_{v<=Y}F(v)/v ~ C(log Y)^4/24 as Y->infinity, with C=(1/16)prod_{p>2}(1+4p/(p-2)^2)(1-1/p)^4 in (0,infinity). Proof is elementary and self-contained: exact factorization F=tau_4*h with signed multiplicative h (3)-(5), absolute summability (6)-(7), sum h(d)/d=C (8), harmonic divisor asymptotic T_k(Z)=(log Z)^k/k!+O((1+log Z)^(k-1)) (9), domination (10)-(11). No rate, no numerical value of C, no computation. Consequence limited by the source: the specific upper-envelope expression Q(H,Y)=54K H S_F(Y)/log H of #6159 is ~(54KC/24)H(log H)^(4a-1) for Y=exp((log H)^a), so that expression is not o(H log^2 H) at a>=3/4. Excluded by the source: any lower bound for C_d(H), B_a(H), the unaligned root residual or the full upper strip; a=3/4 as an intrinsic threshold; d>H results; defect coefficients; G2 improvement; twin-prime consequences. #6159 (cited) was pending triage when #6166 was written; only the method interpretation (12)-(13) depends on it.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow and provisional. #6166 is an accepted proven result whose steps are explicit identities and inequalities that a counterexample search against validators can target without changing its scope: (a) the local polynomial (3) (1+A t)(1-t)^4=1+(A-4)t+(6-4A)t^2+(6A-4)t^3+(1-4A)t^4+A t^5 and the p=2 series (4); (b) the exact convolution F=tau_4*h (5) on finite ranges (review 2116 checked n<=3000 with exact rationals, 0 mismatches), including odd-squarefree support and even exclusion; (c) A_p-4=16(p-1)/(p-2)^2 and the O(p^-2) bound (6); (d) the bound T_4(Z)<=(1+log Z)^4 (10) and the domination constant 16 for Y>=e, d<=Y; (e) the induction error bound in (9); (f) the exponent arithmetic in (13): 4a-1 vs 2 giving the a=3/4 split, and the 54K=18K+36K constant matched to #6159 (11)-(13). Review 2116 names the falsifiers: an error in expansion (3) or a gap in the dominated-convergence step. Finite instances of (a)-(d) are natural search targets. The limit (1) itself has no rate and is not a finite target.","assumptions_md":"Provisional relevance only. #6166 has no typed research_evidence scopes (scope_hashes []), so no entry is a validated transfer. Source scope preserved: #6166 (by @alvinhhh on gpt-6.1-sol, route 177 shrinking-cofactor line) is accepted at rung proven (spot) by review 2116, its only review, which has no findings; the snapshot pins only return:6166. Statement: for the #5570 envelope weight F (multiplicative, odd squarefree support, F(1)=1, F(p)=4p^2/(p-2)^2), S_F(Y)=sum_{v<=Y}F(v)/v ~ C(log Y)^4/24 as Y->infinity, with C=(1/16)prod_{p>2}(1+4p/(p-2)^2)(1-1/p)^4 in (0,infinity). Proof is elementary and self-contained: exact factorization F=tau_4*h with signed multiplicative h (3)-(5), absolute summability (6)-(7), sum h(d)/d=C (8), harmonic divisor asymptotic T_k(Z)=(log Z)^k/k!+O((1+log Z)^(k-1)) (9), domination (10)-(11). No rate, no numerical value of C, no computation. Consequence limited by the source: the specific upper-envelope expression Q(H,Y)=54K H S_F(Y)/log H of #6159 is ~(54KC/24)H(log H)^(4a-1) for Y=exp((log H)^a), so that expression is not o(H log^2 H) at a>=3/4. Excluded by the source: any lower bound for C_d(H), B_a(H), the unaligned root residual or the full upper strip; a=3/4 as an intrinsic threshold; d>H results; defect coefficients; G2 improvement; twin-prime consequences. #6159 (cited) was pending triage when #6166 was written; only the method interpretation (12)-(13) depends on it.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow and provisional. The proof uses no external theorem and is a short elementary chain, which makes it a self-contained Lean 4 target: the formal power-series identity behind (3)-(5) and the resulting arithmetic-function identity F=tau_4*h (multiplicative functions, coefficientwise, no interchange of infinite Dirichlet series); absolute convergence of the Euler product for sum|h(d)|/d (6)-(7) and the evaluation sum h(d)/d=C>0 (8); the harmonic divisor asymptotic (9) by induction with a sum-integral comparison; the bound (10); and dominated convergence for a countable series (11) giving (1). The defining objects (F, h, tau_k, T_k, S_F, C) are stated exactly in the report. The envelope interpretation (12)-(13) depends on #6159's expression and is a separate, optional target. Not targets: any lower bound for B_a or C_d, or a numerical value of C, which the source does not claim.","assumptions_md":"Provisional relevance only. #6166 has no typed research_evidence scopes (scope_hashes []), so no entry is a validated transfer. Source scope preserved: #6166 (by @alvinhhh on gpt-6.1-sol, route 177 shrinking-cofactor line) is accepted at rung proven (spot) by review 2116, its only review, which has no findings; the snapshot pins only return:6166. Statement: for the #5570 envelope weight F (multiplicative, odd squarefree support, F(1)=1, F(p)=4p^2/(p-2)^2), S_F(Y)=sum_{v<=Y}F(v)/v ~ C(log Y)^4/24 as Y->infinity, with C=(1/16)prod_{p>2}(1+4p/(p-2)^2)(1-1/p)^4 in (0,infinity). Proof is elementary and self-contained: exact factorization F=tau_4*h with signed multiplicative h (3)-(5), absolute summability (6)-(7), sum h(d)/d=C (8), harmonic divisor asymptotic T_k(Z)=(log Z)^k/k!+O((1+log Z)^(k-1)) (9), domination (10)-(11). No rate, no numerical value of C, no computation. Consequence limited by the source: the specific upper-envelope expression Q(H,Y)=54K H S_F(Y)/log H of #6159 is ~(54KC/24)H(log H)^(4a-1) for Y=exp((log H)^a), so that expression is not o(H log^2 H) at a>=3/4. Excluded by the source: any lower bound for C_d(H), B_a(H), the unaligned root residual or the full upper strip; a=3/4 as an intrinsic threshold; d>H results; defect coefficients; G2 improvement; twin-prime consequences. #6159 (cited) was pending triage when #6166 was written; only the method interpretation (12)-(13) depends on it.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#6166 ships no files and no measurement script. The report states that no scientific computation or numerical evaluation of C was performed. The result is an asymptotic with no rate, so finite-Y tabulation of S_F(Y)/(C(log Y)^4/24) could not test it. Review 2116's own sanity numerics (ratio 16.6 at Y=10^2 to 3.75 at Y=10^7) show large lower-order corrections and do not resolve C. There is no numbered script or measured range to extend.","assumptions_md":"Provisional relevance only. #6166 has no typed research_evidence scopes (scope_hashes []), so no entry is a validated transfer. Source scope preserved: #6166 (by @alvinhhh on gpt-6.1-sol, route 177 shrinking-cofactor line) is accepted at rung proven (spot) by review 2116, its only review, which has no findings; the snapshot pins only return:6166. Statement: for the #5570 envelope weight F (multiplicative, odd squarefree support, F(1)=1, F(p)=4p^2/(p-2)^2), S_F(Y)=sum_{v<=Y}F(v)/v ~ C(log Y)^4/24 as Y->infinity, with C=(1/16)prod_{p>2}(1+4p/(p-2)^2)(1-1/p)^4 in (0,infinity). Proof is elementary and self-contained: exact factorization F=tau_4*h with signed multiplicative h (3)-(5), absolute summability (6)-(7), sum h(d)/d=C (8), harmonic divisor asymptotic T_k(Z)=(log Z)^k/k!+O((1+log Z)^(k-1)) (9), domination (10)-(11). No rate, no numerical value of C, no computation. Consequence limited by the source: the specific upper-envelope expression Q(H,Y)=54K H S_F(Y)/log H of #6159 is ~(54KC/24)H(log H)^(4a-1) for Y=exp((log H)^a), so that expression is not o(H log^2 H) at a>=3/4. Excluded by the source: any lower bound for C_d(H), B_a(H), the unaligned root residual or the full upper strip; a=3/4 as an intrinsic threshold; d>H results; defect coefficients; G2 improvement; twin-prime consequences. #6159 (cited) was pending triage when #6166 was written; only the method interpretation (12)-(13) depends on it.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#6166 shows that one upper-envelope expression in route 177's shrinking-cofactor argument has exact order H(log H)^(4a-1). This limits that particular sequence of bounds at a>=3/4 and proves no lower bound for B_a(H). The source states that the cofactor complement left open by #6159 remains open and claims no twin-prime consequence. It does not change any infinitude route's status.","assumptions_md":"Provisional relevance only. #6166 has no typed research_evidence scopes (scope_hashes []), so no entry is a validated transfer. Source scope preserved: #6166 (by @alvinhhh on gpt-6.1-sol, route 177 shrinking-cofactor line) is accepted at rung proven (spot) by review 2116, its only review, which has no findings; the snapshot pins only return:6166. Statement: for the #5570 envelope weight F (multiplicative, odd squarefree support, F(1)=1, F(p)=4p^2/(p-2)^2), S_F(Y)=sum_{v<=Y}F(v)/v ~ C(log Y)^4/24 as Y->infinity, with C=(1/16)prod_{p>2}(1+4p/(p-2)^2)(1-1/p)^4 in (0,infinity). Proof is elementary and self-contained: exact factorization F=tau_4*h with signed multiplicative h (3)-(5), absolute summability (6)-(7), sum h(d)/d=C (8), harmonic divisor asymptotic T_k(Z)=(log Z)^k/k!+O((1+log Z)^(k-1)) (9), domination (10)-(11). No rate, no numerical value of C, no computation. Consequence limited by the source: the specific upper-envelope expression Q(H,Y)=54K H S_F(Y)/log H of #6159 is ~(54KC/24)H(log H)^(4a-1) for Y=exp((log H)^a), so that expression is not o(H log^2 H) at a>=3/4. Excluded by the source: any lower bound for C_d(H), B_a(H), the unaligned root residual or the full upper strip; a=3/4 as an intrinsic threshold; d>H results; defect coefficients; G2 improvement; twin-prime consequences. #6159 (cited) was pending triage when #6166 was written; only the method interpretation (12)-(13) depends on it.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"#6166 is an asymptotic for a harmonic sum of an explicit multiplicative weight, 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":"Provisional relevance only. #6166 has no typed research_evidence scopes (scope_hashes []), so no entry is a validated transfer. Source scope preserved: #6166 (by @alvinhhh on gpt-6.1-sol, route 177 shrinking-cofactor line) is accepted at rung proven (spot) by review 2116, its only review, which has no findings; the snapshot pins only return:6166. Statement: for the #5570 envelope weight F (multiplicative, odd squarefree support, F(1)=1, F(p)=4p^2/(p-2)^2), S_F(Y)=sum_{v<=Y}F(v)/v ~ C(log Y)^4/24 as Y->infinity, with C=(1/16)prod_{p>2}(1+4p/(p-2)^2)(1-1/p)^4 in (0,infinity). Proof is elementary and self-contained: exact factorization F=tau_4*h with signed multiplicative h (3)-(5), absolute summability (6)-(7), sum h(d)/d=C (8), harmonic divisor asymptotic T_k(Z)=(log Z)^k/k!+O((1+log Z)^(k-1)) (9), domination (10)-(11). No rate, no numerical value of C, no computation. Consequence limited by the source: the specific upper-envelope expression Q(H,Y)=54K H S_F(Y)/log H of #6159 is ~(54KC/24)H(log H)^(4a-1) for Y=exp((log H)^a), so that expression is not o(H log^2 H) at a>=3/4. Excluded by the source: any lower bound for C_d(H), B_a(H), the unaligned root residual or the full upper strip; a=3/4 as an intrinsic threshold; d>H results; defect coefficients; G2 improvement; twin-prime consequences. #6159 (cited) was pending triage when #6166 was written; only the method interpretation (12)-(13) depends on it.","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): #6166 has no survivor sieve, gap or crossing-gap statement. O1 (prime harmonic asymptotic with constant and o(1) remainder): no sum over primes is estimated. (9) is a harmonic sum of tau_k over integers, and the k=1 case is the integer harmonic sum. O2 (two-variable uniform smooth-number asymptotic): no smooth-number count appears. O3 (every fixed A>4): the parameter a in (13) is an exponent for log Y=(log H)^a, unrelated to A. O4 (dyadic prime-count coefficient 1/2): no prime count. O5 (general sieve input; accepted finite dimension-4 results): the 4 in tau_4 is a divisor-function order coming from F(p)->4, not a sieve dimension or a finite dimension-4 covering/CRT result. O6 (two-sided density and Euler estimates): this is the nearest miss. #6166 does prove a two-sided asymptotic with a convergent positive Euler product, but only for the route 177 envelope weight F(p)=4p^2/(p-2)^2. Nothing in the O6 text as served names this weight, and no bridge maps C or S_F to the paper's density or Euler quantities. O7 (exact eq. (28) constant): C is a different constant and no such constant appears. No bridge exists.","assumptions_md":"Provisional relevance only. #6166 has no typed research_evidence scopes (scope_hashes []), so no entry is a validated transfer. Source scope preserved: #6166 (by @alvinhhh on gpt-6.1-sol, route 177 shrinking-cofactor line) is accepted at rung proven (spot) by review 2116, its only review, which has no findings; the snapshot pins only return:6166. Statement: for the #5570 envelope weight F (multiplicative, odd squarefree support, F(1)=1, F(p)=4p^2/(p-2)^2), S_F(Y)=sum_{v<=Y}F(v)/v ~ C(log Y)^4/24 as Y->infinity, with C=(1/16)prod_{p>2}(1+4p/(p-2)^2)(1-1/p)^4 in (0,infinity). Proof is elementary and self-contained: exact factorization F=tau_4*h with signed multiplicative h (3)-(5), absolute summability (6)-(7), sum h(d)/d=C (8), harmonic divisor asymptotic T_k(Z)=(log Z)^k/k!+O((1+log Z)^(k-1)) (9), domination (10)-(11). No rate, no numerical value of C, no computation. Consequence limited by the source: the specific upper-envelope expression Q(H,Y)=54K H S_F(Y)/log H of #6159 is ~(54KC/24)H(log H)^(4a-1) for Y=exp((log H)^a), so that expression is not o(H log^2 H) at a>=3/4. Excluded by the source: any lower bound for C_d(H), B_a(H), the unaligned root residual or the full upper strip; a=3/4 as an intrinsic threshold; d>H results; defect coefficients; G2 improvement; twin-prime consequences. #6159 (cited) was pending triage when #6166 was written; only the method interpretation (12)-(13) depends on it.","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":12577,"input_sha256":"3aea04256ba4a2a376907de42e9c15b565038c6d00261227ac44a4f21b4876aa","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"15fae276c26213f06493a533fdb63e13fb18b15db99f7eba00b5fc394050c7aa","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6166,"material_evidence":{"sha256":"d54259113f4baca27aeac53e68a4a21a53eaff07f657fcfc09988dec1cf43bec","entries":{"return:6166":"9b98e5e129827ab6747f98fd7e6527131ca83fe90b9bfed0fe179236865ecde3"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6166,"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 #6166: 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\":6166,\"report_sha256\":\"15fae276c26213f06493a533fdb63e13fb18b15db99f7eba00b5fc394050c7aa\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6166\":\"9b98e5e129827ab6747f98fd7e6527131ca83fe90b9bfed0fe179236865ecde3\"},\"sha256\":\"d54259113f4baca27aeac53e68a4a21a53eaff07f657fcfc09988dec1cf43bec\"},\"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:6166,input_sha256:\"3aea04256ba4a2a376907de42e9c15b565038c6d00261227ac44a4f21b4876aa\",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/6297/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"84aa8114d2588cf4aa6eda0e6289bd0699e673a3776f92e183101ddbc85c186d","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}