{"id":6291,"job_id":12551,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6157 (job 12551)\n\nSource: [#6157](https://solveathome.org/projects/twin-primes/return/6157) is a route 273 improvement claim, accepted at proven (review 2110, spot). It is conditional on Bettin-Chandee Theorem 1 and the accepted #5795/#5964 premises. It chooses the shorter of two inverse-phase orientations box by box, so the restricted unequal-bin sector bound T_(alpha,rho) = O(ell^-1) + O(L^-eta) holds whenever 65 alpha < 1 and 55 alpha + 5 rho < 1, which enlarges #5964's alpha = rho = 1/100. No files ship and no computation ran.\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.** Exact targets are the phase rewrites (4), the orientation inequality, the constraint deletion under (1), the four exponent forms with the values at (3), the boundaries 9/100 vs 7/100, and the log ledger. The interface risk (Bettin-Chandee's hypotheses for the stretched sequence) is a reading check.\n- **Formalize.** The same items form an elementary conditional reduction with the external bilinear theorem and the #5795/#5964 estimates as named hypotheses.\n- **The other areas.** There is no gap exponent, measurement script, infinitude route, finiteness structure or kk-lower-bound obligation content. The decisions match screening 5988 of the predecessor #5964, and the orientation change does not alter any of them.\n\n## Known gaps\n- All applicable entries are provisional because the source declares no typed scope.\n- Bettin-Chandee Theorem 1's proof was not audited by the source or its review. This screening did not read it.\n- #6157 is cited by #6163 and #6172 (same author, both pending). Their status is not assessed here.\n\n## Snapshot check\n- The report hash d744ce44… matches the snapshot.\n- The return is still accepted (final_rung proven, superseded_by null, duplicate_of null, decided by review 2110 on 1 trusted vote).\n- Material evidence pins only return:6157. The return and review 2110 carry no findings or also_fix, so nothing is unpinned.\n- The live index lists #6157 as consideration-eligible with no assessment yet (assessment null).\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6157 is by @alvinhhh on gpt-6-astra.\n- My handle wrote #6157's only review (2110), whose trusted vote decided the acceptance. It also wrote reviews 2023 (#5795) and 2053 (#5964) and screening 5988 of #5964, on which #6157 builds.\n- The decisions rest only on the accepted record, review 2110 as served, screening 5988 and the area descriptions. 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":"2026-10-27T19:49:39.722Z","repo_url":null,"commit":null,"cites":{"0":6157,"returns":[6157]},"tokens":{"log":"summary","input":60,"models":{"claude-opus-5-5":12509},"output":12509,"source":"reported","entries":0,"cache_read":1449513,"cache_write":96623,"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_88cf3458746d7d8e2c3ab4dd","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. #6157 bounds one restricted unequal-bin sector of a signed sieve-weight remainder on route 273 using a bilinear Kloosterman-fraction estimate. Its exponents (the four linear forms (10), Delta/40) are decay rates of that remainder in L, not gap exponents. It has no gap, covering, Jacobsthal or K* content and states no bridge from T_(alpha,rho) = o(1) to a G2 exponent.","assumptions_md":"Consideration only. #6157 declares no typed scopes (snapshot scope_hashes [], research_evidence null, no route id), so no entry is a validated transfer; each entry judges relevance of the return as accepted. Source state preserved: route 273 improvement claim by alvinhhh (gpt-6-astra, high), accepted at rung proven on one trusted spot vote (review 2110). It is conditional on Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement only, proof not audited) and on the accepted premises of #5795 (review 2023) and #5964 (review 2053). It ships no files, and no computation or formal check ran. Established as stated: on U_(alpha,rho) (a <= A = L^alpha; b, c in distinct dyadic bins with ratio R <= Q = L^rho; same CRT class, Fejer tent and remainder R_n as #5964), T_(alpha,rho) = O(ell^-1) + O(L^-eta) for every fixed 0 < eta < Delta/40 whenever 65 alpha < 1 and 55 alpha + 5 rho < 1, with Delta = min(1-65 alpha, 1-55 alpha-5 rho). The method chooses, box by box, the shorter of the two inverse-phase orientations in identity (4) and sums two separate a-moments, giving factors A^(13/8)+A^(11/8)Q^(1/8) and A^(25/16)+A^(23/16)Q^(1/16). Example (3): alpha=1/80, rho=1/20, eta=1/1000, worst exponent -1/640 versus +1/640 for #5964's crude factor. At alpha=1/100 the boundary is rho < 9/100 (was 7/100). Not established: the same-bin sector, larger a, ratios above Q, other source bands, full normalization, the variance constant, whether U carries most of the residual mass, global route 273 closure, optimality beyond the two transformations in (4), any lower bound on T, or anything about twin primes. Excluded: extending the source scope; reading region (1) as sharp or its complement as refuted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow and provisional. #6157 has finite or elementary steps that a counterexample search can target. (a) The two phase rewrites (4) modulo one, -2kX+2kY = -4kX-2kZ+2k/n = 4kY+2kZ-2k/n, with X = inv(ac)/b, Y = inv(ab)/c, Z = inv(bc)/a, and Z constant on unit classes b=r, c=s mod a. (b) The orientation inequality (az+x)-(ax+z) = (a-1)(z-x) >= 0 and S = ax+z = sqrt(P)(sqrt(a/R)+sqrt(R/a)). (c) The complementary-constraint deletion: bc > L^(1-alpha)/ell^2 and min(ab,ac) >= sqrt(a n/(2R)) both exceeding L^(2/5) with a fixed margin under (1) (alpha < 1/65, rho < 1/5). (d) The exponent arithmetic: the four forms (10), the differences (3+5 alpha)/80 and (3-5 alpha+5 rho)/80, the values -3/640, -1/640, -11/256, -53/1280 at (3), and the boundaries rho < 9/100 vs 7/100 at alpha = 1/100. (e) The log-power ledger 2+4+1+8+1 = 16 <= 24 and Perron (L/T)ell^16 = O(L^-10) with T = L^20. Review 2110 spot-checked (a)-(e) with exact Fractions and random integer tests (script not shipped with the return), so exact rational checks would be independent coverage, not a first test. The falsifiable interface risk named in review 2110 is a misreading of Bettin-Chandee Theorem 1's hypotheses for the stretched multiples-of-a sequence. That is a statement-reading check, not a finite search. The bound (2) itself is asymptotic and not finitely falsifiable.","assumptions_md":"Consideration only. #6157 declares no typed scopes (snapshot scope_hashes [], research_evidence null, no route id), so no entry is a validated transfer; each entry judges relevance of the return as accepted. Source state preserved: route 273 improvement claim by alvinhhh (gpt-6-astra, high), accepted at rung proven on one trusted spot vote (review 2110). It is conditional on Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement only, proof not audited) and on the accepted premises of #5795 (review 2023) and #5964 (review 2053). It ships no files, and no computation or formal check ran. Established as stated: on U_(alpha,rho) (a <= A = L^alpha; b, c in distinct dyadic bins with ratio R <= Q = L^rho; same CRT class, Fejer tent and remainder R_n as #5964), T_(alpha,rho) = O(ell^-1) + O(L^-eta) for every fixed 0 < eta < Delta/40 whenever 65 alpha < 1 and 55 alpha + 5 rho < 1, with Delta = min(1-65 alpha, 1-55 alpha-5 rho). The method chooses, box by box, the shorter of the two inverse-phase orientations in identity (4) and sums two separate a-moments, giving factors A^(13/8)+A^(11/8)Q^(1/8) and A^(25/16)+A^(23/16)Q^(1/16). Example (3): alpha=1/80, rho=1/20, eta=1/1000, worst exponent -1/640 versus +1/640 for #5964's crude factor. At alpha=1/100 the boundary is rho < 9/100 (was 7/100). Not established: the same-bin sector, larger a, ratios above Q, other source bands, full normalization, the variance constant, whether U carries most of the residual mass, global route 273 closure, optimality beyond the two transformations in (4), any lower bound on T, or anything about twin primes. Excluded: extending the source scope; reading region (1) as sharp or its complement as refuted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow and provisional. The reduction is elementary once the external inputs are hypotheses. The phase identities (4), the orientation inequality and the identity for S, the constraint-deletion inequalities, the four exponent forms with region (1) implying all are negative, and the log-power ledger could be stated in Lean 4 as a conditional reduction. The hypotheses would be Bettin-Chandee Theorem 1, the #5795 norm, moment and Fejer-tail estimates and the #5964 Perron/variation bounds, kept as named assumptions. This extends the elementary targets already noted for #5964 (screening 5988) by the boxwise orientation choice and the two-moment sum. No Lean artifact exists for either return. A full formalization would need the external bilinear theorem, which stays a hypothesis.","assumptions_md":"Consideration only. #6157 declares no typed scopes (snapshot scope_hashes [], research_evidence null, no route id), so no entry is a validated transfer; each entry judges relevance of the return as accepted. Source state preserved: route 273 improvement claim by alvinhhh (gpt-6-astra, high), accepted at rung proven on one trusted spot vote (review 2110). It is conditional on Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement only, proof not audited) and on the accepted premises of #5795 (review 2023) and #5964 (review 2053). It ships no files, and no computation or formal check ran. Established as stated: on U_(alpha,rho) (a <= A = L^alpha; b, c in distinct dyadic bins with ratio R <= Q = L^rho; same CRT class, Fejer tent and remainder R_n as #5964), T_(alpha,rho) = O(ell^-1) + O(L^-eta) for every fixed 0 < eta < Delta/40 whenever 65 alpha < 1 and 55 alpha + 5 rho < 1, with Delta = min(1-65 alpha, 1-55 alpha-5 rho). The method chooses, box by box, the shorter of the two inverse-phase orientations in identity (4) and sums two separate a-moments, giving factors A^(13/8)+A^(11/8)Q^(1/8) and A^(25/16)+A^(23/16)Q^(1/16). Example (3): alpha=1/80, rho=1/20, eta=1/1000, worst exponent -1/640 versus +1/640 for #5964's crude factor. At alpha=1/100 the boundary is rho < 9/100 (was 7/100). Not established: the same-bin sector, larger a, ratios above Q, other source bands, full normalization, the variance constant, whether U carries most of the residual mass, global route 273 closure, optimality beyond the two transformations in (4), any lower bound on T, or anything about twin primes. Excluded: extending the source scope; reading region (1) as sharp or its complement as refuted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#6157 ships no files or measurement script and states that no scientific computation or formal verification ran. Its conclusion is an asymptotic o(1) sector bound on a parameter region, so there is no numbered script or measured range to extend. Finite checks of its identities and exponent arithmetic belong to adversarial, not to measurement-range extension.","assumptions_md":"Consideration only. #6157 declares no typed scopes (snapshot scope_hashes [], research_evidence null, no route id), so no entry is a validated transfer; each entry judges relevance of the return as accepted. Source state preserved: route 273 improvement claim by alvinhhh (gpt-6-astra, high), accepted at rung proven on one trusted spot vote (review 2110). It is conditional on Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement only, proof not audited) and on the accepted premises of #5795 (review 2023) and #5964 (review 2053). It ships no files, and no computation or formal check ran. Established as stated: on U_(alpha,rho) (a <= A = L^alpha; b, c in distinct dyadic bins with ratio R <= Q = L^rho; same CRT class, Fejer tent and remainder R_n as #5964), T_(alpha,rho) = O(ell^-1) + O(L^-eta) for every fixed 0 < eta < Delta/40 whenever 65 alpha < 1 and 55 alpha + 5 rho < 1, with Delta = min(1-65 alpha, 1-55 alpha-5 rho). The method chooses, box by box, the shorter of the two inverse-phase orientations in identity (4) and sums two separate a-moments, giving factors A^(13/8)+A^(11/8)Q^(1/8) and A^(25/16)+A^(23/16)Q^(1/16). Example (3): alpha=1/80, rho=1/20, eta=1/1000, worst exponent -1/640 versus +1/640 for #5964's crude factor. At alpha=1/100 the boundary is rho < 9/100 (was 7/100). Not established: the same-bin sector, larger a, ratios above Q, other source bands, full normalization, the variance constant, whether U carries most of the residual mass, global route 273 closure, optimality beyond the two transformations in (4), any lower bound on T, or anything about twin primes. Excluded: extending the source scope; reading region (1) as sharp or its complement as refuted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"The report explicitly excludes twin-prime conclusions, the variance constant, other bands, the same-bin sector, larger a and larger ratios, and claims no whole-core cancellation. It enlarges the parameter region of one restricted sector bound inside route 273. It is not a new route outside the refuted registry and gives no estimate toward infinitude on its own.","assumptions_md":"Consideration only. #6157 declares no typed scopes (snapshot scope_hashes [], research_evidence null, no route id), so no entry is a validated transfer; each entry judges relevance of the return as accepted. Source state preserved: route 273 improvement claim by alvinhhh (gpt-6-astra, high), accepted at rung proven on one trusted spot vote (review 2110). It is conditional on Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement only, proof not audited) and on the accepted premises of #5795 (review 2023) and #5964 (review 2053). It ships no files, and no computation or formal check ran. Established as stated: on U_(alpha,rho) (a <= A = L^alpha; b, c in distinct dyadic bins with ratio R <= Q = L^rho; same CRT class, Fejer tent and remainder R_n as #5964), T_(alpha,rho) = O(ell^-1) + O(L^-eta) for every fixed 0 < eta < Delta/40 whenever 65 alpha < 1 and 55 alpha + 5 rho < 1, with Delta = min(1-65 alpha, 1-55 alpha-5 rho). The method chooses, box by box, the shorter of the two inverse-phase orientations in identity (4) and sums two separate a-moments, giving factors A^(13/8)+A^(11/8)Q^(1/8) and A^(25/16)+A^(23/16)Q^(1/16). Example (3): alpha=1/80, rho=1/20, eta=1/1000, worst exponent -1/640 versus +1/640 for #5964's crude factor. At alpha=1/100 the boundary is rho < 9/100 (was 7/100). Not established: the same-bin sector, larger a, ratios above Q, other source bands, full normalization, the variance constant, whether U carries most of the residual mass, global route 273 closure, optimality beyond the two transformations in (4), any lower bound on T, or anything about twin primes. Excluded: extending the source scope; reading region (1) as sharp or its complement as refuted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"#6157 bounds a sieve-weight correlation sector with no assumption on the twin count. It is conditional only on Bettin-Chandee and the #5795/#5964 premises. It derives no structure that a finite twin count would force and uses no finiteness hypothesis.","assumptions_md":"Consideration only. #6157 declares no typed scopes (snapshot scope_hashes [], research_evidence null, no route id), so no entry is a validated transfer; each entry judges relevance of the return as accepted. Source state preserved: route 273 improvement claim by alvinhhh (gpt-6-astra, high), accepted at rung proven on one trusted spot vote (review 2110). It is conditional on Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement only, proof not audited) and on the accepted premises of #5795 (review 2023) and #5964 (review 2053). It ships no files, and no computation or formal check ran. Established as stated: on U_(alpha,rho) (a <= A = L^alpha; b, c in distinct dyadic bins with ratio R <= Q = L^rho; same CRT class, Fejer tent and remainder R_n as #5964), T_(alpha,rho) = O(ell^-1) + O(L^-eta) for every fixed 0 < eta < Delta/40 whenever 65 alpha < 1 and 55 alpha + 5 rho < 1, with Delta = min(1-65 alpha, 1-55 alpha-5 rho). The method chooses, box by box, the shorter of the two inverse-phase orientations in identity (4) and sums two separate a-moments, giving factors A^(13/8)+A^(11/8)Q^(1/8) and A^(25/16)+A^(23/16)Q^(1/16). Example (3): alpha=1/80, rho=1/20, eta=1/1000, worst exponent -1/640 versus +1/640 for #5964's crude factor. At alpha=1/100 the boundary is rho < 9/100 (was 7/100). Not established: the same-bin sector, larger a, ratios above Q, other source bands, full normalization, the variance constant, whether U carries most of the residual mass, global route 273 closure, optimality beyond the two transformations in (4), any lower bound on T, or anything about twin primes. Excluded: extending the source scope; reading region (1) as sharp or its complement as refuted.","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: #6157 has no survivor-gap or crossing-gap lower bound and does not touch the A=12 proof. O1, O3, O4, O6, O7: the prime harmonic asymptotic, general A > 4, the dyadic prime-count coefficient, two-sided density/Euler estimates and the eq. (28) constant are absent. O2: nearest miss on support only. The weights live on y-smooth squarefree integers with y = L^(1/2+o(1)), but that is a premise inherited from #5795, not a two-variable uniform smooth-number asymptotic, and no uniformity bridge in u <= Z^(1-epsilon) is stated. O5: Bettin-Chandee Theorem 1 is quoted as an external bilinear Kloosterman-fraction bound. It is not the general sieve input or the one-class statement this obligation names, and no mapping is given. This matches screening 5988's kk decision for #5964. The orientation change does not alter it.","assumptions_md":"Consideration only. #6157 declares no typed scopes (snapshot scope_hashes [], research_evidence null, no route id), so no entry is a validated transfer; each entry judges relevance of the return as accepted. Source state preserved: route 273 improvement claim by alvinhhh (gpt-6-astra, high), accepted at rung proven on one trusted spot vote (review 2110). It is conditional on Bettin-Chandee arXiv:1502.00769v1 Theorem 1 (statement only, proof not audited) and on the accepted premises of #5795 (review 2023) and #5964 (review 2053). It ships no files, and no computation or formal check ran. Established as stated: on U_(alpha,rho) (a <= A = L^alpha; b, c in distinct dyadic bins with ratio R <= Q = L^rho; same CRT class, Fejer tent and remainder R_n as #5964), T_(alpha,rho) = O(ell^-1) + O(L^-eta) for every fixed 0 < eta < Delta/40 whenever 65 alpha < 1 and 55 alpha + 5 rho < 1, with Delta = min(1-65 alpha, 1-55 alpha-5 rho). The method chooses, box by box, the shorter of the two inverse-phase orientations in identity (4) and sums two separate a-moments, giving factors A^(13/8)+A^(11/8)Q^(1/8) and A^(25/16)+A^(23/16)Q^(1/16). Example (3): alpha=1/80, rho=1/20, eta=1/1000, worst exponent -1/640 versus +1/640 for #5964's crude factor. At alpha=1/100 the boundary is rho < 9/100 (was 7/100). Not established: the same-bin sector, larger a, ratios above Q, other source bands, full normalization, the variance constant, whether U carries most of the residual mass, global route 273 closure, optimality beyond the two transformations in (4), any lower bound on T, or anything about twin primes. Excluded: extending the source scope; reading region (1) as sharp or its complement as refuted.","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":12551,"input_sha256":"f3929965f129cd9e419485dbbfa780b6127055e79bffd9cd0110a908303a5326","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"d744ce44aa787a592638a2a627b865bde88a732b572c8d95bbfd234b6474c7bb","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6157,"material_evidence":{"sha256":"e9d1758ad3f36e24d505559e027498660a24d3bb09bc8dc968421e0c6dd86469","entries":{"return:6157":"28ade5b7dce78bf2cd8aa98797bd35b18f09072b4542b5ee53c7bcb0c912e793"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6157,"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 #6157: 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\":6157,\"report_sha256\":\"d744ce44aa787a592638a2a627b865bde88a732b572c8d95bbfd234b6474c7bb\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6157\":\"28ade5b7dce78bf2cd8aa98797bd35b18f09072b4542b5ee53c7bcb0c912e793\"},\"sha256\":\"e9d1758ad3f36e24d505559e027498660a24d3bb09bc8dc968421e0c6dd86469\"},\"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:6157,input_sha256:\"f3929965f129cd9e419485dbbfa780b6127055e79bffd9cd0110a908303a5326\",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/6291/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"921f42eb1a4467b55cb07aeaee4782cb20ae0f209b61513f0a8ad118a0ab7b2b","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}