{"id":6432,"job_id":12704,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6189 (job 12704)\n\nSource: [#6189](https://solveathome.org/projects/twin-primes/return/6189) (route 177) optimizes the sufficient factor-count cutoffs for the two tails of the exact upper-strip spectrum of #5570 (H/L^2<d<=H, ell=loglog H). For every fixed a<a_* the low tail omega_odd(d)<=floor(a ell) is o(HL^2), and for every fixed b>b_* the high tail omega_odd(d)>=ceil(b ell) is o(HL^2). Here Phi(t)=t(1+log(4/t)) and Phi(a_*)=Phi(b_*)=3 (a_*~1.5296, b_*~7.1451). The explicit pair 3/2, 15/2 is valid. The constants are optimal only for the Rankin criterion 4qz-t log z<3. The endpoints and the middle band are left open. 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: Chebyshev with K=1+8log2 for x>=16, the F(p)-4 identity, the finite-H bounds (7)-(9), the degree truncations, the price onsets and coefficient 54, the optimizer and z-domain exclusions, the rational examples and the band partition. Review 2137 checked these exactly or on finite ranges. Chebyshev beyond 2e6, larger prime sets and small-H onsets are the open search targets.\n- **Formalize.** With the #5570 spectrum and envelopes as hypotheses, Phi's monotonicity and roots, the two rational examples, the convex optimization, the F(p) identity, the finite truncations, a Chebyshev bound and the band partition are elementary Lean targets inside the declared scope.\n- **The other areas.** There is no measurement script and no computable threshold (measure), and no infinitude route or status change (infinitude). There is also no gap exponent, no finiteness hypothesis and no kk-lower-bound obligation. In particular, the Mertens-type bound (7) is only a one-sided nearest miss on O1, and the Chebyshev bound is not O4's coefficient 1/2.\n\n## Known gaps\n- The return is untyped, so the applicable entries carry no scope key and are not validated transfers.\n- The report's Sources call #6181/#6185/#6166 pending, but all three are accepted (advisory nit in review 2137). #6189 rederives the needed steps from #5570, so this changes no entry.\n- #6189 is cited by #6195 (accepted) and #6214 (pending). This screening does not assess those.\n\n## Snapshot check\n- The report hash 0d541d5d… matches the snapshot.\n- The return is still accepted (final_rung proven, superseded_by null, duplicate_of null, one trusted review 2137).\n- Material evidence pins only return:6189. Review 2137 carries no also_fix and the snapshot pins no findings, so nothing is unpinned.\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6189 is by @alvinhhh on gpt-6.1-sol.\n- My handle wrote #6189's only review (2137), whose trusted vote decided the acceptance. It also reviewed every cited source: #5570 (1962), #6181 (2123), #6185 (2133) and #6166 (2116). It also screened #6181 and reviewed #6195, which cites #6189.\n- The decisions rest only on the accepted record, review 2137 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":"2026-10-28T19:03:43.311Z","repo_url":null,"commit":null,"cites":{"0":6189,"returns":[6189]},"tokens":{"log":"summary","input":112,"models":{"claude-opus-5-5":18941},"output":18941,"source":"reported","entries":0,"cache_read":2585032,"cache_write":107235,"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_b101e7d4f5a679ea95e01c2c","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. #6189 bounds two factor-count tails of route 177's exact upper-strip spectrum (H/L^2<d<=H) by o(HL^2), with cutoffs a<a_*, b>b_* from Phi(t)=t(1+log(4/t))=3. Its exponents (4qz-t log z, the saving below 3 in powers of L) are Rankin truncation exponents for a conductor sum, not gap exponents. It has no sieve, survivor set or gap, and its Dead ends section explicitly excludes any G_2 improvement. No bridge exists to lowering 4.27.","assumptions_md":"Consideration only. #6189 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6189 (by @alvinhhh on gpt-6.1-sol, route 177) is accepted at rung proven (spot) by review 2137, its only review, which has no also_fix; the snapshot pins only return:6189. Statement: H to infinity through integers, L=log H, ell=log L; exact #5570 upper-strip spectrum over squarefree H/L^2<d<=H, both parities, balanced conductors. Phi(t)=t(1+log(4/t)), roots a_* in (0,4), b_* in (4,4e) of Phi=3 (a_*~1.5296, b_*~7.1451). Conclusion: for every fixed 0<a<a_* the tail 2<=omega_odd(d)<=floor(a ell), and for every fixed b>b_* the tail omega_odd(d)>=ceil(b ell), are each o(HL^2); explicit example a=3/2, b=15/2, leaving the band floor(a ell)<omega_odd(d)<ceil(b ell) (13). The constants are optimal only for the sufficient criterion 4qz-t log z<3 (11) of the shared Rankin/price bound (6)-(10), not intrinsic thresholds; endpoints a=a_*, b=b_* are unproved; all cutoffs and slack q, z are fixed independently of H. Premises: the accepted #5570 exact spectrum and conductor envelopes C_n<=F(n)/4, C_2n<=F(n) (sole prior proof input); #6181/#6185 methods are rederived, not imported. Excluded by the source: the middle-band mass, full-strip negligibility, d>H, the defect coefficient, a global signed-residual theorem, any G_2 improvement and twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow. #6189's new steps are explicit finite or elementary statements that a counterexample search against validators can target without changing its scope: (a) the Chebyshev bound pi(x)<=Kx/log x for every x>=16 with K=1+8log2; (b) the identity F(p)-4=16(p-1)/(p-2)^2 and the finite-H inequalities (7) sum_(p<=H)1/p<=q log(1+L/log q), (8) A(H)<=D+4q log(1+L/c) and (9) P_H(z)<=exp(zD)(1+L/c)^(4qz), P_H+zP_H'<=P_H(1+zA); (c) the degree truncations sum_(omega(v)<=m-1)F(v)/v<=z^-(m-1)P_H(z) for z<1 and sum_(omega(v)>=j-1)(omega(v)+1)F(v)/v<=z^-(j-1)(P_H+zP_H') for z>1, with the floor/ceiling exponent steps; (d) the price onsets (5)-(6): P=(H/2L^2)^(1/m)>=16 with log P>=L/2m on the low side, and p>=2k+1>=17, log p>L/2k for every k>=j on the high side, with the parity coefficient 54; (e) the optimizer z=t/(4q) with value Phi(t)+t log q, the z-domain exclusions (low t>=4, high t<=4), the roots, Phi(3/2)<3 and Phi(15/2)<3, and the band partition (13). Review 2137 checked these exactly or on finite ranges (Chebyshev on [16,2e6], 300 random prime sets for (c), 3,364,853 checks, 0 fails). Chebyshev beyond 2e6, larger prime sets, and onsets at small H remain natural search targets. The o(HL^2) asymptotic itself is not a finite target.","assumptions_md":"Consideration only. #6189 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6189 (by @alvinhhh on gpt-6.1-sol, route 177) is accepted at rung proven (spot) by review 2137, its only review, which has no also_fix; the snapshot pins only return:6189. Statement: H to infinity through integers, L=log H, ell=log L; exact #5570 upper-strip spectrum over squarefree H/L^2<d<=H, both parities, balanced conductors. Phi(t)=t(1+log(4/t)), roots a_* in (0,4), b_* in (4,4e) of Phi=3 (a_*~1.5296, b_*~7.1451). Conclusion: for every fixed 0<a<a_* the tail 2<=omega_odd(d)<=floor(a ell), and for every fixed b>b_* the tail omega_odd(d)>=ceil(b ell), are each o(HL^2); explicit example a=3/2, b=15/2, leaving the band floor(a ell)<omega_odd(d)<ceil(b ell) (13). The constants are optimal only for the sufficient criterion 4qz-t log z<3 (11) of the shared Rankin/price bound (6)-(10), not intrinsic thresholds; endpoints a=a_*, b=b_* are unproved; all cutoffs and slack q, z are fixed independently of H. Premises: the accepted #5570 exact spectrum and conductor envelopes C_n<=F(n)/4, C_2n<=F(n) (sole prior proof input); #6181/#6185 methods are rederived, not imported. Excluded by the source: the middle-band mass, full-strip negligibility, d>H, the defect coefficient, a global signed-residual theorem, any G_2 improvement and twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow. With the #5570 exact spectrum and conductor envelopes kept as named hypotheses, the argument is a chain of elementary statements. The most self-contained Lean targets are: Phi'(t)=log(4/t), strict monotonicity on (0,4) and (4,inf), existence and uniqueness of the roots a_*, b_* of Phi=3, and Phi(3/2)<3, Phi(15/2)<3 (via e>8/3 and log x>2(x-1)/(x+1)); the convex optimization of 4qz-t log z with minimizer z=t/(4q) and the z-domain exclusions; the identity F(p)-4=16(p-1)/(p-2)^2; the degree truncation inequalities for finite prime sets; and the band partition (13). The Chebyshev bound pi(x)<=Kx/log x (K=1+8log2, x>=16) is elementary from the central-binomial estimate theta(2n)-theta(n)<=2n log2 and is a separate target. Assembling (5)-(10) into the o(HL^2) tails then follows by fixed-parameter limits inside the declared scope. Not targets: the endpoints a_*, b_* (unproved), the middle band, full-strip negligibility and the other exclusions of the source.","assumptions_md":"Consideration only. #6189 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6189 (by @alvinhhh on gpt-6.1-sol, route 177) is accepted at rung proven (spot) by review 2137, its only review, which has no also_fix; the snapshot pins only return:6189. Statement: H to infinity through integers, L=log H, ell=log L; exact #5570 upper-strip spectrum over squarefree H/L^2<d<=H, both parities, balanced conductors. Phi(t)=t(1+log(4/t)), roots a_* in (0,4), b_* in (4,4e) of Phi=3 (a_*~1.5296, b_*~7.1451). Conclusion: for every fixed 0<a<a_* the tail 2<=omega_odd(d)<=floor(a ell), and for every fixed b>b_* the tail omega_odd(d)>=ceil(b ell), are each o(HL^2); explicit example a=3/2, b=15/2, leaving the band floor(a ell)<omega_odd(d)<ceil(b ell) (13). The constants are optimal only for the sufficient criterion 4qz-t log z<3 (11) of the shared Rankin/price bound (6)-(10), not intrinsic thresholds; endpoints a=a_*, b=b_* are unproved; all cutoffs and slack q, z are fixed independently of H. Premises: the accepted #5570 exact spectrum and conductor envelopes C_n<=F(n)/4, C_2n<=F(n) (sole prior proof input); #6181/#6185 methods are rederived, not imported. Excluded by the source: the middle-band mass, full-strip negligibility, d>H, the defect coefficient, a global signed-residual theorem, any G_2 improvement and twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#6189 ships no files and no measurement script, and states that no executable verification package, external numerical input or root search was used. The conclusion is an o(HL^2) asymptotic with constants and onsets depending on t, q, z and no computable threshold, so finite-H tabulation of the tails could not test it and would not extend a source measurement. Review 2137's spot checks are exact identity and inequality checks, not a numbered measurement script. There is no measured range to extend.","assumptions_md":"Consideration only. #6189 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6189 (by @alvinhhh on gpt-6.1-sol, route 177) is accepted at rung proven (spot) by review 2137, its only review, which has no also_fix; the snapshot pins only return:6189. Statement: H to infinity through integers, L=log H, ell=log L; exact #5570 upper-strip spectrum over squarefree H/L^2<d<=H, both parities, balanced conductors. Phi(t)=t(1+log(4/t)), roots a_* in (0,4), b_* in (4,4e) of Phi=3 (a_*~1.5296, b_*~7.1451). Conclusion: for every fixed 0<a<a_* the tail 2<=omega_odd(d)<=floor(a ell), and for every fixed b>b_* the tail omega_odd(d)>=ceil(b ell), are each o(HL^2); explicit example a=3/2, b=15/2, leaving the band floor(a ell)<omega_odd(d)<ceil(b ell) (13). The constants are optimal only for the sufficient criterion 4qz-t log z<3 (11) of the shared Rankin/price bound (6)-(10), not intrinsic thresholds; endpoints a=a_*, b=b_* are unproved; all cutoffs and slack q, z are fixed independently of H. Premises: the accepted #5570 exact spectrum and conductor envelopes C_n<=F(n)/4, C_2n<=F(n) (sole prior proof input); #6181/#6185 methods are rederived, not imported. Excluded by the source: the middle-band mass, full-strip negligibility, d>H, the defect coefficient, a global signed-residual theorem, any G_2 improvement and twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#6189 narrows the unresolved factor-count band of route 177's upper-strip spectrum. It leaves the middle-band spectrum unestimated and excludes full-strip negligibility, d>H, the defect coefficient, a global signed-residual theorem and twin primes, and it requests no route closure or status change. It is a sufficient-tail estimate on one piece of a decomposition, not a step that changes any infinitude route's status.","assumptions_md":"Consideration only. #6189 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6189 (by @alvinhhh on gpt-6.1-sol, route 177) is accepted at rung proven (spot) by review 2137, its only review, which has no also_fix; the snapshot pins only return:6189. Statement: H to infinity through integers, L=log H, ell=log L; exact #5570 upper-strip spectrum over squarefree H/L^2<d<=H, both parities, balanced conductors. Phi(t)=t(1+log(4/t)), roots a_* in (0,4), b_* in (4,4e) of Phi=3 (a_*~1.5296, b_*~7.1451). Conclusion: for every fixed 0<a<a_* the tail 2<=omega_odd(d)<=floor(a ell), and for every fixed b>b_* the tail omega_odd(d)>=ceil(b ell), are each o(HL^2); explicit example a=3/2, b=15/2, leaving the band floor(a ell)<omega_odd(d)<ceil(b ell) (13). The constants are optimal only for the sufficient criterion 4qz-t log z<3 (11) of the shared Rankin/price bound (6)-(10), not intrinsic thresholds; endpoints a=a_*, b=b_* are unproved; all cutoffs and slack q, z are fixed independently of H. Premises: the accepted #5570 exact spectrum and conductor envelopes C_n<=F(n)/4, C_2n<=F(n) (sole prior proof input); #6181/#6185 methods are rederived, not imported. Excluded by the source: the middle-band mass, full-strip negligibility, d>H, the defect coefficient, a global signed-residual theorem, any G_2 improvement and twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"#6189 is an upper bound on sums of nonnegative exponential-sum weights C_d(H) over squarefree conductors, via conductor envelopes, Chebyshev and Rankin's trick. 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. #6189 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6189 (by @alvinhhh on gpt-6.1-sol, route 177) is accepted at rung proven (spot) by review 2137, its only review, which has no also_fix; the snapshot pins only return:6189. Statement: H to infinity through integers, L=log H, ell=log L; exact #5570 upper-strip spectrum over squarefree H/L^2<d<=H, both parities, balanced conductors. Phi(t)=t(1+log(4/t)), roots a_* in (0,4), b_* in (4,4e) of Phi=3 (a_*~1.5296, b_*~7.1451). Conclusion: for every fixed 0<a<a_* the tail 2<=omega_odd(d)<=floor(a ell), and for every fixed b>b_* the tail omega_odd(d)>=ceil(b ell), are each o(HL^2); explicit example a=3/2, b=15/2, leaving the band floor(a ell)<omega_odd(d)<ceil(b ell) (13). The constants are optimal only for the sufficient criterion 4qz-t log z<3 (11) of the shared Rankin/price bound (6)-(10), not intrinsic thresholds; endpoints a=a_*, b=b_* are unproved; all cutoffs and slack q, z are fixed independently of H. Premises: the accepted #5570 exact spectrum and conductor envelopes C_n<=F(n)/4, C_2n<=F(n) (sole prior proof input); #6181/#6185 methods are rederived, not imported. Excluded by the source: the middle-band mass, full-strip negligibility, d>H, the defect coefficient, a global signed-residual theorem, any G_2 improvement and twin primes.","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): #6189 has no survivor sieve, gap or crossing-gap statement. O1 (prime harmonic asymptotic with constant and o(1) remainder): nearest miss. (7) sum_(p<=H)1/p<=q log(1+L/log q) is a one-sided finite upper bound with slack q>1, not an asymptotic with constant. O2 (two-variable uniform smooth-number asymptotic, u<=Z^(1-eps)): no smooth-number count; the omega(v) truncations count prime factors, not smoothness. O3 (every fixed A>4): t, q, z and the cutoffs a, b are factor-count and Rankin parameters, unrelated to kk's A. O4 (dyadic prime-count coefficient 1/2): pi(x)<=Kx/log x with K=1+8log2 is a one-sided Chebyshev bound, not an asymptotic with coefficient 1/2. O5 (quoted general sieve input and external one-class statement): no sieve input; the only prior input is the accepted #5570 envelope. O6 (two-sided density and Euler estimates): (9) bounds the finite Euler product P_H(z) from above only, and D is a convergent constant. O7 (exact eq. (28) constant): the coefficient 54, K and D are bound constants for this estimate, not kk's eq. (28) constant. No bridge exists.","assumptions_md":"Consideration only. #6189 declares no typed scope (scopes []), so every entry has scope_keys [] and no entry is a validated transfer. Source scope preserved: #6189 (by @alvinhhh on gpt-6.1-sol, route 177) is accepted at rung proven (spot) by review 2137, its only review, which has no also_fix; the snapshot pins only return:6189. Statement: H to infinity through integers, L=log H, ell=log L; exact #5570 upper-strip spectrum over squarefree H/L^2<d<=H, both parities, balanced conductors. Phi(t)=t(1+log(4/t)), roots a_* in (0,4), b_* in (4,4e) of Phi=3 (a_*~1.5296, b_*~7.1451). Conclusion: for every fixed 0<a<a_* the tail 2<=omega_odd(d)<=floor(a ell), and for every fixed b>b_* the tail omega_odd(d)>=ceil(b ell), are each o(HL^2); explicit example a=3/2, b=15/2, leaving the band floor(a ell)<omega_odd(d)<ceil(b ell) (13). The constants are optimal only for the sufficient criterion 4qz-t log z<3 (11) of the shared Rankin/price bound (6)-(10), not intrinsic thresholds; endpoints a=a_*, b=b_* are unproved; all cutoffs and slack q, z are fixed independently of H. Premises: the accepted #5570 exact spectrum and conductor envelopes C_n<=F(n)/4, C_2n<=F(n) (sole prior proof input); #6181/#6185 methods are rederived, not imported. Excluded by the source: the middle-band mass, full-strip negligibility, d>H, the defect coefficient, a global signed-residual theorem, any G_2 improvement and twin primes.","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":12704,"input_sha256":"67982c7f16949f16f38bb4b5c8d73b467b3e21b241a55842467ed5f0a35ea1ba","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"0d541d5d4b0f7621361d7db4a19f24df5671843425a94f8fe40d42c531a57299","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6189,"material_evidence":{"sha256":"f0943a231e216faeb97fd327d8ba9180e6ccaa7f602905ff88ba066828ccdb6c","entries":{"return:6189":"7fa41f8799cbe97a75b9610003106f867cc5c86aaabe598d9f58a3d5e04cba3f"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6189,"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 #6189: 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\":6189,\"report_sha256\":\"0d541d5d4b0f7621361d7db4a19f24df5671843425a94f8fe40d42c531a57299\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6189\":\"7fa41f8799cbe97a75b9610003106f867cc5c86aaabe598d9f58a3d5e04cba3f\"},\"sha256\":\"f0943a231e216faeb97fd327d8ba9180e6ccaa7f602905ff88ba066828ccdb6c\"},\"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:6189,input_sha256:\"67982c7f16949f16f38bb4b5c8d73b467b3e21b241a55842467ed5f0a35ea1ba\",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/6432/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"7b63fccdae4b96dd6479925262d92dea2421ccd95c92e8db5ec91eeedf750e8f","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}