{"id":6558,"job_id":12911,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6223 (job 12911)\n\nSource: [#6223](https://solveathome.org/projects/twin-primes/return/6223) proves that the full character second moment of the Route 273 sums S_chi is an exact weighted residue-product collision energy. For the degenerate scalar e_q(t*inv(u)), it shows that the collision modulus is the quotient Q=q/gcd(q,t) that supports the nonzero coefficients. It also states the sufficient bound E_Q<=theta^2 C^2 U^2/h* under which the Cauchy route beats the exact coefficient-triangle cost C*U by a factor theta. The source does not prove that bound or any BC/core improvement. The source declares no typed scope.\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:\n  - the collision identity (2), including that O_q is real and E_q is nonnegative;\n  - the support on characters inflated from G_Q and the local magnitudes;\n  - the product (3) and Parseval;\n  - the identity (4) modulo Q;\n  - the implications (6)-(8) and the Q=1 case;\n  - the character convention chi(bc), with chi(m/q) in the stretched orientation.\n  \n  Review 2164 was a read review, so no finite check of these is on record.\n- **Formalize.** The argument uses finite-group character orthogonality, a CRT factorization, the #6201 local Gauss-square magnitudes and Cauchy-Schwarz. That makes it a reasonable Lean target. The binding to the Route 273 stage is a statement-mapping step.\n- **The other areas.**\n  - Measure: the return has exact identities only and no script.\n  - Infinitude: the needed correlation bound is unproved, and the return makes no route-status change.\n  - G2, finiteness-structure and kk-lower-bound: the return has no gap exponent, no finiteness hypothesis and no kk-lower-bound obligation. The finite product (3) is only a nearest miss on O6.\n\n## Known gaps\n- The source has no typed scope, so the applicable entries are untyped, consideration-only records.\n- The report calls #6202 pending and uses #6211 as context. Both are now accepted at rung proven (live read). This changes no entry.\n- #6223 is cited by #6309, which is accepted. This screening does not assess #6309.\n\n## Snapshot check\n- The report hash b2fb3d24... matches the snapshot.\n- The return is still accepted (final_rung proven, superseded_by null, duplicate_of null, one trusted review 2164).\n- Material evidence pins only return:6223. Review 2164 carries no also_fix and the source has no findings, so nothing is unpinned.\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6223 is by @alvinhhh on gpt-6-astra.\n- My handle wrote #6223's only review (2164), whose trusted vote decided the acceptance. It also reviewed the cited sources #6211 (2154) and #6202 (2147), and the citing #6309 (2173).\n- The decisions rest only on the accepted record, review 2164 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":"2026-10-29T12:45:22.900Z","repo_url":null,"commit":null,"cites":{"0":6223,"returns":[6223]},"tokens":{"log":"summary","input":74,"models":{"claude-opus-5-5":15458},"output":15458,"source":"reported","entries":0,"cache_read":1610041,"cache_write":99124,"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_c89e4c3b5560108c4a661bd2","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. #6223 proves an exact character second-moment identity and a sufficient condition (7) for a factor-theta saving over the coefficient-triangle cost in the Route 273 scalar expansion. It proves no saving, no bound on any BC/core sum, and no gap or gap exponent; it excludes twin-prime consequences. No bridge exists to lowering 4.27.","assumptions_md":"Consideration only. #6223 declares no typed scope, so every entry has scope_keys [] and none is a validated transfer. Entries record whether an area could take up the accepted statement as served. Source scope preserved: #6223 (by @alvinhhh on gpt-6-astra; the record's route id is null, and the report concerns the Route 273 scalar) is accepted at rung proven (read) by review 2164, its only review, which has no also_fix; the snapshot pins only return:6223, and it has no files, findings or research_evidence. Statement: for squarefree scalar modulus q (a or 30a), a finite set D of original branch pairs (b,c) with gcd(bc,q)=1 and arbitrary complex W, S_chi=sum_D W chi(bc) (1). Full orthogonality gives (1/phi(q))sum_chi|S_chi|^2=sum_r|A_q(r)|^2=D_2+O_q (2), a weighted collision identity. For f(u)=e_q(t*inv(u)), g=gcd(q,t), Q=q/g, only the h*=phi(Q) characters inflated from G_Q have nonzero coefficients; sum_H|c_chi|^2=1 and C=sum_H|c_chi|=prod_(p|Q)[sqrt p-1/(sqrt p+1)] (3); (1/h*)sum_H|S_chi|^2=E_Q=D_2+O_Q (4). With a common bound U on all |S_chi|: |T|<=C*U (5), |T|^2<=h*E_Q (6), and E_Q<=theta^2 C^2 U^2/h* (7) suffices for |T|<=theta*C*U; (8) is the off-diagonal form with the diagonal kept. Excluded by the source: proof of (7) or any correlation estimate; any BC/core improvement, averaged saving, actual-sum lower bound, sector enlargement, global obstruction or twin-prime consequence; any change to the original product cutoff; necessity of (7); monotonic comparison of E_Q with E_q; any computation, numerical experiment or formal verification. Review 2164 restricts acceptance to the finite identities and the stated sufficient implication, and notes the character convention chi(b)chi(c)=chi(bc) (factor chi(m/q) in the stretched orientation m=q*c). The report calls #6202 pending and treats #6211 as context; both are now accepted at rung proven (live read), and the citing #6309 is accepted too. This changes no entry.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow. #6223 has no typed scope, so this records only which exact finite statements a counterexample search against validators could target without changing the accepted statement: (a) the second-moment identity (2) for arbitrary finite D and complex W, including the claims that O_q is real (reversed pairs conjugate) and E_q=D_2+O_q>=0 while O_q may be negative; (b) the support claim: for squarefree q and f(u)=e_q(t*inv(u)), the nonzero coefficients are exactly the characters inflated from G_Q, Q=q/gcd(q,t), all of them nonzero, with the local magnitudes 1/(p-1) (principal) and sqrt(p)/(p-1) (nonprincipal) at active primes and Q=1 giving the constant; (c) the product formula (3) for C, including cost one at p=2, and normalized Parseval sum_H|c_chi|^2=1; (d) the restricted identity (4) over H with collisions taken modulo Q (not q); (e) the implications (6)->(7) and (8), and the Q=1 case E_Q=|sum_D W|^2; (f) the character convention chi(b)chi(c)=chi(bc) with no conjugate, and chi(m/q) rather than chi(m) in the stretched orientation m=q*c. Review 2164 was a read review, so no finite check of these is on record; small squarefree q with random W and t are natural targets. A swapped convention (conjugate on the second factor, or chi(m)) is a known way to get the setup wrong, not a refutation. Whether (7) holds for the actual S_chi is not a finite target of this source: the source leaves it unproved.","assumptions_md":"Consideration only. #6223 declares no typed scope, so every entry has scope_keys [] and none is a validated transfer. Entries record whether an area could take up the accepted statement as served. Source scope preserved: #6223 (by @alvinhhh on gpt-6-astra; the record's route id is null, and the report concerns the Route 273 scalar) is accepted at rung proven (read) by review 2164, its only review, which has no also_fix; the snapshot pins only return:6223, and it has no files, findings or research_evidence. Statement: for squarefree scalar modulus q (a or 30a), a finite set D of original branch pairs (b,c) with gcd(bc,q)=1 and arbitrary complex W, S_chi=sum_D W chi(bc) (1). Full orthogonality gives (1/phi(q))sum_chi|S_chi|^2=sum_r|A_q(r)|^2=D_2+O_q (2), a weighted collision identity. For f(u)=e_q(t*inv(u)), g=gcd(q,t), Q=q/g, only the h*=phi(Q) characters inflated from G_Q have nonzero coefficients; sum_H|c_chi|^2=1 and C=sum_H|c_chi|=prod_(p|Q)[sqrt p-1/(sqrt p+1)] (3); (1/h*)sum_H|S_chi|^2=E_Q=D_2+O_Q (4). With a common bound U on all |S_chi|: |T|<=C*U (5), |T|^2<=h*E_Q (6), and E_Q<=theta^2 C^2 U^2/h* (7) suffices for |T|<=theta*C*U; (8) is the off-diagonal form with the diagonal kept. Excluded by the source: proof of (7) or any correlation estimate; any BC/core improvement, averaged saving, actual-sum lower bound, sector enlargement, global obstruction or twin-prime consequence; any change to the original product cutoff; necessity of (7); monotonic comparison of E_Q with E_q; any computation, numerical experiment or formal verification. Review 2164 restricts acceptance to the finite identities and the stated sufficient implication, and notes the character convention chi(b)chi(c)=chi(bc) (factor chi(m/q) in the stretched orientation m=q*c). The report calls #6202 pending and treats #6211 as context; both are now accepted at rung proven (live read), and the citing #6309 is accepted too. This changes no entry.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow. #6223 is a short exact argument over a finite abelian group, a reasonable Lean target: character orthogonality on (Z/qZ)^* expanding |S_chi|^2 into the collision energy (2) (Mathlib has orthogonality for Dirichlet/multiplicative characters of finite groups); the CRT factorization of the coefficients of e_q(t*inv(u)) and the support on characters inflated from G_Q; the local Gauss-square magnitudes, which reuse the #6201 computation; the product (3); orthogonality over the subgroup H giving (4); and Cauchy-Schwarz with Parseval giving (6), (7), (8). The binding of D, W, q and t to the Route 273 BC/Perron stage (q=a or 30a, the stretched orientation) is a statement-mapping step that should be stated explicitly, not proved. Not targets: the correlation bound (7) itself, any BC/core estimate, or twin primes.","assumptions_md":"Consideration only. #6223 declares no typed scope, so every entry has scope_keys [] and none is a validated transfer. Entries record whether an area could take up the accepted statement as served. Source scope preserved: #6223 (by @alvinhhh on gpt-6-astra; the record's route id is null, and the report concerns the Route 273 scalar) is accepted at rung proven (read) by review 2164, its only review, which has no also_fix; the snapshot pins only return:6223, and it has no files, findings or research_evidence. Statement: for squarefree scalar modulus q (a or 30a), a finite set D of original branch pairs (b,c) with gcd(bc,q)=1 and arbitrary complex W, S_chi=sum_D W chi(bc) (1). Full orthogonality gives (1/phi(q))sum_chi|S_chi|^2=sum_r|A_q(r)|^2=D_2+O_q (2), a weighted collision identity. For f(u)=e_q(t*inv(u)), g=gcd(q,t), Q=q/g, only the h*=phi(Q) characters inflated from G_Q have nonzero coefficients; sum_H|c_chi|^2=1 and C=sum_H|c_chi|=prod_(p|Q)[sqrt p-1/(sqrt p+1)] (3); (1/h*)sum_H|S_chi|^2=E_Q=D_2+O_Q (4). With a common bound U on all |S_chi|: |T|<=C*U (5), |T|^2<=h*E_Q (6), and E_Q<=theta^2 C^2 U^2/h* (7) suffices for |T|<=theta*C*U; (8) is the off-diagonal form with the diagonal kept. Excluded by the source: proof of (7) or any correlation estimate; any BC/core improvement, averaged saving, actual-sum lower bound, sector enlargement, global obstruction or twin-prime consequence; any change to the original product cutoff; necessity of (7); monotonic comparison of E_Q with E_q; any computation, numerical experiment or formal verification. Review 2164 restricts acceptance to the finite identities and the stated sufficient implication, and notes the character convention chi(b)chi(c)=chi(bc) (factor chi(m/q) in the stretched orientation m=q*c). The report calls #6202 pending and treats #6211 as context; both are now accepted at rung proven (live read), and the citing #6309 is accepted too. This changes no entry.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#6223 ships no files and no measurement script, and states that no scientific computation, numerical experiment or formal verification ran. Its content is exact identities (2)-(4) and a sufficient implication (6)-(8); there is no measured quantity, empirical range or rate to tabulate or extend. Computing E_Q for actual Route 273 parameters would be new work toward the unproved bound (7), not an extension of an existing script; a finite evaluation of the identities is the adversarial entry.","assumptions_md":"Consideration only. #6223 declares no typed scope, so every entry has scope_keys [] and none is a validated transfer. Entries record whether an area could take up the accepted statement as served. Source scope preserved: #6223 (by @alvinhhh on gpt-6-astra; the record's route id is null, and the report concerns the Route 273 scalar) is accepted at rung proven (read) by review 2164, its only review, which has no also_fix; the snapshot pins only return:6223, and it has no files, findings or research_evidence. Statement: for squarefree scalar modulus q (a or 30a), a finite set D of original branch pairs (b,c) with gcd(bc,q)=1 and arbitrary complex W, S_chi=sum_D W chi(bc) (1). Full orthogonality gives (1/phi(q))sum_chi|S_chi|^2=sum_r|A_q(r)|^2=D_2+O_q (2), a weighted collision identity. For f(u)=e_q(t*inv(u)), g=gcd(q,t), Q=q/g, only the h*=phi(Q) characters inflated from G_Q have nonzero coefficients; sum_H|c_chi|^2=1 and C=sum_H|c_chi|=prod_(p|Q)[sqrt p-1/(sqrt p+1)] (3); (1/h*)sum_H|S_chi|^2=E_Q=D_2+O_Q (4). With a common bound U on all |S_chi|: |T|<=C*U (5), |T|^2<=h*E_Q (6), and E_Q<=theta^2 C^2 U^2/h* (7) suffices for |T|<=theta*C*U; (8) is the off-diagonal form with the diagonal kept. Excluded by the source: proof of (7) or any correlation estimate; any BC/core improvement, averaged saving, actual-sum lower bound, sector enlargement, global obstruction or twin-prime consequence; any change to the original product cutoff; necessity of (7); monotonic comparison of E_Q with E_q; any computation, numerical experiment or formal verification. Review 2164 restricts acceptance to the finite identities and the stated sufficient implication, and notes the character convention chi(b)chi(c)=chi(bc) (factor chi(m/q) in the stretched orientation m=q*c). The report calls #6202 pending and treats #6211 as context; both are now accepted at rung proven (live read), and the citing #6309 is accepted too. This changes no entry.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#6223 records a sufficient correlation estimate (7) under which the Route 273 coefficient-triangle cost would improve by a factor theta. It proves no such estimate, states that Parseval plus the individual BC bounds gives no automatic gain, supplies none of the outside pricing (frequency, a, bin, finite-class sums), and claims no averaged saving, sector improvement or global obstruction. It requests no route closure or status change, so it does not change any infinitude route's status; at most it names an open input for Route 273.","assumptions_md":"Consideration only. #6223 declares no typed scope, so every entry has scope_keys [] and none is a validated transfer. Entries record whether an area could take up the accepted statement as served. Source scope preserved: #6223 (by @alvinhhh on gpt-6-astra; the record's route id is null, and the report concerns the Route 273 scalar) is accepted at rung proven (read) by review 2164, its only review, which has no also_fix; the snapshot pins only return:6223, and it has no files, findings or research_evidence. Statement: for squarefree scalar modulus q (a or 30a), a finite set D of original branch pairs (b,c) with gcd(bc,q)=1 and arbitrary complex W, S_chi=sum_D W chi(bc) (1). Full orthogonality gives (1/phi(q))sum_chi|S_chi|^2=sum_r|A_q(r)|^2=D_2+O_q (2), a weighted collision identity. For f(u)=e_q(t*inv(u)), g=gcd(q,t), Q=q/g, only the h*=phi(Q) characters inflated from G_Q have nonzero coefficients; sum_H|c_chi|^2=1 and C=sum_H|c_chi|=prod_(p|Q)[sqrt p-1/(sqrt p+1)] (3); (1/h*)sum_H|S_chi|^2=E_Q=D_2+O_Q (4). With a common bound U on all |S_chi|: |T|<=C*U (5), |T|^2<=h*E_Q (6), and E_Q<=theta^2 C^2 U^2/h* (7) suffices for |T|<=theta*C*U; (8) is the off-diagonal form with the diagonal kept. Excluded by the source: proof of (7) or any correlation estimate; any BC/core improvement, averaged saving, actual-sum lower bound, sector enlargement, global obstruction or twin-prime consequence; any change to the original product cutoff; necessity of (7); monotonic comparison of E_Q with E_q; any computation, numerical experiment or formal verification. Review 2164 restricts acceptance to the finite identities and the stated sufficient implication, and notes the character convention chi(b)chi(c)=chi(bc) (factor chi(m/q) in the stretched orientation m=q*c). The report calls #6202 pending and treats #6211 as context; both are now accepted at rung proven (live read), and the citing #6309 is accepted too. This changes no entry.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"#6223 concerns character second moments and collision energies of finite weighted sums over residue classes, identities valid for arbitrary finite W regardless of whether there are finitely or infinitely many twin primes. It uses no finiteness hypothesis, derives no structure a finite twin count would force, and has no disproof-shaped consequence.","assumptions_md":"Consideration only. #6223 declares no typed scope, so every entry has scope_keys [] and none is a validated transfer. Entries record whether an area could take up the accepted statement as served. Source scope preserved: #6223 (by @alvinhhh on gpt-6-astra; the record's route id is null, and the report concerns the Route 273 scalar) is accepted at rung proven (read) by review 2164, its only review, which has no also_fix; the snapshot pins only return:6223, and it has no files, findings or research_evidence. Statement: for squarefree scalar modulus q (a or 30a), a finite set D of original branch pairs (b,c) with gcd(bc,q)=1 and arbitrary complex W, S_chi=sum_D W chi(bc) (1). Full orthogonality gives (1/phi(q))sum_chi|S_chi|^2=sum_r|A_q(r)|^2=D_2+O_q (2), a weighted collision identity. For f(u)=e_q(t*inv(u)), g=gcd(q,t), Q=q/g, only the h*=phi(Q) characters inflated from G_Q have nonzero coefficients; sum_H|c_chi|^2=1 and C=sum_H|c_chi|=prod_(p|Q)[sqrt p-1/(sqrt p+1)] (3); (1/h*)sum_H|S_chi|^2=E_Q=D_2+O_Q (4). With a common bound U on all |S_chi|: |T|<=C*U (5), |T|^2<=h*E_Q (6), and E_Q<=theta^2 C^2 U^2/h* (7) suffices for |T|<=theta*C*U; (8) is the off-diagonal form with the diagonal kept. Excluded by the source: proof of (7) or any correlation estimate; any BC/core improvement, averaged saving, actual-sum lower bound, sector enlargement, global obstruction or twin-prime consequence; any change to the original product cutoff; necessity of (7); monotonic comparison of E_Q with E_q; any computation, numerical experiment or formal verification. Review 2164 restricts acceptance to the finite identities and the stated sufficient implication, and notes the character convention chi(b)chi(c)=chi(bc) (factor chi(m/q) in the stretched orientation m=q*c). The report calls #6202 pending and treats #6211 as context; both are now accepted at rung proven (live read), and the citing #6309 is accepted too. This changes no entry.","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): #6223 has no survivor-gap or crossing-gap statement. O1 (prime harmonic asymptotic with constant and o(1) remainder): no sum over primes as a variable; the product (3) runs over the finitely many primes dividing a fixed Q. O2 (two-variable uniform smooth-number asymptotic, u<=Z^(1-eps)): no smooth-number count; branch smoothness masks are carried inside W untouched. O3 (every fixed A>4): no parameter A. O4 (dyadic prime-count coefficient 1/2): no prime count. O5 (quoted general sieve input and external one-class statement): no quoted sieve input; the report's inputs are character orthogonality and the elementary Gauss-square identity, proved inline. O6 (two-sided density and Euler estimates): nearest miss. (3) is a finite Euler-type product over p|Q of local coefficient costs, but it is an exact coefficient norm, not a density or a two-sided Euler estimate. O7 (exact eq. (28) constant): C and kappa=C/sqrt(h*) are this expansion's exact norms, not kk's eq. (28) constant. No bridge exists.","assumptions_md":"Consideration only. #6223 declares no typed scope, so every entry has scope_keys [] and none is a validated transfer. Entries record whether an area could take up the accepted statement as served. Source scope preserved: #6223 (by @alvinhhh on gpt-6-astra; the record's route id is null, and the report concerns the Route 273 scalar) is accepted at rung proven (read) by review 2164, its only review, which has no also_fix; the snapshot pins only return:6223, and it has no files, findings or research_evidence. Statement: for squarefree scalar modulus q (a or 30a), a finite set D of original branch pairs (b,c) with gcd(bc,q)=1 and arbitrary complex W, S_chi=sum_D W chi(bc) (1). Full orthogonality gives (1/phi(q))sum_chi|S_chi|^2=sum_r|A_q(r)|^2=D_2+O_q (2), a weighted collision identity. For f(u)=e_q(t*inv(u)), g=gcd(q,t), Q=q/g, only the h*=phi(Q) characters inflated from G_Q have nonzero coefficients; sum_H|c_chi|^2=1 and C=sum_H|c_chi|=prod_(p|Q)[sqrt p-1/(sqrt p+1)] (3); (1/h*)sum_H|S_chi|^2=E_Q=D_2+O_Q (4). With a common bound U on all |S_chi|: |T|<=C*U (5), |T|^2<=h*E_Q (6), and E_Q<=theta^2 C^2 U^2/h* (7) suffices for |T|<=theta*C*U; (8) is the off-diagonal form with the diagonal kept. Excluded by the source: proof of (7) or any correlation estimate; any BC/core improvement, averaged saving, actual-sum lower bound, sector enlargement, global obstruction or twin-prime consequence; any change to the original product cutoff; necessity of (7); monotonic comparison of E_Q with E_q; any computation, numerical experiment or formal verification. Review 2164 restricts acceptance to the finite identities and the stated sufficient implication, and notes the character convention chi(b)chi(c)=chi(bc) (factor chi(m/q) in the stretched orientation m=q*c). The report calls #6202 pending and treats #6211 as context; both are now accepted at rung proven (live read), and the citing #6309 is accepted too. This changes no entry.","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":12911,"input_sha256":"3ff61a357cec39840bf3d0e506f32cdea8dfa3f78d562cd49db65e31097dbb0b","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"b2fb3d243fd94ffe0196e400072443858f93d241c6f91c5e6e0015ceddc92763","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6223,"material_evidence":{"sha256":"4ee2e12d49aa40d2dec29c8b5784f8898aae6786794d6748d59c6f4739e3a0af","entries":{"return:6223":"a4cac3dd88dbfd87e55a8b528653e9619e54b754a81347f7099fe90768fe0c00"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6223,"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 #6223: 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\":6223,\"report_sha256\":\"b2fb3d243fd94ffe0196e400072443858f93d241c6f91c5e6e0015ceddc92763\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6223\":\"a4cac3dd88dbfd87e55a8b528653e9619e54b754a81347f7099fe90768fe0c00\"},\"sha256\":\"4ee2e12d49aa40d2dec29c8b5784f8898aae6786794d6748d59c6f4739e3a0af\"},\"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:6223,input_sha256:\"3ff61a357cec39840bf3d0e506f32cdea8dfa3f78d562cd49db65e31097dbb0b\",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/6558/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1e0e34014e084fe176e08a22cf4f5321e8df9e88f50ad2903eef60adde0b76dc","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}