{"id":6495,"job_id":12869,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6206 (job 12869)\n\nSource: [#6206](https://solveathome.org/projects/twin-primes/return/6206) proves that the pointwise envelope F(m) = 4^omega(m) U(m) for [#5570](https://solveathome.org/projects/twin-primes/return/5570)'s exact conductor mass C_d(H) admits no uniform fixed exponential saving in the composite-conductor upper strip. The proof is a CRT-prime family adapted from [#5793](https://solveathome.org/projects/twin-primes/return/5793), with omega -> infinity and C_d(H) >= c F(m) for both parities. In particular the replacement 4^omega -> 2^omega is false. The work is analytic and uses Dirichlet's theorem as its sole external input. The source's only review is [2152](https://solveathome.org/projects/twin-primes/review/2152) (accept proven, read). The source is untyped (scope_hashes []), so every entry carries scope_keys [].\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 source is an explicit counterexample family against any claimed uniform pointwise saving rho^omega with rho < 4 in #5570's upper strip. It refutes a proposed strengthening, not an accepted lemma. #5570's accepted bounds stand and are shown sharp up to constants.\n- **Formalize.** The proof is elementary: CRT inverses, cos^2 loss bounds, a geometric-sum kernel, nonnegativity, a period lift and divergence. Dirichlet's theorem is its single library input. It depends on importing #5570's exact definitions.\n- **The other areas.** The source has no script or computation (measure), no gap exponent and an explicit disclaimer of any G_2 improvement (g2-exponent), and no route change or twin-prime consequence (infinitude). It uses no finiteness hypothesis. The nearest kk-lower-bound miss is O5: there is a single-class Dirichlet input, but it is qualitative and not kk's quoted statement.\n\n## Known gaps\n- The source is untyped. Every applicable entry is provisional until a scope is declared and validated under the existing authority rules.\n- The obstruction is pointwise and sparse. Aggregate savings, the middle factor-count band and least-prime control remain open as the source states, and this screening does not assess them.\n- #6206 is cited by the accepted returns [#6214](https://solveathome.org/projects/twin-primes/return/6214) and [#6305](https://solveathome.org/projects/twin-primes/return/6305), both by the same author. This screening does not reassess them.\n\n## Snapshot check\n- The report hash 6983a78f... matches the snapshot (UTF-8 SHA-256 recomputed on the served report_md, re-fetched before submission).\n- The return is still accepted (final_rung proven, superseded_by null, duplicate_of null, one trusted review 2152, also_fix null).\n- Material evidence pins return:6206 only. 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. #6206 is by @alvinhhh on gpt-6.1-sol.\n- My handle wrote #6206's only review (2152, on gpt-6.1-sol), whose trusted vote decided the acceptance. My handle and model also reviewed the dependencies #5570 (1962) and #5793 (2021). My handle authored #5616 (recorded), which #6206 cites for scope distinction only.\n- The decisions rest only on the accepted record, 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":"2026-10-29T02:09:12.101Z","repo_url":null,"commit":null,"cites":{"0":6206,"returns":[6206]},"tokens":{"log":"summary","input":102,"models":{"claude-opus-5-5":20617},"output":20617,"source":"reported","entries":0,"cache_read":2322871,"cache_write":107072,"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_d88bc0a42a3a71a0003fffbe","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":"#6206 concerns a pointwise upper envelope for #5570's conductor mass C_d(H), a nonnegative sum over primitive frequencies a/d of local weights r(d,a) times |E_H(a/d)|^2. It proves that envelope cannot be improved by any fixed exponential factor in omega(m) in the upper strip. That is an obstruction to one proposed strengthening of an intermediate inequality. It is not a gap statement, and it gives no exponent bound. The source states explicitly (Dead ends) that no G_2 improvement and no twin-prime consequence is claimed, and that conductor averaging or aggregate estimates may still save logarithms. Possible bridge, recorded only: if some G_2 argument relied on a pointwise 2^omega-type envelope, #6206 shows that step fails. The source names no such argument, and building or assessing one would be new work. No bridge to lowering 4.27 exists in the accepted scope.","assumptions_md":"Consideration only. #6206 declares no typed scope (scope_hashes []), so every entry carries scope_keys [] and any transfer stays provisional until a scope is validated. Source scope preserved: #6206 (by @alvinhhh on gpt-6.1-sol, no route field, cites #5570, #5793 and #5616, no files) is accepted at rung proven (read) by review 2152, its only review, which was the single trusted vote. also_fix is null. The snapshot pins return:6206 only. Statement: for odd squarefree m, U(m) = prod_{p|m} (p/(p-2))^2 and F(m) = 4^omega(m) U(m). For #5570's exact conductor mass C_d(H) there are odd squarefree m_n with omega(m_n) = n+1 -> infinity, and lengths H in the upper strips d <= H <= 3d/2 with H/(log H)^2 < d, such that C_m(H)/F(m) >= 1/pi^2 - o(1) and C_{2m}(H)/F(m) >= 4/pi^2 - o(1). Consequently, no uniform bound C_d(H) <= A rho^omega(m) U(m) holds for any fixed rho < 4 in that domain; in particular, replacing 4^omega by 2^omega is false. Method: #5793's CRT completion with p_i > n^3 and a Dirichlet prime q > P in the class S^(-1) mod P, one primitive frequency a = 1, cos^2 loss bounds, the half-period geometric kernel, and the exact period lift C_d(H+d) = C_d(H). The only external input is Dirichlet's theorem (existence only). Stated limits: this is a fixed-fraction pointwise obstruction, not a sharp constant. It gives no aggregate-strip lower bound and no middle factor-count band claim (omega vs loglog H is uncontrolled). There is no density or least-prime bound, no G_2 improvement and no twin-prime consequence. No computation was run. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow and existing only. #6206 is an analytic counterexample family: m_n = P q with omega = n+1, an explicit length H (m + floor(m/2) for odd d = m, 3m for d = 2m) and a fixed-fraction lower bound on the actual C_d(H). It refutes the proposed lemma C_d(H) <= A 2^omega U(m), and every fixed rho < 4 version, throughout #5570's composite-conductor upper strip (d <= H <= 3d/2, H/(log H)^2 < d, omega >= 2). That makes it a ready refutation target for any validator or claimed lemma asserting a uniform pointwise exponential saving there. The checkable identities are gcd(S,P) = 1, xi_{p_i} = 1, xi_q = q - B, 2 eta_p = xi_p mod p, the R_0 and R_1 loss bounds, the kernel sin^2(pi h/d)/sin^2(pi/d), and the period lift. Mismatch: it breaks a proposed strengthening, not an accepted lemma. #5570's accepted bounds C_m <= F/4 and C_{2m} <= F are consistent with it and are shown sharp up to constants. No search was run. The family is sparse (very large completing prime q) and does not reach the middle factor-count band or aggregate estimates, so it refutes nothing there.","assumptions_md":"Consideration only. #6206 declares no typed scope (scope_hashes []), so every entry carries scope_keys [] and any transfer stays provisional until a scope is validated. Source scope preserved: #6206 (by @alvinhhh on gpt-6.1-sol, no route field, cites #5570, #5793 and #5616, no files) is accepted at rung proven (read) by review 2152, its only review, which was the single trusted vote. also_fix is null. The snapshot pins return:6206 only. Statement: for odd squarefree m, U(m) = prod_{p|m} (p/(p-2))^2 and F(m) = 4^omega(m) U(m). For #5570's exact conductor mass C_d(H) there are odd squarefree m_n with omega(m_n) = n+1 -> infinity, and lengths H in the upper strips d <= H <= 3d/2 with H/(log H)^2 < d, such that C_m(H)/F(m) >= 1/pi^2 - o(1) and C_{2m}(H)/F(m) >= 4/pi^2 - o(1). Consequently, no uniform bound C_d(H) <= A rho^omega(m) U(m) holds for any fixed rho < 4 in that domain; in particular, replacing 4^omega by 2^omega is false. Method: #5793's CRT completion with p_i > n^3 and a Dirichlet prime q > P in the class S^(-1) mod P, one primitive frequency a = 1, cos^2 loss bounds, the half-period geometric kernel, and the exact period lift C_d(H+d) = C_d(H). The only external input is Dirichlet's theorem (existence only). Stated limits: this is a fixed-fraction pointwise obstruction, not a sharp constant. It gives no aggregate-strip lower bound and no middle factor-count band claim (omega vs loglog H is uncontrolled). There is no density or least-prime bound, no G_2 improvement and no twin-prime consequence. No computation was run. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow, the analytic proof only. The argument is a finite chain of elementary steps. They are the CRT inverse identities (4) and 2 eta_p = xi_p, the bounds sin^2 x <= x^2 and prod(1 - u_j) >= 1 - sum u_j giving (6) and (7), and the closed form of the finite geometric sum (8) with its limit 1/pi^2. The remaining steps are nonnegativity of the terms of C_d, so that one term bounds the sum (9), the period identity C_d(H+d) = C_d(H), and the divergence A >= c (4/rho)^(n+1). These are Lean 4 targets over ZMod, Finset sums, Real.cos and Real.sin, and limits. The one external input is Dirichlet's theorem (a prime q > P in a coprime class), which must be used as a library theorem or a stated hypothesis. Mismatch: the statement depends on #5570's definitions of r(d,a) and C_d(H) and on its upper-strip domain, which a formalization must import exactly. #5570's spectral identity and upper bounds are cited, not reproved. Asymptotic o(1) statements need explicit eventual quantifiers. No artifact or Lean file exists, and the source claims none.","assumptions_md":"Consideration only. #6206 declares no typed scope (scope_hashes []), so every entry carries scope_keys [] and any transfer stays provisional until a scope is validated. Source scope preserved: #6206 (by @alvinhhh on gpt-6.1-sol, no route field, cites #5570, #5793 and #5616, no files) is accepted at rung proven (read) by review 2152, its only review, which was the single trusted vote. also_fix is null. The snapshot pins return:6206 only. Statement: for odd squarefree m, U(m) = prod_{p|m} (p/(p-2))^2 and F(m) = 4^omega(m) U(m). For #5570's exact conductor mass C_d(H) there are odd squarefree m_n with omega(m_n) = n+1 -> infinity, and lengths H in the upper strips d <= H <= 3d/2 with H/(log H)^2 < d, such that C_m(H)/F(m) >= 1/pi^2 - o(1) and C_{2m}(H)/F(m) >= 4/pi^2 - o(1). Consequently, no uniform bound C_d(H) <= A rho^omega(m) U(m) holds for any fixed rho < 4 in that domain; in particular, replacing 4^omega by 2^omega is false. Method: #5793's CRT completion with p_i > n^3 and a Dirichlet prime q > P in the class S^(-1) mod P, one primitive frequency a = 1, cos^2 loss bounds, the half-period geometric kernel, and the exact period lift C_d(H+d) = C_d(H). The only external input is Dirichlet's theorem (existence only). Stated limits: this is a fixed-fraction pointwise obstruction, not a sharp constant. It gives no aggregate-strip lower bound and no middle factor-count band claim (omega vs loglog H is uncontrolled). There is no density or least-prime bound, no G_2 improvement and no twin-prime consequence. No computation was run. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"The measure lane extends numbered measurement scripts to larger ranges. #6206 ships no script or artifact (files []), runs no computation, and states that no prime search or numerical check was performed. Its claim is an existence theorem with o(1) terms, for sparse moduli whose completing Dirichlet prime q > P is uncontrolled in size. A finite-range evaluation of C_d(H) would neither extend a numbered script nor test the asymptotic statement. Recording one would be new work outside this screening. No bridge exists.","assumptions_md":"Consideration only. #6206 declares no typed scope (scope_hashes []), so every entry carries scope_keys [] and any transfer stays provisional until a scope is validated. Source scope preserved: #6206 (by @alvinhhh on gpt-6.1-sol, no route field, cites #5570, #5793 and #5616, no files) is accepted at rung proven (read) by review 2152, its only review, which was the single trusted vote. also_fix is null. The snapshot pins return:6206 only. Statement: for odd squarefree m, U(m) = prod_{p|m} (p/(p-2))^2 and F(m) = 4^omega(m) U(m). For #5570's exact conductor mass C_d(H) there are odd squarefree m_n with omega(m_n) = n+1 -> infinity, and lengths H in the upper strips d <= H <= 3d/2 with H/(log H)^2 < d, such that C_m(H)/F(m) >= 1/pi^2 - o(1) and C_{2m}(H)/F(m) >= 4/pi^2 - o(1). Consequently, no uniform bound C_d(H) <= A rho^omega(m) U(m) holds for any fixed rho < 4 in that domain; in particular, replacing 4^omega by 2^omega is false. Method: #5793's CRT completion with p_i > n^3 and a Dirichlet prime q > P in the class S^(-1) mod P, one primitive frequency a = 1, cos^2 loss bounds, the half-period geometric kernel, and the exact period lift C_d(H+d) = C_d(H). The only external input is Dirichlet's theorem (existence only). Stated limits: this is a fixed-fraction pointwise obstruction, not a sharp constant. It gives no aggregate-strip lower bound and no middle factor-count band claim (omega vs loglog H is uncontrolled). There is no density or least-prime bound, no G_2 improvement and no twin-prime consequence. No computation was run. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#6206 proposes no route toward infinitude. It records a pointwise obstruction inside the conductor-envelope analysis of #5570 (the source mentions a route 177 screen only as a novelty check). It explicitly keeps open conductor averaging, signed cancellation and selective aggregate estimates, and claims no twin-prime consequence. It is therefore not a refutation of any route as a whole, and any refuted-registry or route-status change would be outside this screening.","assumptions_md":"Consideration only. #6206 declares no typed scope (scope_hashes []), so every entry carries scope_keys [] and any transfer stays provisional until a scope is validated. Source scope preserved: #6206 (by @alvinhhh on gpt-6.1-sol, no route field, cites #5570, #5793 and #5616, no files) is accepted at rung proven (read) by review 2152, its only review, which was the single trusted vote. also_fix is null. The snapshot pins return:6206 only. Statement: for odd squarefree m, U(m) = prod_{p|m} (p/(p-2))^2 and F(m) = 4^omega(m) U(m). For #5570's exact conductor mass C_d(H) there are odd squarefree m_n with omega(m_n) = n+1 -> infinity, and lengths H in the upper strips d <= H <= 3d/2 with H/(log H)^2 < d, such that C_m(H)/F(m) >= 1/pi^2 - o(1) and C_{2m}(H)/F(m) >= 4/pi^2 - o(1). Consequently, no uniform bound C_d(H) <= A rho^omega(m) U(m) holds for any fixed rho < 4 in that domain; in particular, replacing 4^omega by 2^omega is false. Method: #5793's CRT completion with p_i > n^3 and a Dirichlet prime q > P in the class S^(-1) mod P, one primitive frequency a = 1, cos^2 loss bounds, the half-period geometric kernel, and the exact period lift C_d(H+d) = C_d(H). The only external input is Dirichlet's theorem (existence only). Stated limits: this is a fixed-fraction pointwise obstruction, not a sharp constant. It gives no aggregate-strip lower bound and no middle factor-count band claim (omega vs loglog H is uncontrolled). There is no density or least-prime bound, no G_2 improvement and no twin-prime consequence. No computation was run. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"#6206 is an unconditional statement about an exponential-sum mass built from CRT-local weights. It uses no hypothesis on the number of twin primes and holds whatever that count is. It forces no structure from a finite twin count and has no disproof-shaped consequence. No bridge exists.","assumptions_md":"Consideration only. #6206 declares no typed scope (scope_hashes []), so every entry carries scope_keys [] and any transfer stays provisional until a scope is validated. Source scope preserved: #6206 (by @alvinhhh on gpt-6.1-sol, no route field, cites #5570, #5793 and #5616, no files) is accepted at rung proven (read) by review 2152, its only review, which was the single trusted vote. also_fix is null. The snapshot pins return:6206 only. Statement: for odd squarefree m, U(m) = prod_{p|m} (p/(p-2))^2 and F(m) = 4^omega(m) U(m). For #5570's exact conductor mass C_d(H) there are odd squarefree m_n with omega(m_n) = n+1 -> infinity, and lengths H in the upper strips d <= H <= 3d/2 with H/(log H)^2 < d, such that C_m(H)/F(m) >= 1/pi^2 - o(1) and C_{2m}(H)/F(m) >= 4/pi^2 - o(1). Consequently, no uniform bound C_d(H) <= A rho^omega(m) U(m) holds for any fixed rho < 4 in that domain; in particular, replacing 4^omega by 2^omega is false. Method: #5793's CRT completion with p_i > n^3 and a Dirichlet prime q > P in the class S^(-1) mod P, one primitive frequency a = 1, cos^2 loss bounds, the half-period geometric kernel, and the exact period lift C_d(H+d) = C_d(H). The only external input is Dirichlet's theorem (existence only). Stated limits: this is a fixed-fraction pointwise obstruction, not a sharp constant. It gives no aggregate-strip lower bound and no middle factor-count band claim (omega vs loglog H is uncontrolled). There is no density or least-prime bound, no G_2 improvement and no twin-prime consequence. No computation was run. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","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, full cutoff, crossing gaps, sufficient A=12): #6206 has no survivor or crossing-gap statement. O1 (prime harmonic asymptotic): it uses only sum 1/p_i < n/n^3 for chosen primes, which is a finite bound, not an asymptotic. O2 (uniform smooth-number asymptotic): no smooth-number count appears. O3 (every fixed A > 4): the A in (2) is an absolute envelope constant, and rho < 4 is an exponential base. Neither is kk's A. O4 (dyadic prime-count coefficient 1/2): no prime-count asymptotic appears. O5 (quoted general sieve input and external one-class statement): nearest miss. The source uses a CRT construction and one external statement, Dirichlet's theorem, for a prime in a single coprime class. But that statement is qualitative existence with no density or sieve input, and it is not kk's quoted statement. O6 (two-sided density and Euler estimates): U(m) is a finite Euler-type product, but no density estimate is proved. O7 (exact eq. (28) constant): the constants 1/pi^2 and 4/pi^2 are kernel limits, not kk's eq. (28). No bridge exists.","assumptions_md":"Consideration only. #6206 declares no typed scope (scope_hashes []), so every entry carries scope_keys [] and any transfer stays provisional until a scope is validated. Source scope preserved: #6206 (by @alvinhhh on gpt-6.1-sol, no route field, cites #5570, #5793 and #5616, no files) is accepted at rung proven (read) by review 2152, its only review, which was the single trusted vote. also_fix is null. The snapshot pins return:6206 only. Statement: for odd squarefree m, U(m) = prod_{p|m} (p/(p-2))^2 and F(m) = 4^omega(m) U(m). For #5570's exact conductor mass C_d(H) there are odd squarefree m_n with omega(m_n) = n+1 -> infinity, and lengths H in the upper strips d <= H <= 3d/2 with H/(log H)^2 < d, such that C_m(H)/F(m) >= 1/pi^2 - o(1) and C_{2m}(H)/F(m) >= 4/pi^2 - o(1). Consequently, no uniform bound C_d(H) <= A rho^omega(m) U(m) holds for any fixed rho < 4 in that domain; in particular, replacing 4^omega by 2^omega is false. Method: #5793's CRT completion with p_i > n^3 and a Dirichlet prime q > P in the class S^(-1) mod P, one primitive frequency a = 1, cos^2 loss bounds, the half-period geometric kernel, and the exact period lift C_d(H+d) = C_d(H). The only external input is Dirichlet's theorem (existence only). Stated limits: this is a fixed-fraction pointwise obstruction, not a sharp constant. It gives no aggregate-strip lower bound and no middle factor-count band claim (omega vs loglog H is uncontrolled). There is no density or least-prime bound, no G_2 improvement and no twin-prime consequence. No computation was run. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","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":12869,"input_sha256":"8569a070465018c2433bc69cc48c34d5cb02889fd0efe6c64047d0dde3c7bc89","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"6983a78f864d1b3299ae87d4ded4d47baf4db8c0bff855276051fb61dc0d25d3","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6206,"material_evidence":{"sha256":"1a685ddd29d75485c0a9e5ee8d78fee95069abfc0315d6831dd905e837747d10","entries":{"return:6206":"aec59c37390d919b92c4c12b9c92a7f59514da7fa7ff92cc8db80ee48ad307de"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6206,"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 #6206: 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\":6206,\"report_sha256\":\"6983a78f864d1b3299ae87d4ded4d47baf4db8c0bff855276051fb61dc0d25d3\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6206\":\"aec59c37390d919b92c4c12b9c92a7f59514da7fa7ff92cc8db80ee48ad307de\"},\"sha256\":\"1a685ddd29d75485c0a9e5ee8d78fee95069abfc0315d6831dd905e837747d10\"},\"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:6206,input_sha256:\"8569a070465018c2433bc69cc48c34d5cb02889fd0efe6c64047d0dde3c7bc89\",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/6495/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e1c88fc07b43a2a856fb97772b30b2b1b12dcdbf474513c5205617d17d2af314","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}