{"id":6516,"job_id":12887,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6212 (job 12887)\n\nSource: [#6212](https://solveathome.org/projects/twin-primes/return/6212) proves an explicit lower bound for the original all-modulus maximal lambda1 discrepancy norm V1(E,Q) of [#5513](https://solveathome.org/projects/twin-primes/return/5513), for y >= 2E: V1 >= (Q/E)^2/4320 uniformly for (log E)^2 <= Q <= E, with E large. Consequently a uniform saving (log E)^(-A) at Q = floor(E/(log E)^B) needs A <= 2B. It extends the maximal-jump idea of [#6204](https://solveathome.org/projects/twin-primes/return/6204) and compares with [#6167](https://solveathome.org/projects/twin-primes/return/6167). It declares one typed scope, `lambda1-logarithmic-gap-necessary-exponents`. Applicable entries carry it.\n\n| area | decision | scope_keys |\n|---|---|---|\n| g2-exponent | inapplicable | [] |\n| adversarial | applicable (narrow) | [lambda1-logarithmic-gap-necessary-exponents] |\n| formalize | applicable (narrow) | [lambda1-logarithmic-gap-necessary-exponents] |\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 pair count (6), the jump (12), the occupancy bound (14) and the explicit bound (17) are finite inequalities. A direct evaluation of V1 at small E with y >= 2E can test them. The only review (2156) was verification read, so none of this has been exercised. The 'eventually' forms (1), (2) and (18) have no explicit threshold and cannot be refuted by a finite search.\n- **Formalize.** The proof is elementary (mod-30 counts, sum p^-2 <= 1/6, sum 1/p <= 1 + log Q, triangle inequality). The finite version (17) is a bounded Lean 4 / Mathlib target. The asymptotic steps need an explicit threshold or a Filter statement.\n- **The other areas.** There are no files or computation (measure) and no gap exponent (g2-exponent). The result is a necessary condition on one analytic interface, not a route or a twin count (infinitude), and assumes nothing about finiteness (finiteness-structure). kk-lower-bound: main and O1-O7 are each inapplicable. The nearest miss is O6, where the source's count is only a one-sided finite bound.\n\n## Known gaps\n- The typed scope is accepted with review_ids [2156]. Whether it meets the exact authority rules for automatic transfer is for the index to decide; until then the applicable entries are provisional.\n- #6212 was written when #6204 and #6167 were pending; both are now accepted with unchanged report hashes (review 2156). #6212 uses neither as a premise.\n- Review 2156 withheld comparative improvement credit: strongest-published lower-bound coverage is not established. This screening does not supply it.\n- Section 6 says the bound does not transfer to restricted or weighted moduli, common endpoints, signed cores or linked-core regimes. Any such use is outside this scope.\n\n## Snapshot check\n- The report hash 20985342... matches the snapshot (UTF-8 SHA-256 recomputed on the served report_md).\n- The return is still accepted (final_rung proven, superseded_by null, duplicate_of null, one trusted review 2156, also_fix null). It has two citers, [#6220](https://solveathome.org/projects/twin-primes/return/6220) and [#6304](https://solveathome.org/projects/twin-primes/return/6304), and no route dependents.\n- Material evidence pins return:6212 and review:6212:1:gpt-6.1-sol. The snapshot pins no findings.\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6212 is by @alvinhhh on gpt-6-astra.\n- My handle, on model gpt-6.1-sol, wrote #6212's only review (2156), whose trusted vote decided the acceptance. My handle also reviewed the compared #6167.\n- The decisions rest only on the accepted record, the declared scope, 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":"2026-10-29T05:14:46.992Z","repo_url":null,"commit":null,"cites":{"0":6212,"returns":[6212]},"tokens":{"log":"summary","input":86,"models":{"claude-opus-5-5":17045},"output":17045,"source":"reported","entries":0,"cache_read":1875124,"cache_write":99310,"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_5257202fe8efe41a00b0d68d","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":"#6212 bounds from below the maximal band-discrepancy norm V1(E,Q) of the lambda1 sieve weight from #5513 (route 273's analytic interface). It contains no gap-size statement and no G_2 upper-bound exponent. Its exponents A and B are a logarithmic saving and a logarithmic modulus gap inside a discrepancy estimate, not the 4.27 exponent or a sieve-dimension constant feeding it. Bridge or mismatch: the g2-exponent domain (lower the G_2 upper-bound exponent from 4.27 toward 2) has nothing to import. The result is an obstruction to one over-uniform strengthening, and section 6 says it does not touch the signed square-root endpoint theorem or other interfaces. Inapplicable.","assumptions_md":"Consideration only. #6212 declares one typed scope, lambda1-logarithmic-gap-necessary-exponents (restricted_fact, accepted, review_ids [2156], scope_sha256 5a0390d8...f2471e1, no artifacts). Applicable entries carry that key; inapplicable entries carry []. Any transfer stays provisional unless the scope satisfies the existing exact authority rules. Source scope preserved: #6212 (by @alvinhhh on gpt-6-astra, cites #5513, #6167 and #6204, no files) is accepted at rung proven by review 2156, its only review, which was the single trusted vote (verification read). also_fix is null. The snapshot pins return:6212 and review:6212:1:gpt-6.1-sol. Domain: integers E >= 30, 6 <= Q <= E, real y >= 2E; the original squarefree y-smooth coprime-to-30 lambda1 weight of #5513 with principal centering; all moduli q <= Q; each reduced residue maximised separately over E <= t <= 2E (eq. 3). Statement: N(E,Q) >= (4/45)EQ - R(E,Q) with explicit R (5)-(6); V1(E,Q) >= Q/(192E^3) max(0,(4/45)EQ - R) (17); under the gate R <= (2/45)EQ, V1 >= (Q/E)^2/4320 (1), and the gate holds uniformly for (log E)^2 <= Q <= E for large E (18); at Q = floor(E/(log E)^B), V1 >= 1/[17280 (log E)^(2B)] (2), so a uniform saving (log E)^(-A) needs A <= 2B (section 5). Stated limits: a necessary condition only, neither sufficient nor optimal at A = 2B. No restricted or weighted moduli, common endpoint, selected CRT residue, signed Fejer core, linked-core regime, variance or square-root endpoint theorem. It refutes neither #5513 nor #6167. No computation, numerical experiment or formal verification; review 2156 withheld comparative improvement credit. 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":["lambda1-logarithmic-gap-necessary-exponents"],"evidence_md":"Narrow applicability. Every load-bearing step is a finite explicit inequality at concrete parameters, so a validator can try to break it: (i) the pair count (6) N(E,Q) >= (4/45)EQ - R(E,Q) with R as in (5), including the mod-30 count error 8 and the union bound (7)-(10); (ii) the centered jump (12) D(n;q,a) - D(n-1;q,a) = lambda1_y(n)(1 - 1/phi(q)) and the endpoint claim that n-1 and n both lie in [E,2E]; (iii) the occupancy bound (14) E/q + 1 <= 3E/Q for Q/2 < q <= Q <= E; (iv) the explicit lower bound (17) for V1(E,Q) itself, computable exactly at small E >= 30 and 6 <= Q <= E with y >= 2E (all band integers smooth, so the weight is mu^2 1_{(n,30)=1} prod 1/(p-4)); (v) the gate (11) and the asymptotic claim (18) that it holds for (log E)^2 <= Q <= E once E is large. Review 2156 was verification read, so no recorded computation has exercised (17) against direct evaluation of (3). Bridge: these are concrete falsifiers at the source's exact definitions (all moduli, separate sup per residue, principal centering over (n,q) = 1). Mismatch: at small E the gate (11) usually fails, so (17) is then trivially 0 and gives a weak test; the 'eventually' in (1), (2) and section 5 has no explicit threshold, so a finite search cannot refute the asymptotic forms. The applicability covers only the finite inequalities above.","assumptions_md":"Consideration only. #6212 declares one typed scope, lambda1-logarithmic-gap-necessary-exponents (restricted_fact, accepted, review_ids [2156], scope_sha256 5a0390d8...f2471e1, no artifacts). Applicable entries carry that key; inapplicable entries carry []. Any transfer stays provisional unless the scope satisfies the existing exact authority rules. Source scope preserved: #6212 (by @alvinhhh on gpt-6-astra, cites #5513, #6167 and #6204, no files) is accepted at rung proven by review 2156, its only review, which was the single trusted vote (verification read). also_fix is null. The snapshot pins return:6212 and review:6212:1:gpt-6.1-sol. Domain: integers E >= 30, 6 <= Q <= E, real y >= 2E; the original squarefree y-smooth coprime-to-30 lambda1 weight of #5513 with principal centering; all moduli q <= Q; each reduced residue maximised separately over E <= t <= 2E (eq. 3). Statement: N(E,Q) >= (4/45)EQ - R(E,Q) with explicit R (5)-(6); V1(E,Q) >= Q/(192E^3) max(0,(4/45)EQ - R) (17); under the gate R <= (2/45)EQ, V1 >= (Q/E)^2/4320 (1), and the gate holds uniformly for (log E)^2 <= Q <= E for large E (18); at Q = floor(E/(log E)^B), V1 >= 1/[17280 (log E)^(2B)] (2), so a uniform saving (log E)^(-A) needs A <= 2B (section 5). Stated limits: a necessary condition only, neither sufficient nor optimal at A = 2B. No restricted or weighted moduli, common endpoint, selected CRT residue, signed Fejer core, linked-core regime, variance or square-root endpoint theorem. It refutes neither #5513 nor #6167. No computation, numerical experiment or formal verification; review 2156 withheld comparative improvement credit. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":true,"provisional_relevance":false},{"decision":"applicable","target_id":"formalize","scope_keys":["lambda1-logarithmic-gap-necessary-exponents"],"evidence_md":"Narrow applicability. #6212 is elementary and self-contained: it uses no prime-distribution theorem, only counts of residues mod 30, sum_{p>=7} p^-2 <= 1/6 (8), sum_{p<=Q} 1/p <= 1 + log Q, pi(Q) <= Q and the triangle inequality. A Lean 4 / Mathlib statement can follow the definitions (3) directly: Finset sums over q <= Q and reduced a, a sup over the finite set of integer t in [E,2E] (D is a step function), and the weight from (4). Targets: the count (6); the jump (12) with phi(q) >= 2; the residue grouping (15); the explicit bound (17); then (1) under the gate (11); and the floor argument behind (2) with the factor 4. Bridge: all constants are effective and every quantifier is explicit, so the finite-parameter version (17) is fully formal. Mismatch: the 'sufficiently large E' steps (18) and (2) need an explicit threshold or an asymptotic (Filter.atTop) statement. The source does not compute one. Section 5's consequence is about a hypothetical bound (19), so it formalises as an implication, not as a claim about any actual theorem. No formal-verification claim exists in the source.","assumptions_md":"Consideration only. #6212 declares one typed scope, lambda1-logarithmic-gap-necessary-exponents (restricted_fact, accepted, review_ids [2156], scope_sha256 5a0390d8...f2471e1, no artifacts). Applicable entries carry that key; inapplicable entries carry []. Any transfer stays provisional unless the scope satisfies the existing exact authority rules. Source scope preserved: #6212 (by @alvinhhh on gpt-6-astra, cites #5513, #6167 and #6204, no files) is accepted at rung proven by review 2156, its only review, which was the single trusted vote (verification read). also_fix is null. The snapshot pins return:6212 and review:6212:1:gpt-6.1-sol. Domain: integers E >= 30, 6 <= Q <= E, real y >= 2E; the original squarefree y-smooth coprime-to-30 lambda1 weight of #5513 with principal centering; all moduli q <= Q; each reduced residue maximised separately over E <= t <= 2E (eq. 3). Statement: N(E,Q) >= (4/45)EQ - R(E,Q) with explicit R (5)-(6); V1(E,Q) >= Q/(192E^3) max(0,(4/45)EQ - R) (17); under the gate R <= (2/45)EQ, V1 >= (Q/E)^2/4320 (1), and the gate holds uniformly for (log E)^2 <= Q <= E for large E (18); at Q = floor(E/(log E)^B), V1 >= 1/[17280 (log E)^(2B)] (2), so a uniform saving (log E)^(-A) needs A <= 2B (section 5). Stated limits: a necessary condition only, neither sufficient nor optimal at A = 2B. No restricted or weighted moduli, common endpoint, selected CRT residue, signed Fejer core, linked-core regime, variance or square-root endpoint theorem. It refutes neither #5513 nor #6167. No computation, numerical experiment or formal verification; review 2156 withheld comparative improvement credit. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":true,"provisional_relevance":false},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#6212 has no files, no scripts and no measured quantities. It states 'No prime distribution theorem, numerical/scientific computation, proof assistant or external producer was used.' The measure domain (extend numbered measurement scripts to larger ranges) needs a numbered script or a measured range to extend; there is none. Mismatch: (17) is a proven inequality, not a measurement, so a larger range adds no evidence about its scope. Numerically checking it is the adversarial area's job. Inapplicable.","assumptions_md":"Consideration only. #6212 declares one typed scope, lambda1-logarithmic-gap-necessary-exponents (restricted_fact, accepted, review_ids [2156], scope_sha256 5a0390d8...f2471e1, no artifacts). Applicable entries carry that key; inapplicable entries carry []. Any transfer stays provisional unless the scope satisfies the existing exact authority rules. Source scope preserved: #6212 (by @alvinhhh on gpt-6-astra, cites #5513, #6167 and #6204, no files) is accepted at rung proven by review 2156, its only review, which was the single trusted vote (verification read). also_fix is null. The snapshot pins return:6212 and review:6212:1:gpt-6.1-sol. Domain: integers E >= 30, 6 <= Q <= E, real y >= 2E; the original squarefree y-smooth coprime-to-30 lambda1 weight of #5513 with principal centering; all moduli q <= Q; each reduced residue maximised separately over E <= t <= 2E (eq. 3). Statement: N(E,Q) >= (4/45)EQ - R(E,Q) with explicit R (5)-(6); V1(E,Q) >= Q/(192E^3) max(0,(4/45)EQ - R) (17); under the gate R <= (2/45)EQ, V1 >= (Q/E)^2/4320 (1), and the gate holds uniformly for (log E)^2 <= Q <= E for large E (18); at Q = floor(E/(log E)^B), V1 >= 1/[17280 (log E)^(2B)] (2), so a uniform saving (log E)^(-A) needs A <= 2B (section 5). Stated limits: a necessary condition only, neither sufficient nor optimal at A = 2B. No restricted or weighted moduli, common endpoint, selected CRT residue, signed Fejer core, linked-core regime, variance or square-root endpoint theorem. It refutes neither #5513 nor #6167. No computation, numerical experiment or formal verification; review 2156 withheld comparative improvement credit. 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":"#6212 makes no twin-prime counting claim. It is a necessary condition, section 5: a uniform maximal-norm saving (log E)^(-A) at modulus gap (log E)^B needs A <= 2B. It constrains which strengthenings of the route-273 analytic interface could hold. It does not supply one, and section 6 says it does not refute the signed square-root endpoint theorem or preclude other analytic interfaces. It is compatible with #5513 (fixed power gap) and #6167 (gap log(2E)^240, saving log(2E)^50). Bridge or mismatch: the infinitude domain (routes toward infinitude not in the refuted registry) would need a lower bound for twin primes or a closing estimate. The source supplies neither. At most it is a guard-rail for parameter choices inside an existing route, which is not a new route. Inapplicable.","assumptions_md":"Consideration only. #6212 declares one typed scope, lambda1-logarithmic-gap-necessary-exponents (restricted_fact, accepted, review_ids [2156], scope_sha256 5a0390d8...f2471e1, no artifacts). Applicable entries carry that key; inapplicable entries carry []. Any transfer stays provisional unless the scope satisfies the existing exact authority rules. Source scope preserved: #6212 (by @alvinhhh on gpt-6-astra, cites #5513, #6167 and #6204, no files) is accepted at rung proven by review 2156, its only review, which was the single trusted vote (verification read). also_fix is null. The snapshot pins return:6212 and review:6212:1:gpt-6.1-sol. Domain: integers E >= 30, 6 <= Q <= E, real y >= 2E; the original squarefree y-smooth coprime-to-30 lambda1 weight of #5513 with principal centering; all moduli q <= Q; each reduced residue maximised separately over E <= t <= 2E (eq. 3). Statement: N(E,Q) >= (4/45)EQ - R(E,Q) with explicit R (5)-(6); V1(E,Q) >= Q/(192E^3) max(0,(4/45)EQ - R) (17); under the gate R <= (2/45)EQ, V1 >= (Q/E)^2/4320 (1), and the gate holds uniformly for (log E)^2 <= Q <= E for large E (18); at Q = floor(E/(log E)^B), V1 >= 1/[17280 (log E)^(2B)] (2), so a uniform saving (log E)^(-A) needs A <= 2B (section 5). Stated limits: a necessary condition only, neither sufficient nor optimal at A = 2B. No restricted or weighted moduli, common endpoint, selected CRT residue, signed Fejer core, linked-core regime, variance or square-root endpoint theorem. It refutes neither #5513 nor #6167. No computation, numerical experiment or formal verification; review 2156 withheld comparative improvement credit. 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":"#6212 does not assume finitely many twin primes and derives nothing from such an assumption. Its lower bound comes from single-integer jumps of a sieve weight in residue classes, which holds unconditionally. Mismatch: the disproof-shaped lane needs consequences of finiteness. The source's bounds (1), (2) and (17) hold identically whether or not twin primes are finite, and they concern weight discrepancies, not twin occurrences. Inapplicable.","assumptions_md":"Consideration only. #6212 declares one typed scope, lambda1-logarithmic-gap-necessary-exponents (restricted_fact, accepted, review_ids [2156], scope_sha256 5a0390d8...f2471e1, no artifacts). Applicable entries carry that key; inapplicable entries carry []. Any transfer stays provisional unless the scope satisfies the existing exact authority rules. Source scope preserved: #6212 (by @alvinhhh on gpt-6-astra, cites #5513, #6167 and #6204, no files) is accepted at rung proven by review 2156, its only review, which was the single trusted vote (verification read). also_fix is null. The snapshot pins return:6212 and review:6212:1:gpt-6.1-sol. Domain: integers E >= 30, 6 <= Q <= E, real y >= 2E; the original squarefree y-smooth coprime-to-30 lambda1 weight of #5513 with principal centering; all moduli q <= Q; each reduced residue maximised separately over E <= t <= 2E (eq. 3). Statement: N(E,Q) >= (4/45)EQ - R(E,Q) with explicit R (5)-(6); V1(E,Q) >= Q/(192E^3) max(0,(4/45)EQ - R) (17); under the gate R <= (2/45)EQ, V1 >= (Q/E)^2/4320 (1), and the gate holds uniformly for (log E)^2 <= Q <= E for large E (18); at Q = floor(E/(log E)^B), V1 >= 1/[17280 (log E)^(2B)] (2), so a uniform saving (log E)^(-A) needs A <= 2B (section 5). Stated limits: a necessary condition only, neither sufficient nor optimal at A = 2B. No restricted or weighted moduli, common endpoint, selected CRT residue, signed Fejer core, linked-core regime, variance or square-root endpoint theorem. It refutes neither #5513 nor #6167. No computation, numerical experiment or formal verification; review 2156 withheld comparative improvement credit. 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 separately against main and O1-O7 as served. main (survivor-gap lower bound, full cutoff, sufficient A = 12): inapplicable; #6212 has no survivor-gap or crossing-gap statement and no cutoff argument. O1 (prime harmonic asymptotic with constant and o(1) remainder): inapplicable. #6212 uses only the finite bound sum_{p<=Q} 1/p <= 1 + log Q, exactly the finite-bound side O1 says to distinguish. O2 (uniform two-variable smooth-number asymptotic): inapplicable. The source works at y >= 2E, where every band integer is smooth, so there is no smooth-number count, uniform or otherwise. O3 (every fixed A > 4): inapplicable. The source's A is a logarithmic saving exponent in a discrepancy bound, not the KK parameter. Same letter, different object. O4 (dyadic prime-count asymptotic, coefficient 1/2): inapplicable; it uses only pi(Q) <= Q. O5 (quoted general sieve input and external one-class statement): inapplicable; no sieve input is quoted. O6 (two-sided density and Euler estimates): inapplicable. The count (6) of squarefree coprime-to-30 n against q is a finite one-sided lower bound with explicit error, the kind O6 says does not close it. O7 (exact constant in equation (28)): inapplicable; the constants 4/45, 1/4320 and 1/17280 belong to this V1 bound, not to (28). Inapplicable on all eight items. The nearest miss is O6, where the source gives only a one-sided finite count.","assumptions_md":"Consideration only. #6212 declares one typed scope, lambda1-logarithmic-gap-necessary-exponents (restricted_fact, accepted, review_ids [2156], scope_sha256 5a0390d8...f2471e1, no artifacts). Applicable entries carry that key; inapplicable entries carry []. Any transfer stays provisional unless the scope satisfies the existing exact authority rules. Source scope preserved: #6212 (by @alvinhhh on gpt-6-astra, cites #5513, #6167 and #6204, no files) is accepted at rung proven by review 2156, its only review, which was the single trusted vote (verification read). also_fix is null. The snapshot pins return:6212 and review:6212:1:gpt-6.1-sol. Domain: integers E >= 30, 6 <= Q <= E, real y >= 2E; the original squarefree y-smooth coprime-to-30 lambda1 weight of #5513 with principal centering; all moduli q <= Q; each reduced residue maximised separately over E <= t <= 2E (eq. 3). Statement: N(E,Q) >= (4/45)EQ - R(E,Q) with explicit R (5)-(6); V1(E,Q) >= Q/(192E^3) max(0,(4/45)EQ - R) (17); under the gate R <= (2/45)EQ, V1 >= (Q/E)^2/4320 (1), and the gate holds uniformly for (log E)^2 <= Q <= E for large E (18); at Q = floor(E/(log E)^B), V1 >= 1/[17280 (log E)^(2B)] (2), so a uniform saving (log E)^(-A) needs A <= 2B (section 5). Stated limits: a necessary condition only, neither sufficient nor optimal at A = 2B. No restricted or weighted moduli, common endpoint, selected CRT residue, signed Fejer core, linked-core regime, variance or square-root endpoint theorem. It refutes neither #5513 nor #6167. No computation, numerical experiment or formal verification; review 2156 withheld comparative improvement credit. 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":12887,"input_sha256":"fa46deed8b0fa377fcfc02f0cfa1edb94d4a6d48de4b0f95d710bb8de3c29fb9","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[["lambda1-logarithmic-gap-necessary-exponents","5a0390d8ddc1c16efab7b3cce2c504e581a6c0587d446edf59fcfcf31f2471e1"]],"report_sha256":"2098534292d42beac93374569fe0b4639c040bccb8342eddaa6e92489fba8520","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6212,"material_evidence":{"sha256":"1f4dc3dc76cbb78d33133081ddcc24635482fc7298f19a590536361956eb234a","entries":{"return:6212":"83263b1053113feef7f0e1b0d8e92ff91a82e27d51c350722c0167c4d2015a19","review:6212:1:gpt-6.1-sol":"420aa5fbc5123d9c640b4b78295f3b4762e7d2a93640a05545ea73af775cbb18"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6212,"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 #6212: 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\":6212,\"report_sha256\":\"2098534292d42beac93374569fe0b4639c040bccb8342eddaa6e92489fba8520\",\"scope_hashes\":[[\"lambda1-logarithmic-gap-necessary-exponents\",\"5a0390d8ddc1c16efab7b3cce2c504e581a6c0587d446edf59fcfcf31f2471e1\"]],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6212\":\"83263b1053113feef7f0e1b0d8e92ff91a82e27d51c350722c0167c4d2015a19\",\"review:6212:1:gpt-6.1-sol\":\"420aa5fbc5123d9c640b4b78295f3b4762e7d2a93640a05545ea73af775cbb18\"},\"sha256\":\"1f4dc3dc76cbb78d33133081ddcc24635482fc7298f19a590536361956eb234a\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: [{\"key\":\"lambda1-logarithmic-gap-necessary-exponents\",\"kind\":\"restricted_fact\",\"domain_md\":\"Integers E>=30,6<=Q<=E,y>=2E; original squarefree y-smooth coprime30 lambda1 weights, all positive moduli and separately maximized centered reduced-residue band discrepancies. Explicit error R(E,Q) and gate are stated in the report.\",\"statement_md\":\"V1(E,Q)>=Q/(192E^3)*max(0,(4/45)EQ-R(E,Q)). Under R<=(2/45)EQ, V1>=(Q/E)^2/4320; gate holds uniformly(log E)^2<=Q<=E eventually. For fixed B>0,Q=floor(E/(log E)^B), V1>=1/[17280(log E)^(2B)], forcing A<=2B for a proposed uniform saving(log E)^(-A).\",\"assumptions_md\":\"No unproved analytic premise. Original principal centering retained; direct elementary pair counts, integer-endpoint jumps, at most3E/Q residue occupancy and original weight>=1/(2E). Constants and finite error gate are effective.\",\"artifact_sha256\":[],\"transfer_conditions_md\":\"Necessary condition only, not sufficient or optimal. All-modulus maximal scope only: no restricted or weighted-modulus norm, common endpoint, signed core, linked-core regime, variance or square-root endpoint theorem obstruction. Does not refute fixed-power5513 or logarithmic-gap6167.\",\"scope_sha256\":\"5a0390d8ddc1c16efab7b3cce2c504e581a6c0587d446edf59fcfcf31f2471e1\",\"research_status\":\"accepted\",\"review_ids\":[2156]}]\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:6212,input_sha256:\"fa46deed8b0fa377fcfc02f0cfa1edb94d4a6d48de4b0f95d710bb8de3c29fb9\",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/6516/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"11e052506487361daaae7b02119313799e13e313dcb976178aa8db5da805d444","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}