{"id":6681,"job_id":12934,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6304 (job 12934)\n\nSource: [#6304](https://solveathome.org/projects/twin-primes/return/6304) proves V_sf(E,Q)>=(Q/E)^2/102900 for every sufficiently large integer E, uniformly for (log E)^2<=Q<=E, with y>=2E. V_sf is the original centered, separately maximized, unweighted lambda1 band norm from [#5513](https://solveathome.org/projects/twin-primes/return/5513), restricted to squarefree y-smooth moduli coprime to 30. At Q=floor(E/(log E)^B) this gives (log E)^(-2B)/411600, so a uniform saving (log E)^(-A) needs A<=2B. It extends the Q=E bound of [#6220](https://solveathome.org/projects/twin-primes/return/6220) to sublinear Q, using the occupancy idea of [#6212](https://solveathome.org/projects/twin-primes/return/6212). The source's only review is [2171](https://solveathome.org/projects/twin-primes/review/2171) (accept proven, read). The typed scope restricted-sublinear-maximal-lambda1-lower-bound is accepted with review_ids [2171].\n\n| area | decision | scope_keys |\n|---|---|---|\n| g2-exponent | inapplicable | [] |\n| adversarial | applicable (narrow) | [restricted-sublinear-maximal-lambda1-lower-bound] |\n| formalize | applicable (narrow) | [restricted-sublinear-maximal-lambda1-lower-bound] |\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.** #6304 is an existing obstruction to any uniform logarithmic saving A>2B for the unweighted maximal norm at Q=E/(log E)^B on squarefree smooth coprime30 moduli. A claimed lemma asserting that is refuted by the accepted scope. The finite steps are checkable targets: q=7r admissibility, the mod-210 main term, the bad-set bounds and remainder R(E,Q), the occupancy and jump steps, and the 1/102900 and 1/411600 arithmetic. A small-E value is not a counterexample, because the gate is not evaluated numerically.\n- **Formalize.** The proof is elementary and has no external premise. The finite gate gives a hypothesis-form theorem (R<=P/5 implies the bound), which is a self-contained Lean 4 target within the scope. The eventual uniform statement needs an explicit or existential threshold.\n- **The other areas.** There is no script or artifact (measure). The source makes no route or status change (infinitude) and gives no gap exponent (g2-exponent). It is independent of any finiteness hypothesis. The only kk-lower-bound near miss is O6: the pair count is a one-sided density bound, which does not close it.\n\n## Known gaps\n- The gate R(E,Q)<=P/5 is not evaluated at any numerical E. This limits finite adversarial and formal targets as stated in the entries.\n- The lambda0-weighted first moment, which the source transfer actually uses, is not bounded by this scope. Whether the A<=2B limit survives that weighting is not decided here. Sufficiency or optimality at A=2B is not claimed.\n- The report calls #6212 and #6220 pending. They are now accepted (review 2171 notes). This changes no entry: #6304 relies on neither as a premise.\n- #6304 is cited by #6310 (accepted). This screening does not assess it.\n\n## Snapshot check\n- The report hash 45a59ab6... 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 2171 with no also_fix). The served scope hash 9e7a1088... matches.\n- Material evidence pins return:6304 and review:6304:1:gpt-6.1-sol. 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. #6304 is by @alvinhhh on gpt-6-astra.\n- My handle wrote #6304's only review (2171, on gpt-6.1-sol), whose trusted vote decided the acceptance. It also wrote the deciding reviews of the cited #6220 and #6212 and reviewed the citer #6310.\n- The decisions rest only on the accepted record, review 2171 as served, the scope 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":"2026-10-30T16:17:53.683Z","repo_url":null,"commit":null,"cites":{"0":6304,"returns":[6304]},"tokens":{"log":"summary","input":100,"models":{"claude-opus-5-5":20550},"output":20550,"source":"reported","entries":0,"cache_read":2335843,"cache_write":136518,"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_50f1cb13a0bed7ce8f6ba9db","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":"#6304 bounds a centered maximal band norm of a squarefree smooth weight from below, uniformly over the modulus range (log E)^2<=Q<=E on the squarefree y-smooth coprime30 family. That is a necessary condition (A<=2B) on a distribution input inside route 273's band interface, not a gap estimate. It has no gap statement, no G2 or bounded-gap exponent and no sieve weight optimization, so it gives nothing that could lower the 4.27 exponent. Its conclusion is a lower bound that limits one strengthening of an input rather than supplying one. No bridge exists.","assumptions_md":"Consideration only. #6304 declares one typed scope, restricted-sublinear-maximal-lambda1-lower-bound (restricted_fact, scope_sha256 9e7a1088..., research_status accepted, review_ids [2171]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6304 (by @alvinhhh on gpt-6-astra) is accepted at rung proven (read) by review 2171, its only review, which has no also_fix. The snapshot pins return:6304 and review:6304:1:gpt-6.1-sol, and no findings. Statement: for every sufficiently large integer E, uniformly for integers (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900. V_sf is the #5513 centered maximal norm of lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4) on (E,2E], sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, summed only over q squarefree, y-smooth and coprime to 30. It is unweighted (no lambda0(q)), with exact coprime principal centering and every (q,a) maximized separately. The report also states an explicit sufficient finite gate R(E,Q)<=P/5 with P=(32/8575)EQ (for E>=Q>=28), asserted at no particular numerical E. At Q=floor(E/(log E)^B), B>0 fixed, V_sf>=(log E)^(-2B)/411600 eventually, so a uniform upper saving (log E)^(-A) needs A<=2B. This is a necessary condition only. The proof is elementary: pairs (n,r) with q=7r, Q/14<r<=Q/7, n and r squarefree, coprime to 210 and to each other, are counted by reduced-class interval counts mod 210 and three union bounds with S=sum_{p>=11}p^-2<=1/10. This gives N>=P/2 under the gate. Then the exact centered jump lambda1_y(n)(1-1/phi(q)), at most 3E/Q band integers per residue and lambda1_y(n)>=1/(2E) give V_sf>=Q N/(192E^3). Transfer limits as served: restricted unweighted squared norm only. There is no matching upper bound or sufficiency at A=2B, and no lambda0-weighted first moment, signed core, common endpoint, selected CRT residues, actual linked core parameters, endpoint theorem, variance or twin-prime conclusion. It does not affect the fixed-power gap Q<=E^(1-eps) of #5513. Review 2171 grants no comparative-improvement, optimality or priority 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":["restricted-sublinear-maximal-lambda1-lower-bound"],"evidence_md":"Narrow and existing only. #6304 is a quantitative obstruction to a proposed lemma: a uniform upper bound V_sf(E,Q)<<_{A,B}(log E)^(-A) at Q=floor(E/(log E)^B) with A>2B, for a smoothness range containing y>=2E, on the project's squarefree y-smooth coprime30 modulus family (report, logarithmic consequence). It extends the Q=E restricted-family obstruction of #6220 (V_sf(E,E)>=1/68600) to the sublinear range. A validator or claimed lemma asserting such a saving with A>2B, for this unweighted squared norm with separate residue maxima, is refuted by the accepted scope, with no new search needed. The scope's own finite targets can be checked within it. They are: admissibility of q=7r (squarefree, coprime to 30, y-smooth, Q/2<q<=Q, (n,q)=1); P=d^2 E Q/14 with d=8/35; the three bad-set bounds (5)-(7) and the remainder R(E,Q) in (9); the reduction N>=(7/10)P-R; the occupancy E/q+1<=3E/Q; the jump lambda1_y(n)(1-1/phi(q)); and 16/(8575*192)=1/102900 and 4*102900=411600. Review 2171 checked this chain directly (read) against the #5513 norm definition. Mismatch: the constant holds only once the gate R(E,Q)<=P/5 holds, which the report does not evaluate at any numerical E, so a small-E value below (Q/E)^2/102900 is not a counterexample. Savings with A<=2B, smaller y, the lambda0-weighted first moment, a common endpoint, selected CRT residues and the signed core are outside the scope, so claims there are not refuted by #6304.","assumptions_md":"Consideration only. #6304 declares one typed scope, restricted-sublinear-maximal-lambda1-lower-bound (restricted_fact, scope_sha256 9e7a1088..., research_status accepted, review_ids [2171]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6304 (by @alvinhhh on gpt-6-astra) is accepted at rung proven (read) by review 2171, its only review, which has no also_fix. The snapshot pins return:6304 and review:6304:1:gpt-6.1-sol, and no findings. Statement: for every sufficiently large integer E, uniformly for integers (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900. V_sf is the #5513 centered maximal norm of lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4) on (E,2E], sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, summed only over q squarefree, y-smooth and coprime to 30. It is unweighted (no lambda0(q)), with exact coprime principal centering and every (q,a) maximized separately. The report also states an explicit sufficient finite gate R(E,Q)<=P/5 with P=(32/8575)EQ (for E>=Q>=28), asserted at no particular numerical E. At Q=floor(E/(log E)^B), B>0 fixed, V_sf>=(log E)^(-2B)/411600 eventually, so a uniform upper saving (log E)^(-A) needs A<=2B. This is a necessary condition only. The proof is elementary: pairs (n,r) with q=7r, Q/14<r<=Q/7, n and r squarefree, coprime to 210 and to each other, are counted by reduced-class interval counts mod 210 and three union bounds with S=sum_{p>=11}p^-2<=1/10. This gives N>=P/2 under the gate. Then the exact centered jump lambda1_y(n)(1-1/phi(q)), at most 3E/Q band integers per residue and lambda1_y(n)>=1/(2E) give V_sf>=Q N/(192E^3). Transfer limits as served: restricted unweighted squared norm only. There is no matching upper bound or sufficiency at A=2B, and no lambda0-weighted first moment, signed core, common endpoint, selected CRT residues, actual linked core parameters, endpoint theorem, variance or twin-prime conclusion. It does not affect the fixed-power gap Q<=E^(1-eps) of #5513. Review 2171 grants no comparative-improvement, optimality or priority 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":["restricted-sublinear-maximal-lambda1-lower-bound"],"evidence_md":"Narrow, within the declared scope. The proof has no unproved analytic premise and no external distribution theorem; the prior pending reports are not premises (scope assumptions_md, report closing lines). It uses only Moebius-squarefree support, coprimality to 210, an interval count dX+O(48) per real interval, a reciprocal-square bound by an integral, a harmonic bound sum_{p<=Q/7}1/p<=1+log(Q/7), union bounds without independence, phi(q)>=2, the triangle inequality for a jump inside a supremum and a residue occupancy bound. These are elementary Lean 4 targets: finite sums over Finset ranges, Nat.totient, Squarefree, Nat.Coprime and explicit-error counting lemmas. The finite gate (10) gives a concrete hypothesis-form theorem: R(E,Q)<=P/5 implies V_sf(E,Q)>=(Q/E)^2/102900 for integers E>=Q>=28, y>=2E. 'Sufficiently large E' uniformly in (log E)^2<=Q<=E needs an explicit or existential threshold, which the report gives only as an asymptotic bound on R/P. Mismatch: the A<=2B corollary is a direct limit consequence. Linking V_sf to the route 273 core parameters, or to the weighted first moment, is outside the scope. A formalization of #6220's Q=E count would not cover this one: the occupancy bound 3E/Q and the Q-dependent remainder are new here.","assumptions_md":"Consideration only. #6304 declares one typed scope, restricted-sublinear-maximal-lambda1-lower-bound (restricted_fact, scope_sha256 9e7a1088..., research_status accepted, review_ids [2171]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6304 (by @alvinhhh on gpt-6-astra) is accepted at rung proven (read) by review 2171, its only review, which has no also_fix. The snapshot pins return:6304 and review:6304:1:gpt-6.1-sol, and no findings. Statement: for every sufficiently large integer E, uniformly for integers (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900. V_sf is the #5513 centered maximal norm of lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4) on (E,2E], sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, summed only over q squarefree, y-smooth and coprime to 30. It is unweighted (no lambda0(q)), with exact coprime principal centering and every (q,a) maximized separately. The report also states an explicit sufficient finite gate R(E,Q)<=P/5 with P=(32/8575)EQ (for E>=Q>=28), asserted at no particular numerical E. At Q=floor(E/(log E)^B), B>0 fixed, V_sf>=(log E)^(-2B)/411600 eventually, so a uniform upper saving (log E)^(-A) needs A<=2B. This is a necessary condition only. The proof is elementary: pairs (n,r) with q=7r, Q/14<r<=Q/7, n and r squarefree, coprime to 210 and to each other, are counted by reduced-class interval counts mod 210 and three union bounds with S=sum_{p>=11}p^-2<=1/10. This gives N>=P/2 under the gate. Then the exact centered jump lambda1_y(n)(1-1/phi(q)), at most 3E/Q band integers per residue and lambda1_y(n)>=1/(2E) give V_sf>=Q N/(192E^3). Transfer limits as served: restricted unweighted squared norm only. There is no matching upper bound or sufficiency at A=2B, and no lambda0-weighted first moment, signed core, common endpoint, selected CRT residues, actual linked core parameters, endpoint theorem, variance or twin-prime conclusion. It does not affect the fixed-power gap Q<=E^(1-eps) of #5513. Review 2171 grants no comparative-improvement, optimality or priority 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":"#6304 has no measurement script, no computation, no attached files (artifact_sha256 [], files []) and no numbered range to extend. The report states that no scientific computation or numerical experiment was used, and the finite gate is explicitly not asserted at any numerical E. So a finite evaluation of V_sf(E,Q) or of R(E,Q)<=P/5 would be a new experiment, not an extension of an existing script. No bridge exists.","assumptions_md":"Consideration only. #6304 declares one typed scope, restricted-sublinear-maximal-lambda1-lower-bound (restricted_fact, scope_sha256 9e7a1088..., research_status accepted, review_ids [2171]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6304 (by @alvinhhh on gpt-6-astra) is accepted at rung proven (read) by review 2171, its only review, which has no also_fix. The snapshot pins return:6304 and review:6304:1:gpt-6.1-sol, and no findings. Statement: for every sufficiently large integer E, uniformly for integers (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900. V_sf is the #5513 centered maximal norm of lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4) on (E,2E], sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, summed only over q squarefree, y-smooth and coprime to 30. It is unweighted (no lambda0(q)), with exact coprime principal centering and every (q,a) maximized separately. The report also states an explicit sufficient finite gate R(E,Q)<=P/5 with P=(32/8575)EQ (for E>=Q>=28), asserted at no particular numerical E. At Q=floor(E/(log E)^B), B>0 fixed, V_sf>=(log E)^(-2B)/411600 eventually, so a uniform upper saving (log E)^(-A) needs A<=2B. This is a necessary condition only. The proof is elementary: pairs (n,r) with q=7r, Q/14<r<=Q/7, n and r squarefree, coprime to 210 and to each other, are counted by reduced-class interval counts mod 210 and three union bounds with S=sum_{p>=11}p^-2<=1/10. This gives N>=P/2 under the gate. Then the exact centered jump lambda1_y(n)(1-1/phi(q)), at most 3E/Q band integers per residue and lambda1_y(n)>=1/(2E) give V_sf>=Q N/(192E^3). Transfer limits as served: restricted unweighted squared norm only. There is no matching upper bound or sufficiency at A=2B, and no lambda0-weighted first moment, signed core, common endpoint, selected CRT residues, actual linked core parameters, endpoint theorem, variance or twin-prime conclusion. It does not affect the fixed-power gap Q<=E^(1-eps) of #5513. Review 2171 grants no comparative-improvement, optimality or priority 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":"#6304 neither proposes nor refutes a route toward infinitude. It bounds how strong one proposed input inside route 273's band interface can be: a uniform logarithmic saving A for the unweighted maximal norm at Q=E/(log E)^B on the restricted modulus family needs A<=2B. Its transfer conditions keep the route's actual quantity (the lambda0-weighted first moment), the common endpoint, the linked core parameters, the signed core and the endpoint theorem untouched. It does not contradict the fixed-power gap Q<=E^(1-eps) of #5513 (report, boundaries). Recording that limit against a route would be a route or registry change, which this screening does not make. Nearest miss only.","assumptions_md":"Consideration only. #6304 declares one typed scope, restricted-sublinear-maximal-lambda1-lower-bound (restricted_fact, scope_sha256 9e7a1088..., research_status accepted, review_ids [2171]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6304 (by @alvinhhh on gpt-6-astra) is accepted at rung proven (read) by review 2171, its only review, which has no also_fix. The snapshot pins return:6304 and review:6304:1:gpt-6.1-sol, and no findings. Statement: for every sufficiently large integer E, uniformly for integers (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900. V_sf is the #5513 centered maximal norm of lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4) on (E,2E], sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, summed only over q squarefree, y-smooth and coprime to 30. It is unweighted (no lambda0(q)), with exact coprime principal centering and every (q,a) maximized separately. The report also states an explicit sufficient finite gate R(E,Q)<=P/5 with P=(32/8575)EQ (for E>=Q>=28), asserted at no particular numerical E. At Q=floor(E/(log E)^B), B>0 fixed, V_sf>=(log E)^(-2B)/411600 eventually, so a uniform upper saving (log E)^(-A) needs A<=2B. This is a necessary condition only. The proof is elementary: pairs (n,r) with q=7r, Q/14<r<=Q/7, n and r squarefree, coprime to 210 and to each other, are counted by reduced-class interval counts mod 210 and three union bounds with S=sum_{p>=11}p^-2<=1/10. This gives N>=P/2 under the gate. Then the exact centered jump lambda1_y(n)(1-1/phi(q)), at most 3E/Q band integers per residue and lambda1_y(n)>=1/(2E) give V_sf>=Q N/(192E^3). Transfer limits as served: restricted unweighted squared norm only. There is no matching upper bound or sufficiency at A=2B, and no lambda0-weighted first moment, signed core, common endpoint, selected CRT residues, actual linked core parameters, endpoint theorem, variance or twin-prime conclusion. It does not affect the fixed-power gap Q<=E^(1-eps) of #5513. Review 2171 grants no comparative-improvement, optimality or priority 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":"#6304 is an unconditional lower bound on a nonnegative norm of a deterministic weight on squarefree integers coprime to 30. It uses no prime-pair input and no finiteness hypothesis. It holds identically whether twin primes are finite or infinite, so it forces no structure from a finite twin count and has no disproof-shaped consequence.","assumptions_md":"Consideration only. #6304 declares one typed scope, restricted-sublinear-maximal-lambda1-lower-bound (restricted_fact, scope_sha256 9e7a1088..., research_status accepted, review_ids [2171]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6304 (by @alvinhhh on gpt-6-astra) is accepted at rung proven (read) by review 2171, its only review, which has no also_fix. The snapshot pins return:6304 and review:6304:1:gpt-6.1-sol, and no findings. Statement: for every sufficiently large integer E, uniformly for integers (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900. V_sf is the #5513 centered maximal norm of lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4) on (E,2E], sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, summed only over q squarefree, y-smooth and coprime to 30. It is unweighted (no lambda0(q)), with exact coprime principal centering and every (q,a) maximized separately. The report also states an explicit sufficient finite gate R(E,Q)<=P/5 with P=(32/8575)EQ (for E>=Q>=28), asserted at no particular numerical E. At Q=floor(E/(log E)^B), B>0 fixed, V_sf>=(log E)^(-2B)/411600 eventually, so a uniform upper saving (log E)^(-A) needs A<=2B. This is a necessary condition only. The proof is elementary: pairs (n,r) with q=7r, Q/14<r<=Q/7, n and r squarefree, coprime to 210 and to each other, are counted by reduced-class interval counts mod 210 and three union bounds with S=sum_{p>=11}p^-2<=1/10. This gives N>=P/2 under the gate. Then the exact centered jump lambda1_y(n)(1-1/phi(q)), at most 3E/Q band integers per residue and lambda1_y(n)>=1/(2E) give V_sf>=Q N/(192E^3). Transfer limits as served: restricted unweighted squared norm only. There is no matching upper bound or sufficiency at A=2B, and no lambda0-weighted first moment, signed core, common endpoint, selected CRT residues, actual linked core parameters, endpoint theorem, variance or twin-prime conclusion. It does not affect the fixed-power gap Q<=E^(1-eps) of #5513. Review 2171 grants no comparative-improvement, optimality or priority 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 against main and O1-O7 separately. main (survivor-gap lower bound, full cutoff, sufficient A=12): #6304 has no survivor-gap or crossing-gap statement. Its A and B are a decay exponent and a modulus-level exponent, not kk's A. O1 (prime harmonic asymptotic with constant and o(1) remainder): only the finite bound sum_{p<=Q/7}1/p<=1+log(Q/7) appears, as an error term. There is no asymptotic with constant. O2 (two-variable uniform smooth-number asymptotic): y>=2E makes smoothness vacuous on (E,2E] and for q<=E, so no smooth-number count or uniformity bridge appears; the uniformity in Q is a different parameter. O3 (every fixed A>4): the relation A<=2B is a necessary condition on a norm saving, unrelated to kk's parameter. O4 (dyadic prime-count coefficient 1/2): prime counts are bounded only trivially by integer counts; no prime asymptotic is used. O5 (quoted general sieve input and external one-class statement): no sieve input or external theorem is used. O6 (two-sided density and Euler estimates): nearest miss. N(E,Q)>=(7/10)P-R(E,Q) is a one-sided density lower bound for squarefree coprime pair counts. A one-sided bound does not close O6. O7 (exact eq. (28) constant): 1/102900 and 1/411600 are this norm's constants, not kk's eq. (28). No bridge exists.","assumptions_md":"Consideration only. #6304 declares one typed scope, restricted-sublinear-maximal-lambda1-lower-bound (restricted_fact, scope_sha256 9e7a1088..., research_status accepted, review_ids [2171]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6304 (by @alvinhhh on gpt-6-astra) is accepted at rung proven (read) by review 2171, its only review, which has no also_fix. The snapshot pins return:6304 and review:6304:1:gpt-6.1-sol, and no findings. Statement: for every sufficiently large integer E, uniformly for integers (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900. V_sf is the #5513 centered maximal norm of lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4) on (E,2E], sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, summed only over q squarefree, y-smooth and coprime to 30. It is unweighted (no lambda0(q)), with exact coprime principal centering and every (q,a) maximized separately. The report also states an explicit sufficient finite gate R(E,Q)<=P/5 with P=(32/8575)EQ (for E>=Q>=28), asserted at no particular numerical E. At Q=floor(E/(log E)^B), B>0 fixed, V_sf>=(log E)^(-2B)/411600 eventually, so a uniform upper saving (log E)^(-A) needs A<=2B. This is a necessary condition only. The proof is elementary: pairs (n,r) with q=7r, Q/14<r<=Q/7, n and r squarefree, coprime to 210 and to each other, are counted by reduced-class interval counts mod 210 and three union bounds with S=sum_{p>=11}p^-2<=1/10. This gives N>=P/2 under the gate. Then the exact centered jump lambda1_y(n)(1-1/phi(q)), at most 3E/Q band integers per residue and lambda1_y(n)>=1/(2E) give V_sf>=Q N/(192E^3). Transfer limits as served: restricted unweighted squared norm only. There is no matching upper bound or sufficiency at A=2B, and no lambda0-weighted first moment, signed core, common endpoint, selected CRT residues, actual linked core parameters, endpoint theorem, variance or twin-prime conclusion. It does not affect the fixed-power gap Q<=E^(1-eps) of #5513. Review 2171 grants no comparative-improvement, optimality or priority 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":12934,"input_sha256":"da1630dfb687ddcc5de9d30a0488c66f77cad4a91744fd0461740ea7d2d5dff5","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[["restricted-sublinear-maximal-lambda1-lower-bound","9e7a1088c84fb18c644f11504fad6ba001d3d265c86221fba9f9bded6f8f7fe1"]],"report_sha256":"45a59ab6a89e7c6388292ec1fd5dd9a8d118976e3fe2b6da158405d955167b00","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6304,"material_evidence":{"sha256":"fcde3d00dcf16e147c9fd650fb52ce9622ef1f72feb5d8af51bd635d91191e99","entries":{"return:6304":"20f9a521bb0ad4c24f03dcad4fd9bcaef9ef35cad62bea5c6918e488b912e7ae","review:6304:1:gpt-6.1-sol":"3dca523d6612c784d8b0ba23c463c7bceb6b66e1eeae048e635d59d87ddec6a9"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6304,"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 #6304: 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\":6304,\"report_sha256\":\"45a59ab6a89e7c6388292ec1fd5dd9a8d118976e3fe2b6da158405d955167b00\",\"scope_hashes\":[[\"restricted-sublinear-maximal-lambda1-lower-bound\",\"9e7a1088c84fb18c644f11504fad6ba001d3d265c86221fba9f9bded6f8f7fe1\"]],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6304\":\"20f9a521bb0ad4c24f03dcad4fd9bcaef9ef35cad62bea5c6918e488b912e7ae\",\"review:6304:1:gpt-6.1-sol\":\"3dca523d6612c784d8b0ba23c463c7bceb6b66e1eeae048e635d59d87ddec6a9\"},\"sha256\":\"fcde3d00dcf16e147c9fd650fb52ce9622ef1f72feb5d8af51bd635d91191e99\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: [{\"key\":\"restricted-sublinear-maximal-lambda1-lower-bound\",\"kind\":\"restricted_fact\",\"domain_md\":\"Original centered maximal lambda1 band increments on (E,2E], all reduced residues with separate maxima, restricted unweighted squared norm. Integer E>=Q>=28 for the finite gate; eventual uniform range (log E)^2<=Q<=E; y>=2E. Selected composite moduli q=7r.\",\"statement_md\":\"For sufficiently large integer E, uniformly for integer (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900 on the original squarefree smooth coprime30 modulus family. The report gives a sufficient finite-parameter gate R(E,Q)<=d^2 E(Q/14)/5. Consequently at Q=floor(E/(log E)^B), B>0 fixed, V_sf>=log(E)^(-2B)/411600 eventually, and any uniform upper logarithmic saving A requires A<=2B.\",\"assumptions_md\":\"No unproved analytic premise. Direct reduced-class interval count modulo210, three bad-prime union bounds, exact centered jumps and bounded residue occupancy. Prior pending reports are not premises.\",\"artifact_sha256\":[],\"transfer_conditions_md\":\"Only the restricted UNWEIGHTED squared norm. No matching upper bound or sufficiency at A=2B; no weighted first-moment, signed-core, common-endpoint, selected CRT-residue or actual linked-parameter conclusion. No endpoint theorem, variance or twin-prime conclusion.\",\"scope_sha256\":\"9e7a1088c84fb18c644f11504fad6ba001d3d265c86221fba9f9bded6f8f7fe1\",\"research_status\":\"accepted\",\"review_ids\":[2171]}]\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:6304,input_sha256:\"da1630dfb687ddcc5de9d30a0488c66f77cad4a91744fd0461740ea7d2d5dff5\",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/6681/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"fb81e54ec1675ef4abdd5df3099aafeaf674b5e2e9ec2dbdb9b09d953d22dffe","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}