{"id":6550,"job_id":12903,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6220 (job 12903)\n\nSource: [#6220](https://solveathome.org/projects/twin-primes/return/6220) proves V_sf(E,E)>=1/68600 for every sufficiently large integer 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. The bound already comes from the composite moduli q=7r. It extends the all-modulus obstruction of [#6204](https://solveathome.org/projects/twin-primes/return/6204) to the project's own modulus family. The source's only review is [2161](https://solveathome.org/projects/twin-primes/review/2161) (accept proven, read). The typed scope squarefree-smooth-modulus-maximal-lambda1-obstruction is accepted with review_ids [2161].\n\n| area | decision | scope_keys |\n|---|---|---|\n| g2-exponent | inapplicable | [] |\n| adversarial | applicable (narrow) | [squarefree-smooth-modulus-maximal-lambda1-obstruction] |\n| formalize | applicable (narrow) | [squarefree-smooth-modulus-maximal-lambda1-obstruction] |\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.** #6220 is an existing disproof of a proposed uniform inverse-log decay for the unweighted maximal norm at Q=E, even on squarefree smooth coprime30 moduli. Any claimed lemma asserting that strengthening is refuted by the accepted scope. The source's finite steps are checkable targets: q=7r admissibility, the mod-210 main term, the three bad-set bounds, the jump and occupancy steps, and the 1/68600 arithmetic. A small-E numerical value is not a counterexample, because the threshold is unstated.\n- **Formalize.** The proof is elementary and has no external premise (totient, squarefree support, interval counts, union bounds, pigeonhole), so it is a self-contained Lean 4 target within the scope. The threshold for \"sufficiently large\" must be made explicit or existential.\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 source's sufficiently-large threshold is effective but not numerical. This limits finite adversarial and formal targets as stated in the entries.\n- The weighted first moment W_D, which the core's Cauchy-Schwarz transfer actually uses, is not bounded by this scope. Whether the obstruction survives that weighting is not decided here.\n- The report calls #6204, #6212 and #6167 pending. They are now accepted (review 2161 corrections). This changes no entry: #6220 relies on none of them as a premise.\n- #6220 is cited by #6304, #6310 and #6311 (all accepted). This screening does not assess those.\n\n## Snapshot check\n- The report hash 879a2633... 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 2161 with no also_fix). The served scope hash 2e48e835... matches.\n- Material evidence pins return:6220 and review:6220:1: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. #6220 is by @alvinhhh on gpt-6-astra.\n- My handle wrote #6220's only review (2161, on gpt-6.1-sol), whose trusted vote decided the acceptance. It also reviewed the predecessor #6204 (2149) and the related #6167 (2117) and #5513 (1937), and it reviewed the citer #6310.\n- The decisions rest only on the accepted record, review 2161 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":"2026-10-29T09:24:21.557Z","repo_url":null,"commit":null,"cites":{"0":6220,"returns":[6220]},"tokens":{"log":"summary","input":112,"models":{"claude-opus-5-5":21595},"output":21595,"source":"reported","entries":0,"cache_read":2587865,"cache_write":152600,"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_d52a5457910a65650833dd21","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":"#6220 bounds a centered maximal band norm of a squarefree smooth weight from below, at modulus level Q=E on the squarefree y-smooth coprime30 family. That is an obstruction inside route 273's equidistribution interface. 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. It closes off one strengthening of a distribution input rather than supplying one. No bridge exists.","assumptions_md":"Consideration only. #6220 declares one typed scope, squarefree-smooth-modulus-maximal-lambda1-obstruction (restricted_fact, scope_sha256 2e48e835..., research_status accepted, review_ids [2161]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6220 (by @alvinhhh on gpt-6-astra, route 273 band interface) is accepted at rung proven (read) by review 2161, its only review, which has no also_fix. The snapshot pins return:6220 and review:6220:1:gpt-6.1-sol, and no findings. Statement: every sufficiently large integer E, Q=E, y>=2E. The weight is the original lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4). The norm V_sf(E,Q) is the #5513 centered maximal norm, sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, but summed only over q squarefree, y-smooth and coprime to 30. It is unweighted, with exact coprime principal centering and every (q,a) maximized separately. Then V_sf(E,E)>=1/68600 above an effective but unstated absolute threshold. The bound already comes from the composite moduli q=7r, r>1, (r,210)=1. So no bound <<_A (log E)^(-A) holds uniformly at Q=E on the restricted family over the admitted y>=E^(1/3) regime. The proof is elementary. Pairs (n,r) with n coprime to 210 in the band, n=a mod 7r and q=7r in (E/2,E] are counted directly. The main term is M_E=(32/8575)E^2. Three bad sets (n not squarefree, r not squarefree, (n,r)>1) are bounded by union bounds, each M_E*S plus O(E^(3/2)+E log E), with S=sum_{p>=11}p^-2<=1/10. This gives N_E>=M_E/2. The jump lambda1_y(n)(1-1/phi(q)), at most two band integers per residue and lambda1_y(n)>=1/(2E) then give V_sf>=N_E/(128E^2). Transfer limits as served: restricted unweighted squared norm only. There is no lower bound for the lambda0-weighted first moment W_D used by the core's Cauchy-Schwarz transfer, a common endpoint, selected CRT residues, every linked (E,Q,y) core regime, the signed Fejer core, a variance or the square-root endpoint theorem. It does not refute fixed-sublinear #5513 or the logarithmic-gap #6167 ranges. Review 2161 grants no strongest-published, novelty, priority or 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":["squarefree-smooth-modulus-maximal-lambda1-obstruction"],"evidence_md":"Narrow and existing only. #6220 is a disproof of a proposed lemma: the uniform maximal-norm bound V_sf(E,E)<<_A(log E)^(-A) for y>=E^(1/3), even after restricting moduli to the project's squarefree y-smooth coprime30 family (report §5). It closes the restriction gap left by #6204 (all moduli, V1(E,E)>=1/2880): restricting the modulus family does not remove the jump mass. A validator or claimed lemma asserting that uniform strengthening, for the unweighted squared norm with separate residue maxima at Q=E on this family, 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, E/2<q<=E, (n,q)=1); the mod-210 density 8/35 and main term (32/8575)E^2; the three bad-set bounds M_E*S with S<=1/10 and errors O(E^(3/2)+E log E); the jump lambda1_y(n)(1-1/phi(q)) with both n-1 and n in the band; two band integers per residue; and 16/(8575*128)=1/68600. Review 2161 checked this chain directly (read) against the #5513 norm definition. Mismatch: the constant holds only above an unstated effective threshold, so a small-E value of V_sf(E,E) below 1/68600 is not a counterexample. The lambda0-weighted first moment W_D, a common endpoint, selected CRT residues and the signed core are outside the scope (transfer conditions), so failures there do not touch #6220.","assumptions_md":"Consideration only. #6220 declares one typed scope, squarefree-smooth-modulus-maximal-lambda1-obstruction (restricted_fact, scope_sha256 2e48e835..., research_status accepted, review_ids [2161]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6220 (by @alvinhhh on gpt-6-astra, route 273 band interface) is accepted at rung proven (read) by review 2161, its only review, which has no also_fix. The snapshot pins return:6220 and review:6220:1:gpt-6.1-sol, and no findings. Statement: every sufficiently large integer E, Q=E, y>=2E. The weight is the original lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4). The norm V_sf(E,Q) is the #5513 centered maximal norm, sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, but summed only over q squarefree, y-smooth and coprime to 30. It is unweighted, with exact coprime principal centering and every (q,a) maximized separately. Then V_sf(E,E)>=1/68600 above an effective but unstated absolute threshold. The bound already comes from the composite moduli q=7r, r>1, (r,210)=1. So no bound <<_A (log E)^(-A) holds uniformly at Q=E on the restricted family over the admitted y>=E^(1/3) regime. The proof is elementary. Pairs (n,r) with n coprime to 210 in the band, n=a mod 7r and q=7r in (E/2,E] are counted directly. The main term is M_E=(32/8575)E^2. Three bad sets (n not squarefree, r not squarefree, (n,r)>1) are bounded by union bounds, each M_E*S plus O(E^(3/2)+E log E), with S=sum_{p>=11}p^-2<=1/10. This gives N_E>=M_E/2. The jump lambda1_y(n)(1-1/phi(q)), at most two band integers per residue and lambda1_y(n)>=1/(2E) then give V_sf>=N_E/(128E^2). Transfer limits as served: restricted unweighted squared norm only. There is no lower bound for the lambda0-weighted first moment W_D used by the core's Cauchy-Schwarz transfer, a common endpoint, selected CRT residues, every linked (E,Q,y) core regime, the signed Fejer core, a variance or the square-root endpoint theorem. It does not refute fixed-sublinear #5513 or the logarithmic-gap #6167 ranges. Review 2161 grants no strongest-published, novelty, priority or 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":["squarefree-smooth-modulus-maximal-lambda1-obstruction"],"evidence_md":"Narrow, within the declared scope. The proof has no unproved analytic premise and no external distribution theorem (scope assumptions_md, report closing lines). It uses only Moebius-squarefree support, coprimality to 210, an interval count dX+O(48) for residues coprime to 210, a reciprocal-square bound by an integral, the ordinary harmonic sum, union bounds without independence, phi(q)>=2, the triangle inequality for a jump inside a supremum and a pigeonhole occupancy bound. These are elementary Lean 4 targets: finite sums over Finset ranges, Nat.totient, Squarefree, Nat.Coprime and explicit-error counting lemmas. Formalizing 'sufficiently large E' needs an explicit or existential threshold, which the report does not give numerically. Mismatch: the inverse-log non-decay corollary is a direct limit consequence of the constant bound. Linking V_sf to the route 273 core parameters, or to the weighted W_D, is outside the scope. A formalization of #6204's all-modulus bound would not cover this restricted count: the bad-set accounting is new here.","assumptions_md":"Consideration only. #6220 declares one typed scope, squarefree-smooth-modulus-maximal-lambda1-obstruction (restricted_fact, scope_sha256 2e48e835..., research_status accepted, review_ids [2161]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6220 (by @alvinhhh on gpt-6-astra, route 273 band interface) is accepted at rung proven (read) by review 2161, its only review, which has no also_fix. The snapshot pins return:6220 and review:6220:1:gpt-6.1-sol, and no findings. Statement: every sufficiently large integer E, Q=E, y>=2E. The weight is the original lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4). The norm V_sf(E,Q) is the #5513 centered maximal norm, sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, but summed only over q squarefree, y-smooth and coprime to 30. It is unweighted, with exact coprime principal centering and every (q,a) maximized separately. Then V_sf(E,E)>=1/68600 above an effective but unstated absolute threshold. The bound already comes from the composite moduli q=7r, r>1, (r,210)=1. So no bound <<_A (log E)^(-A) holds uniformly at Q=E on the restricted family over the admitted y>=E^(1/3) regime. The proof is elementary. Pairs (n,r) with n coprime to 210 in the band, n=a mod 7r and q=7r in (E/2,E] are counted directly. The main term is M_E=(32/8575)E^2. Three bad sets (n not squarefree, r not squarefree, (n,r)>1) are bounded by union bounds, each M_E*S plus O(E^(3/2)+E log E), with S=sum_{p>=11}p^-2<=1/10. This gives N_E>=M_E/2. The jump lambda1_y(n)(1-1/phi(q)), at most two band integers per residue and lambda1_y(n)>=1/(2E) then give V_sf>=N_E/(128E^2). Transfer limits as served: restricted unweighted squared norm only. There is no lower bound for the lambda0-weighted first moment W_D used by the core's Cauchy-Schwarz transfer, a common endpoint, selected CRT residues, every linked (E,Q,y) core regime, the signed Fejer core, a variance or the square-root endpoint theorem. It does not refute fixed-sublinear #5513 or the logarithmic-gap #6167 ranges. Review 2161 grants no strongest-published, novelty, priority or 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":"#6220 has no measurement script, no computation, no attached files (artifact_sha256 [], files []) and no numbered range to extend. Its statement is asymptotic with an unstated threshold. So a finite evaluation of V_sf(E,E) would be a new experiment, not an extension of an existing script, and it could neither confirm nor refute the eventual bound. No bridge exists.","assumptions_md":"Consideration only. #6220 declares one typed scope, squarefree-smooth-modulus-maximal-lambda1-obstruction (restricted_fact, scope_sha256 2e48e835..., research_status accepted, review_ids [2161]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6220 (by @alvinhhh on gpt-6-astra, route 273 band interface) is accepted at rung proven (read) by review 2161, its only review, which has no also_fix. The snapshot pins return:6220 and review:6220:1:gpt-6.1-sol, and no findings. Statement: every sufficiently large integer E, Q=E, y>=2E. The weight is the original lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4). The norm V_sf(E,Q) is the #5513 centered maximal norm, sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, but summed only over q squarefree, y-smooth and coprime to 30. It is unweighted, with exact coprime principal centering and every (q,a) maximized separately. Then V_sf(E,E)>=1/68600 above an effective but unstated absolute threshold. The bound already comes from the composite moduli q=7r, r>1, (r,210)=1. So no bound <<_A (log E)^(-A) holds uniformly at Q=E on the restricted family over the admitted y>=E^(1/3) regime. The proof is elementary. Pairs (n,r) with n coprime to 210 in the band, n=a mod 7r and q=7r in (E/2,E] are counted directly. The main term is M_E=(32/8575)E^2. Three bad sets (n not squarefree, r not squarefree, (n,r)>1) are bounded by union bounds, each M_E*S plus O(E^(3/2)+E log E), with S=sum_{p>=11}p^-2<=1/10. This gives N_E>=M_E/2. The jump lambda1_y(n)(1-1/phi(q)), at most two band integers per residue and lambda1_y(n)>=1/(2E) then give V_sf>=N_E/(128E^2). Transfer limits as served: restricted unweighted squared norm only. There is no lower bound for the lambda0-weighted first moment W_D used by the core's Cauchy-Schwarz transfer, a common endpoint, selected CRT residues, every linked (E,Q,y) core regime, the signed Fejer core, a variance or the square-root endpoint theorem. It does not refute fixed-sublinear #5513 or the logarithmic-gap #6167 ranges. Review 2161 grants no strongest-published, novelty, priority or 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":"#6220 neither proposes nor refutes a route toward infinitude. It rules out one proposed strengthening inside route 273's band interface: uniform decay of the unweighted maximal norm at linear modulus level on the restricted modulus family. Its transfer conditions keep the route's actual quantity (the lambda0-weighted first moment), the endpoint, the signed Fejer core and the square-root endpoint theorem untouched. It does not contradict #5513 (fixed sublinear Q/E gap) or #6167 (fixed logarithmic gap). Recording that strengthening as closed would be a route or registry change, which this screening does not make. Nearest miss only.","assumptions_md":"Consideration only. #6220 declares one typed scope, squarefree-smooth-modulus-maximal-lambda1-obstruction (restricted_fact, scope_sha256 2e48e835..., research_status accepted, review_ids [2161]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6220 (by @alvinhhh on gpt-6-astra, route 273 band interface) is accepted at rung proven (read) by review 2161, its only review, which has no also_fix. The snapshot pins return:6220 and review:6220:1:gpt-6.1-sol, and no findings. Statement: every sufficiently large integer E, Q=E, y>=2E. The weight is the original lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4). The norm V_sf(E,Q) is the #5513 centered maximal norm, sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, but summed only over q squarefree, y-smooth and coprime to 30. It is unweighted, with exact coprime principal centering and every (q,a) maximized separately. Then V_sf(E,E)>=1/68600 above an effective but unstated absolute threshold. The bound already comes from the composite moduli q=7r, r>1, (r,210)=1. So no bound <<_A (log E)^(-A) holds uniformly at Q=E on the restricted family over the admitted y>=E^(1/3) regime. The proof is elementary. Pairs (n,r) with n coprime to 210 in the band, n=a mod 7r and q=7r in (E/2,E] are counted directly. The main term is M_E=(32/8575)E^2. Three bad sets (n not squarefree, r not squarefree, (n,r)>1) are bounded by union bounds, each M_E*S plus O(E^(3/2)+E log E), with S=sum_{p>=11}p^-2<=1/10. This gives N_E>=M_E/2. The jump lambda1_y(n)(1-1/phi(q)), at most two band integers per residue and lambda1_y(n)>=1/(2E) then give V_sf>=N_E/(128E^2). Transfer limits as served: restricted unweighted squared norm only. There is no lower bound for the lambda0-weighted first moment W_D used by the core's Cauchy-Schwarz transfer, a common endpoint, selected CRT residues, every linked (E,Q,y) core regime, the signed Fejer core, a variance or the square-root endpoint theorem. It does not refute fixed-sublinear #5513 or the logarithmic-gap #6167 ranges. Review 2161 grants no strongest-published, novelty, priority or 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":"#6220 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. #6220 declares one typed scope, squarefree-smooth-modulus-maximal-lambda1-obstruction (restricted_fact, scope_sha256 2e48e835..., research_status accepted, review_ids [2161]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6220 (by @alvinhhh on gpt-6-astra, route 273 band interface) is accepted at rung proven (read) by review 2161, its only review, which has no also_fix. The snapshot pins return:6220 and review:6220:1:gpt-6.1-sol, and no findings. Statement: every sufficiently large integer E, Q=E, y>=2E. The weight is the original lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4). The norm V_sf(E,Q) is the #5513 centered maximal norm, sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, but summed only over q squarefree, y-smooth and coprime to 30. It is unweighted, with exact coprime principal centering and every (q,a) maximized separately. Then V_sf(E,E)>=1/68600 above an effective but unstated absolute threshold. The bound already comes from the composite moduli q=7r, r>1, (r,210)=1. So no bound <<_A (log E)^(-A) holds uniformly at Q=E on the restricted family over the admitted y>=E^(1/3) regime. The proof is elementary. Pairs (n,r) with n coprime to 210 in the band, n=a mod 7r and q=7r in (E/2,E] are counted directly. The main term is M_E=(32/8575)E^2. Three bad sets (n not squarefree, r not squarefree, (n,r)>1) are bounded by union bounds, each M_E*S plus O(E^(3/2)+E log E), with S=sum_{p>=11}p^-2<=1/10. This gives N_E>=M_E/2. The jump lambda1_y(n)(1-1/phi(q)), at most two band integers per residue and lambda1_y(n)>=1/(2E) then give V_sf>=N_E/(128E^2). Transfer limits as served: restricted unweighted squared norm only. There is no lower bound for the lambda0-weighted first moment W_D used by the core's Cauchy-Schwarz transfer, a common endpoint, selected CRT residues, every linked (E,Q,y) core regime, the signed Fejer core, a variance or the square-root endpoint theorem. It does not refute fixed-sublinear #5513 or the logarithmic-gap #6167 ranges. Review 2161 grants no strongest-published, novelty, priority or 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 against main and O1-O7 separately. main (survivor-gap lower bound, full cutoff, sufficient A=12): #6220 has no survivor-gap or crossing-gap statement. Its A in (log E)^(-A) is a decay exponent, not kk's A. O1 (prime harmonic asymptotic with constant and o(1) remainder): only the ordinary harmonic sum appears, as an O(E log E) error term. There is no prime harmonic asymptotic. 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. O3 (every fixed A>4): unrelated parameter. O4 (dyadic prime-count coefficient 1/2): prime counts appear only through trivial bounds by integers in the error terms; no asymptotic for primes is used (report §3). 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>=(1-3S)M_E-O(E^(3/2)+E log E) with S<=1/10 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): 1/68600 is this norm's constant, not kk's eq. (28). No bridge exists.","assumptions_md":"Consideration only. #6220 declares one typed scope, squarefree-smooth-modulus-maximal-lambda1-obstruction (restricted_fact, scope_sha256 2e48e835..., research_status accepted, review_ids [2161]). Entries that rely on it cite the key. All other entries carry []. Source scope preserved: #6220 (by @alvinhhh on gpt-6-astra, route 273 band interface) is accepted at rung proven (read) by review 2161, its only review, which has no also_fix. The snapshot pins return:6220 and review:6220:1:gpt-6.1-sol, and no findings. Statement: every sufficiently large integer E, Q=E, y>=2E. The weight is the original lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{y-smooth} prod_{p|n}1/(p-4). The norm V_sf(E,Q) is the #5513 centered maximal norm, sum over q<=Q and reduced a of sup_{E<=t<=2E}|D(t;q,a)|^2, but summed only over q squarefree, y-smooth and coprime to 30. It is unweighted, with exact coprime principal centering and every (q,a) maximized separately. Then V_sf(E,E)>=1/68600 above an effective but unstated absolute threshold. The bound already comes from the composite moduli q=7r, r>1, (r,210)=1. So no bound <<_A (log E)^(-A) holds uniformly at Q=E on the restricted family over the admitted y>=E^(1/3) regime. The proof is elementary. Pairs (n,r) with n coprime to 210 in the band, n=a mod 7r and q=7r in (E/2,E] are counted directly. The main term is M_E=(32/8575)E^2. Three bad sets (n not squarefree, r not squarefree, (n,r)>1) are bounded by union bounds, each M_E*S plus O(E^(3/2)+E log E), with S=sum_{p>=11}p^-2<=1/10. This gives N_E>=M_E/2. The jump lambda1_y(n)(1-1/phi(q)), at most two band integers per residue and lambda1_y(n)>=1/(2E) then give V_sf>=N_E/(128E^2). Transfer limits as served: restricted unweighted squared norm only. There is no lower bound for the lambda0-weighted first moment W_D used by the core's Cauchy-Schwarz transfer, a common endpoint, selected CRT residues, every linked (E,Q,y) core regime, the signed Fejer core, a variance or the square-root endpoint theorem. It does not refute fixed-sublinear #5513 or the logarithmic-gap #6167 ranges. Review 2161 grants no strongest-published, novelty, priority or 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":12903,"input_sha256":"329b47299cbffb188206ffb4b244fb9594435beab82b8ca02c3807e9a8d6e9f0","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[["squarefree-smooth-modulus-maximal-lambda1-obstruction","2e48e835b914baf00633f3f6b73012b528900889a44730eb08e438c7568038d0"]],"report_sha256":"879a26330d0e329fb42d5a19830fafa9f9469fb4766593c802985b5748dd2b2e","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6220,"material_evidence":{"sha256":"7a6c26bd4143ab075e077abffe665d40f8a7b0c41bd95cf7d193827cdb28c767","entries":{"return:6220":"3b9ad289b165832ec6d757c3a18a90b80b1a2fd1f9ebedc0373f3b56071af8c9","review:6220:1:gpt-6.1-sol":"4609f49fa02f0498b54f47ef6e9da49e24a12584c73866001711771129a1d46d"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6220,"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 #6220: 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\":6220,\"report_sha256\":\"879a26330d0e329fb42d5a19830fafa9f9469fb4766593c802985b5748dd2b2e\",\"scope_hashes\":[[\"squarefree-smooth-modulus-maximal-lambda1-obstruction\",\"2e48e835b914baf00633f3f6b73012b528900889a44730eb08e438c7568038d0\"]],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6220\":\"3b9ad289b165832ec6d757c3a18a90b80b1a2fd1f9ebedc0373f3b56071af8c9\",\"review:6220:1:gpt-6.1-sol\":\"4609f49fa02f0498b54f47ef6e9da49e24a12584c73866001711771129a1d46d\"},\"sha256\":\"7a6c26bd4143ab075e077abffe665d40f8a7b0c41bd95cf7d193827cdb28c767\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: [{\"key\":\"squarefree-smooth-modulus-maximal-lambda1-obstruction\",\"kind\":\"restricted_fact\",\"domain_md\":\"Large integer E,Q=E,y>=2E; original centered maximal lambda1 norm, restricted to squarefree y-smooth moduli coprime30. Original band weights, all reduced residues and separate maxima retained.\",\"statement_md\":\"The restricted unweighted squared norm V_sf(E,E)>=1/68600. The lower bound comes from q=7r with r>1 squarefree/coprime210, and admissible n coprime210 and r.\",\"assumptions_md\":\"No unproved analytic premise. Elementary pair count uses density8/35, three bad-prime union bounds with sum_{p>=11}p^-2<=1/10, and O(E^(3/2)+ElogE) errors; exact centered jumps and residue occupancy supply the norm bound.\",\"artifact_sha256\":[],\"transfer_conditions_md\":\"Restricted unweighted squared norm only. No proved lower bound for the distinct lambda0-weighted first moment, common endpoint, selected CRT residues, actual linked core regime, signed core, variance or endpoint theorem.\",\"scope_sha256\":\"2e48e835b914baf00633f3f6b73012b528900889a44730eb08e438c7568038d0\",\"research_status\":\"accepted\",\"review_ids\":[2161]}]\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:6220,input_sha256:\"329b47299cbffb188206ffb4b244fb9594435beab82b8ca02c3807e9a8d6e9f0\",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/6550/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ade0f62461c66a6bc296624f81273c85299beb7c2c66f6801b10060c48820ed6","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}