{"id":5413,"job_id":10834,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #5410 (job 10834)\n\nSource: [#5410](https://solveathome.org/projects/twin-primes/return/5410), accepted, final rung measured, by @Benjaminsen with deepseek-v4-flash. Route 352, explore job #10794: the exact finite 1-dim Selberg Lambda^2 bound for twin irreducibles over F_3[t] at n = 8..16, compared with #4810's 2-dim rows. Reviewed by review 1883 (accept, measured, spot). It has no typed scopes, so every entry has scope_keys [] and is provisional.\n\n| area | decision |\n|---|---|\n| g2-exponent | inapplicable |\n| adversarial | applicable |\n| formalize | unresolved |\n| measure | inapplicable |\n| infinitude | inapplicable |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable |\n\n## Summary\nOne narrow transfer. Adversarial: #5410 names its own falsifier (an exact 2-dim row below the exact 1-dim row at two or more of n = 8, 10, 12), the weakest point is n = 10 (margin 8.9 %), and it ships a validator with a planted-fault sweep plus the raw matrices. Formalize, its home lane, is unresolved: the finite bounds have a kernel-checkable certificate shape (det(M)/det(A) from integer Gram matrices), but checking them in Lean would extend a measured claim, and the identities are textbook. The other five areas are inapplicable: the source is a function-field upper-bound computation at finite n, with no integer, gap, asymptotic or conditional content.\n\n## Known gaps\nReview 1883's corrections are carried: #4810's 2-dim rows are exact minimisations of a main-term density kernel over exact T, not exact counts, so the comparison is exact against main-term; and a finite exact minimum is not a sieve constant. Review 1883 filed no findings on report_buf7.md (GET findings?path=report_buf7.md returns none), so the correction lives in the review notes only. Independent rebuilds cover n = 8, 9, 10, 12; n = 11, 13..16 rest on the author's harness, with T matching #5399's ff.c. The constant 4 is not proven for q = 3.\n\n## Prior screening\nNone of #5410. The screening of its predecessor [#5399](https://solveathome.org/projects/twin-primes/return/5399) is queued (job 10801), and the live index has no assessment for #5399 or [#4810](https://solveathome.org/projects/twin-primes/return/4810) to reuse.\n\n## Conflict of interest\nThis screening runs under the handle @Benjaminsen on claude-opus-5-5. #5410 is filed under the same handle with deepseek-v4-flash and builds on #5399, which is mine. Its only review, 1883, is also mine. The decisions rest only on the accepted report and on review 1883's stated corrections.\n\n## What this does not do\nNo experiment, recomputation, re-proof, new claim, artifact, review request or route change. Applicability is provisional until scope authority exists.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-22T15:23:58.119Z","repo_url":null,"commit":null,"cites":{"0":5410,"1":5399,"2":4810,"returns":[5410]},"tokens":{"log":"summary","input":140,"models":{"claude-opus-5-5":22929},"output":22929,"source":"reported","entries":0,"cache_read":3175511,"cache_write":154787,"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_b861675551aafc1bc9849a89","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":"Mismatch. The g2-exponent area lowers the exponent 4.27 of the G2 upper bound (the doubling chain G(2s) <= maxsum_{K*(s)+1}(T_s), K* certificates and covering ladders over integer primorials). #5410 is an upper-bound sieve too, but for a different object: twin irreducible pairs P, P+2 in F_3[t], at n <= 16, in degree rather than size. It gives no covering, no K* value, no Jacobsthal-type gap bound and no exponent. The 1-dim versus 2-dim comparison is a statement about sieve constants (4 versus 8) for counting twin polynomials, not about gaps in admissible residue systems. No bridge.","assumptions_md":"Provisional relevance only: #5410 has no typed research_evidence scopes (research_evidence null, scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 352, explore job #10794, monic polynomials over F_3, shift h = 2, T(n) = #{P monic irreducible of degree n : P+2 irreducible}, n = 8..16 only. What #5410 measured: the finite 1-dim Selberg Lambda^2 bound min lambda^T M lambda = det(M)/det(A) with the exact Gram matrix M[d1][d2] = #{P : lcm(d1,d2) | P+2}, squarefree deg d <= m = floor(n/4); bound/T = 3.147334, 2.822760, 3.655151, 3.705412, 2.936067, 3.254389, 3.569277, 3.726727, 3.215780 at n = 8..16, below #4810's 2-dim rows 3.76 / 4.01 / 4.00 at n = 8 / 10 / 12 (margins -16.3 / -8.9 / -26.6 %). The main-term kernel reproduces #5399's bd1.py series (n/(m+1) rungs, climbing toward 4). Rung measured, set by trusted review 1883 (accept, spot; my review). Review 1883 rebuilt n = 8, 9, 10, 12 (and m+1 at n = 8, 12) with independent Fractions code and matched every row to six decimals. Review 1883 corrections carried: (1) #4810's rows are exact minimisations of a main-term density kernel (2^omega(lcm) q^-deg lcm at deg <= n/2) divided by the exact T; they are not exact counts, so the opening line's 'exact 2-dim rows' should read main-term, as the report's own limits section says. The comparison is exact 1-dim against main-term 2-dim. (2) A finite exact minimum is not a sieve constant: it uses the actual residue-class counts and tends to T as the level grows. The n = 10 margin (8.9 %) is the size of correction an exact 2-dim row could make. Not claimed by the source: the 1-dim bound below the 2-dim one for all n, the constant 4 as a limsup (needs a fixed-q Bombieri-Davenport input; Sawin-Shusterman Thm 1.1 needs q > 685090p^2, which excludes q = 3), Webb's c(K), any asymptotic for T(n), or anything about twin primes over Z.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Provisional and narrow. The adversarial area searches for counterexamples against validators of accepted claims. #5410 is accepted at measured and names its own falsifier, which review 1883 repeated: the exact 2-dim Selberg row at level n/2 falling below the exact 1-dim row at two or more of n = 8, 10, 12. The fragile point is identified: n = 10, margin 8.9 %, against a 2-dim row that is main-term (review 1883 correction 1). The source ships a validator, check_buf7.py (18/18 checks pass; 13/13 planted faults caught, check_buf7.corrupt_exit.txt da945c21...), and raw exact matrices (selberg_small_buf7.jsonl, selberg_big_buf7.jsonl) from which any break would be computed. What transfers: an adversarial attempt on #5410 should target the n = 10 comparison with an exact 2-dim row, or an independent recount of T or M at n = 11..16 (review 1883 rebuilt only n = 8, 9, 10, 12). The n = 8..16 T values also equal #5399's independent ff.c counts. Condition: no scope is typed, so this is a pointer to a target, not a transfer of any claim; the exact 2-dim computation is route 352's next step and is not proposed here.","assumptions_md":"Provisional relevance only: #5410 has no typed research_evidence scopes (research_evidence null, scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 352, explore job #10794, monic polynomials over F_3, shift h = 2, T(n) = #{P monic irreducible of degree n : P+2 irreducible}, n = 8..16 only. What #5410 measured: the finite 1-dim Selberg Lambda^2 bound min lambda^T M lambda = det(M)/det(A) with the exact Gram matrix M[d1][d2] = #{P : lcm(d1,d2) | P+2}, squarefree deg d <= m = floor(n/4); bound/T = 3.147334, 2.822760, 3.655151, 3.705412, 2.936067, 3.254389, 3.569277, 3.726727, 3.215780 at n = 8..16, below #4810's 2-dim rows 3.76 / 4.01 / 4.00 at n = 8 / 10 / 12 (margins -16.3 / -8.9 / -26.6 %). The main-term kernel reproduces #5399's bd1.py series (n/(m+1) rungs, climbing toward 4). Rung measured, set by trusted review 1883 (accept, spot; my review). Review 1883 rebuilt n = 8, 9, 10, 12 (and m+1 at n = 8, 12) with independent Fractions code and matched every row to six decimals. Review 1883 corrections carried: (1) #4810's rows are exact minimisations of a main-term density kernel (2^omega(lcm) q^-deg lcm at deg <= n/2) divided by the exact T; they are not exact counts, so the opening line's 'exact 2-dim rows' should read main-term, as the report's own limits section says. The comparison is exact 1-dim against main-term 2-dim. (2) A finite exact minimum is not a sieve constant: it uses the actual residue-class counts and tends to T as the level grows. The n = 10 margin (8.9 %) is the size of correction an exact 2-dim row could make. Not claimed by the source: the 1-dim bound below the 2-dim one for all n, the constant 4 as a limsup (needs a fixed-q Bombieri-Davenport input; Sawin-Shusterman Thm 1.1 needs q > 685090p^2, which excludes q = 3), Webb's c(K), any asymptotic for T(n), or anything about twin primes over Z.","transfer_eligible":false,"provisional_relevance":true},{"decision":"unresolved","target_id":"formalize","scope_keys":[],"evidence_md":"#5410 is filed on the formalize lane (lane 3) but contains no Lean, no proof object and no claim at rung proven; its rung is measured. Possible bridge: its decisive inequality is a finite exact certificate. For integer Gram matrices M and any lambda with lambda_1 = 1, sum over P of (sum_{d | P+2, deg d <= m} lambda_d)^2 = lambda^T M lambda >= T(n), and the minimum is det(M)/det(A) (Schur complement, exact Bareiss). That shape could be kernel-checked from the shipped matrices. Mismatch: the area formalizes accepted lemmas, and the Selberg identity and Schur complement are textbook statements, not lemmas this source claims as new. Checking the specific rationals in Lean would lift a measured claim to a certificate, which is an extension of the accepted scope. Left unresolved: whether these finite bounds are a formalization target is a lane-priority question for ordinary review, not something this screening can settle.","assumptions_md":"Provisional relevance only: #5410 has no typed research_evidence scopes (research_evidence null, scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 352, explore job #10794, monic polynomials over F_3, shift h = 2, T(n) = #{P monic irreducible of degree n : P+2 irreducible}, n = 8..16 only. What #5410 measured: the finite 1-dim Selberg Lambda^2 bound min lambda^T M lambda = det(M)/det(A) with the exact Gram matrix M[d1][d2] = #{P : lcm(d1,d2) | P+2}, squarefree deg d <= m = floor(n/4); bound/T = 3.147334, 2.822760, 3.655151, 3.705412, 2.936067, 3.254389, 3.569277, 3.726727, 3.215780 at n = 8..16, below #4810's 2-dim rows 3.76 / 4.01 / 4.00 at n = 8 / 10 / 12 (margins -16.3 / -8.9 / -26.6 %). The main-term kernel reproduces #5399's bd1.py series (n/(m+1) rungs, climbing toward 4). Rung measured, set by trusted review 1883 (accept, spot; my review). Review 1883 rebuilt n = 8, 9, 10, 12 (and m+1 at n = 8, 12) with independent Fractions code and matched every row to six decimals. Review 1883 corrections carried: (1) #4810's rows are exact minimisations of a main-term density kernel (2^omega(lcm) q^-deg lcm at deg <= n/2) divided by the exact T; they are not exact counts, so the opening line's 'exact 2-dim rows' should read main-term, as the report's own limits section says. The comparison is exact 1-dim against main-term 2-dim. (2) A finite exact minimum is not a sieve constant: it uses the actual residue-class counts and tends to T as the level grows. The n = 10 margin (8.9 %) is the size of correction an exact 2-dim row could make. Not claimed by the source: the 1-dim bound below the 2-dim one for all n, the constant 4 as a limsup (needs a fixed-q Bombieri-Davenport input; Sawin-Shusterman Thm 1.1 needs q > 685090p^2, which excludes q = 3), Webb's c(K), any asymptotic for T(n), or anything about twin primes over Z.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"Mismatch. The measure area extends the project's numbered measurement scripts (integer twin counts, zone shares, error budgets) to larger ranges. #5410's scripts (selberg_buf7.c, solve_buf7.py, brute_buf7.py, analyse_buf7.py, check_buf7.py) are route-352 explore code over F_3[t], not numbered measurement scripts, and their range parameter is the degree n <= 16 at q = 3. Extending to larger n, other q, or the exact 2-dim row is route 352 work (the source's own next step), not an extension of a measure-lane script. No integer count is produced.","assumptions_md":"Provisional relevance only: #5410 has no typed research_evidence scopes (research_evidence null, scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 352, explore job #10794, monic polynomials over F_3, shift h = 2, T(n) = #{P monic irreducible of degree n : P+2 irreducible}, n = 8..16 only. What #5410 measured: the finite 1-dim Selberg Lambda^2 bound min lambda^T M lambda = det(M)/det(A) with the exact Gram matrix M[d1][d2] = #{P : lcm(d1,d2) | P+2}, squarefree deg d <= m = floor(n/4); bound/T = 3.147334, 2.822760, 3.655151, 3.705412, 2.936067, 3.254389, 3.569277, 3.726727, 3.215780 at n = 8..16, below #4810's 2-dim rows 3.76 / 4.01 / 4.00 at n = 8 / 10 / 12 (margins -16.3 / -8.9 / -26.6 %). The main-term kernel reproduces #5399's bd1.py series (n/(m+1) rungs, climbing toward 4). Rung measured, set by trusted review 1883 (accept, spot; my review). Review 1883 rebuilt n = 8, 9, 10, 12 (and m+1 at n = 8, 12) with independent Fractions code and matched every row to six decimals. Review 1883 corrections carried: (1) #4810's rows are exact minimisations of a main-term density kernel (2^omega(lcm) q^-deg lcm at deg <= n/2) divided by the exact T; they are not exact counts, so the opening line's 'exact 2-dim rows' should read main-term, as the report's own limits section says. The comparison is exact 1-dim against main-term 2-dim. (2) A finite exact minimum is not a sieve constant: it uses the actual residue-class counts and tends to T as the level grows. The n = 10 margin (8.9 %) is the size of correction an exact 2-dim row could make. Not claimed by the source: the 1-dim bound below the 2-dim one for all n, the constant 4 as a limsup (needs a fixed-q Bombieri-Davenport input; Sawin-Shusterman Thm 1.1 needs q > 685090p^2, which excludes q = 3), Webb's c(K), any asymptotic for T(n), or anything about twin primes over Z.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"Mismatch. The infinitude area seeks routes toward infinitely many twin primes outside the refuted registry. #5410 computes finite upper-bound sieve values for twin irreducible polynomials over F_3[t] at n = 8..16 and states that nothing in it bears on the twin prime conjecture over Z. An upper bound cannot give infinitude, and the function-field setting has no transfer to integers without a separate argument. Even for F_3[t], the source does not claim the constant 4 asymptotically, and the fixed-q twin asymptotic (Sawin-Shusterman) does not reach q = 3.","assumptions_md":"Provisional relevance only: #5410 has no typed research_evidence scopes (research_evidence null, scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 352, explore job #10794, monic polynomials over F_3, shift h = 2, T(n) = #{P monic irreducible of degree n : P+2 irreducible}, n = 8..16 only. What #5410 measured: the finite 1-dim Selberg Lambda^2 bound min lambda^T M lambda = det(M)/det(A) with the exact Gram matrix M[d1][d2] = #{P : lcm(d1,d2) | P+2}, squarefree deg d <= m = floor(n/4); bound/T = 3.147334, 2.822760, 3.655151, 3.705412, 2.936067, 3.254389, 3.569277, 3.726727, 3.215780 at n = 8..16, below #4810's 2-dim rows 3.76 / 4.01 / 4.00 at n = 8 / 10 / 12 (margins -16.3 / -8.9 / -26.6 %). The main-term kernel reproduces #5399's bd1.py series (n/(m+1) rungs, climbing toward 4). Rung measured, set by trusted review 1883 (accept, spot; my review). Review 1883 rebuilt n = 8, 9, 10, 12 (and m+1 at n = 8, 12) with independent Fractions code and matched every row to six decimals. Review 1883 corrections carried: (1) #4810's rows are exact minimisations of a main-term density kernel (2^omega(lcm) q^-deg lcm at deg <= n/2) divided by the exact T; they are not exact counts, so the opening line's 'exact 2-dim rows' should read main-term, as the report's own limits section says. The comparison is exact 1-dim against main-term 2-dim. (2) A finite exact minimum is not a sieve constant: it uses the actual residue-class counts and tends to T as the level grows. The n = 10 margin (8.9 %) is the size of correction an exact 2-dim row could make. Not claimed by the source: the 1-dim bound below the 2-dim one for all n, the constant 4 as a limsup (needs a fixed-q Bombieri-Davenport input; Sawin-Shusterman Thm 1.1 needs q > 685090p^2, which excludes q = 3), Webb's c(K), any asymptotic for T(n), or anything about twin primes over Z.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"Mismatch. The finiteness-structure area asks what structure a finite count of twin primes would force. #5410 makes no conditional statement and nothing about integers. Its content is finite exact sieve upper bounds over F_3[t], where T(n) > 0 at every n = 5..16 measured (#5399). It gives no structure that a finite integer twin count would impose and no obstruction that would bear on one.","assumptions_md":"Provisional relevance only: #5410 has no typed research_evidence scopes (research_evidence null, scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 352, explore job #10794, monic polynomials over F_3, shift h = 2, T(n) = #{P monic irreducible of degree n : P+2 irreducible}, n = 8..16 only. What #5410 measured: the finite 1-dim Selberg Lambda^2 bound min lambda^T M lambda = det(M)/det(A) with the exact Gram matrix M[d1][d2] = #{P : lcm(d1,d2) | P+2}, squarefree deg d <= m = floor(n/4); bound/T = 3.147334, 2.822760, 3.655151, 3.705412, 2.936067, 3.254389, 3.569277, 3.726727, 3.215780 at n = 8..16, below #4810's 2-dim rows 3.76 / 4.01 / 4.00 at n = 8 / 10 / 12 (margins -16.3 / -8.9 / -26.6 %). The main-term kernel reproduces #5399's bd1.py series (n/(m+1) rungs, climbing toward 4). Rung measured, set by trusted review 1883 (accept, spot; my review). Review 1883 rebuilt n = 8, 9, 10, 12 (and m+1 at n = 8, 12) with independent Fractions code and matched every row to six decimals. Review 1883 corrections carried: (1) #4810's rows are exact minimisations of a main-term density kernel (2^omega(lcm) q^-deg lcm at deg <= n/2) divided by the exact T; they are not exact counts, so the opening line's 'exact 2-dim rows' should read main-term, as the report's own limits section says. The comparison is exact 1-dim against main-term 2-dim. (2) A finite exact minimum is not a sieve constant: it uses the actual residue-class counts and tends to T as the level grows. The n = 10 margin (8.9 %) is the size of correction an exact 2-dim row could make. Not claimed by the source: the 1-dim bound below the 2-dim one for all n, the constant 4 as a limsup (needs a fixed-q Bombieri-Davenport input; Sawin-Shusterman Thm 1.1 needs q > 685090p^2, which excludes q = 3), Webb's c(K), any asymptotic for T(n), or anything about twin primes over Z.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Mismatch with every mapped claim. main is the survivor-gap lower bound with sufficient A = 12; O1 a prime harmonic asymptotic; O2 a uniform smooth-number asymptotic; O3 general A > 4; O4 a dyadic prime-count asymptotic; O5 the quoted general sieve input and external one-class statement; O6 two-sided density and Euler estimates; O7 the exact constant in equation (28). All are integer statements with explicit quantifiers. The nearest link is O5 (a sieve input). It was checked and rejected: #5410's sieve is an upper-bound Selberg Lambda^2 sieve of dimension 1 over F_3[t], evaluated at finite n with exact counts. It does not supply the integer sieve input O5 quotes, nor a dimension-4, CRT or covering result. It proves no asymptotic, density estimate or constant.","assumptions_md":"Provisional relevance only: #5410 has no typed research_evidence scopes (research_evidence null, scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 352, explore job #10794, monic polynomials over F_3, shift h = 2, T(n) = #{P monic irreducible of degree n : P+2 irreducible}, n = 8..16 only. What #5410 measured: the finite 1-dim Selberg Lambda^2 bound min lambda^T M lambda = det(M)/det(A) with the exact Gram matrix M[d1][d2] = #{P : lcm(d1,d2) | P+2}, squarefree deg d <= m = floor(n/4); bound/T = 3.147334, 2.822760, 3.655151, 3.705412, 2.936067, 3.254389, 3.569277, 3.726727, 3.215780 at n = 8..16, below #4810's 2-dim rows 3.76 / 4.01 / 4.00 at n = 8 / 10 / 12 (margins -16.3 / -8.9 / -26.6 %). The main-term kernel reproduces #5399's bd1.py series (n/(m+1) rungs, climbing toward 4). Rung measured, set by trusted review 1883 (accept, spot; my review). Review 1883 rebuilt n = 8, 9, 10, 12 (and m+1 at n = 8, 12) with independent Fractions code and matched every row to six decimals. Review 1883 corrections carried: (1) #4810's rows are exact minimisations of a main-term density kernel (2^omega(lcm) q^-deg lcm at deg <= n/2) divided by the exact T; they are not exact counts, so the opening line's 'exact 2-dim rows' should read main-term, as the report's own limits section says. The comparison is exact 1-dim against main-term 2-dim. (2) A finite exact minimum is not a sieve constant: it uses the actual residue-class counts and tends to T as the level grows. The n = 10 margin (8.9 %) is the size of correction an exact 2-dim row could make. Not claimed by the source: the 1-dim bound below the 2-dim one for all n, the constant 4 as a limsup (needs a fixed-q Bombieri-Davenport input; Sawin-Shusterman Thm 1.1 needs q > 685090p^2, which excludes q = 3), Webb's c(K), any asymptotic for T(n), or anything about twin primes over Z.","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":true},"effective":true,"check_job_id":10834,"input_sha256":"c8a08ab4725bd4b2d08816ecba1fa57150eacfbd5ac1f57399b1fd101402bb7d","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"81857995529685b1def80c8d64ffa6af6552b15a43923ccdcada2dd738ee2659","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5410,"material_evidence":{"sha256":"62d8095a1cc20117f0609a50fae7569ec9030651793f56daeebbdaed961e91e0","entries":{"return:5410":"fdd588518f68bcb1175fc73e73dccaf2cb50863c108d22c95711657fe4c2730f","artifact:5410:118401e7aab9d82374d0f595e21f7e60cf50cd68aea28d1a4142224a6ddf0e63":"118401e7aab9d82374d0f595e21f7e60cf50cd68aea28d1a4142224a6ddf0e63","artifact:5410:11a0e2c335255ae14e92e02d6b30efdd2947c57ca362fb26346ecc1f8ccf87cb":"11a0e2c335255ae14e92e02d6b30efdd2947c57ca362fb26346ecc1f8ccf87cb","artifact:5410:28abcd5f8c32ca2c2e1f646a1e3342ac8d93781c33133ae30bfdfcb4da0b992a":"28abcd5f8c32ca2c2e1f646a1e3342ac8d93781c33133ae30bfdfcb4da0b992a","artifact:5410:4bdaf53070f06a0dbe63571213f517694c8b0a32f3167f258145ded220acdb0d":"4bdaf53070f06a0dbe63571213f517694c8b0a32f3167f258145ded220acdb0d","artifact:5410:5f223d582ac0fb2928878bcbb43ff8a63e11b47dcbf89111906525a42b989218":"5f223d582ac0fb2928878bcbb43ff8a63e11b47dcbf89111906525a42b989218","artifact:5410:81857995529685b1def80c8d64ffa6af6552b15a43923ccdcada2dd738ee2659":"81857995529685b1def80c8d64ffa6af6552b15a43923ccdcada2dd738ee2659","artifact:5410:9fab8501a49b7b71e4d822054f62b98c59ce6949f9e1be1e9faca64ae5fdfb80":"9fab8501a49b7b71e4d822054f62b98c59ce6949f9e1be1e9faca64ae5fdfb80","artifact:5410:b9bde3311ce2582ddb1cdae7570a500e3802835fb8b06664575e23a8735a18c3":"b9bde3311ce2582ddb1cdae7570a500e3802835fb8b06664575e23a8735a18c3","artifact:5410:bc47e9be3280d5561abcad62551ea53289ad75dfa8b84de05eb11759371cb2f0":"bc47e9be3280d5561abcad62551ea53289ad75dfa8b84de05eb11759371cb2f0","artifact:5410:c038c7bc489c8110c707b740cc312e8a3d948bc44cf3340ae8bf8289c9c61cc6":"c038c7bc489c8110c707b740cc312e8a3d948bc44cf3340ae8bf8289c9c61cc6","artifact:5410:c248e7972094cfdb9ed5b8fb7101ef9fe216587668c6f1672f0513531df0ded5":"c248e7972094cfdb9ed5b8fb7101ef9fe216587668c6f1672f0513531df0ded5","artifact:5410:c32a812ae86e7d176168a674884cc3c01ba36c0db65ed827f5079f0a4b9d3978":"c32a812ae86e7d176168a674884cc3c01ba36c0db65ed827f5079f0a4b9d3978","artifact:5410:c4f5ab8269e42a47ae0a5fb76ab02edcd8eedaa37704d1eae13c9961e79ba6eb":"c4f5ab8269e42a47ae0a5fb76ab02edcd8eedaa37704d1eae13c9961e79ba6eb","artifact:5410:da945c21a147109290942b8ec8d60c22d633fd05f18ae13a1431c19a38189ef0":"da945c21a147109290942b8ec8d60c22d633fd05f18ae13a1431c19a38189ef0","artifact:5410:de9cc4470e607ee943c6f1934d734ed661dd190c69abce58394139235b0554ca":"de9cc4470e607ee943c6f1934d734ed661dd190c69abce58394139235b0554ca","artifact:5410:e0365b6af3286dd5f0b2f11926c647674283c660b03add4f2ec9079c5e475fc7":"e0365b6af3286dd5f0b2f11926c647674283c660b03add4f2ec9079c5e475fc7","artifact:5410:e74d06dc789316a6fbcbd237de08eedeedb9b3aa843c02f08df98e8b9c0655dd":"e74d06dc789316a6fbcbd237de08eedeedb9b3aa843c02f08df98e8b9c0655dd"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5410,"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 #5410: 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\":5410,\"report_sha256\":\"81857995529685b1def80c8d64ffa6af6552b15a43923ccdcada2dd738ee2659\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5410\":\"fdd588518f68bcb1175fc73e73dccaf2cb50863c108d22c95711657fe4c2730f\",\"artifact:5410:118401e7aab9d82374d0f595e21f7e60cf50cd68aea28d1a4142224a6ddf0e63\":\"118401e7aab9d82374d0f595e21f7e60cf50cd68aea28d1a4142224a6ddf0e63\",\"artifact:5410:11a0e2c335255ae14e92e02d6b30efdd2947c57ca362fb26346ecc1f8ccf87cb\":\"11a0e2c335255ae14e92e02d6b30efdd2947c57ca362fb26346ecc1f8ccf87cb\",\"artifact:5410:28abcd5f8c32ca2c2e1f646a1e3342ac8d93781c33133ae30bfdfcb4da0b992a\":\"28abcd5f8c32ca2c2e1f646a1e3342ac8d93781c33133ae30bfdfcb4da0b992a\",\"artifact:5410:4bdaf53070f06a0dbe63571213f517694c8b0a32f3167f258145ded220acdb0d\":\"4bdaf53070f06a0dbe63571213f517694c8b0a32f3167f258145ded220acdb0d\",\"artifact:5410:5f223d582ac0fb2928878bcbb43ff8a63e11b47dcbf89111906525a42b989218\":\"5f223d582ac0fb2928878bcbb43ff8a63e11b47dcbf89111906525a42b989218\",\"artifact:5410:81857995529685b1def80c8d64ffa6af6552b15a43923ccdcada2dd738ee2659\":\"81857995529685b1def80c8d64ffa6af6552b15a43923ccdcada2dd738ee2659\",\"artifact:5410:9fab8501a49b7b71e4d822054f62b98c59ce6949f9e1be1e9faca64ae5fdfb80\":\"9fab8501a49b7b71e4d822054f62b98c59ce6949f9e1be1e9faca64ae5fdfb80\",\"artifact:5410:b9bde3311ce2582ddb1cdae7570a500e3802835fb8b06664575e23a8735a18c3\":\"b9bde3311ce2582ddb1cdae7570a500e3802835fb8b06664575e23a8735a18c3\",\"artifact:5410:bc47e9be3280d5561abcad62551ea53289ad75dfa8b84de05eb11759371cb2f0\":\"bc47e9be3280d5561abcad62551ea53289ad75dfa8b84de05eb11759371cb2f0\",\"artifact:5410:c038c7bc489c8110c707b740cc312e8a3d948bc44cf3340ae8bf8289c9c61cc6\":\"c038c7bc489c8110c707b740cc312e8a3d948bc44cf3340ae8bf8289c9c61cc6\",\"artifact:5410:c248e7972094cfdb9ed5b8fb7101ef9fe216587668c6f1672f0513531df0ded5\":\"c248e7972094cfdb9ed5b8fb7101ef9fe216587668c6f1672f0513531df0ded5\",\"artifact:5410:c32a812ae86e7d176168a674884cc3c01ba36c0db65ed827f5079f0a4b9d3978\":\"c32a812ae86e7d176168a674884cc3c01ba36c0db65ed827f5079f0a4b9d3978\",\"artifact:5410:c4f5ab8269e42a47ae0a5fb76ab02edcd8eedaa37704d1eae13c9961e79ba6eb\":\"c4f5ab8269e42a47ae0a5fb76ab02edcd8eedaa37704d1eae13c9961e79ba6eb\",\"artifact:5410:da945c21a147109290942b8ec8d60c22d633fd05f18ae13a1431c19a38189ef0\":\"da945c21a147109290942b8ec8d60c22d633fd05f18ae13a1431c19a38189ef0\",\"artifact:5410:de9cc4470e607ee943c6f1934d734ed661dd190c69abce58394139235b0554ca\":\"de9cc4470e607ee943c6f1934d734ed661dd190c69abce58394139235b0554ca\",\"artifact:5410:e0365b6af3286dd5f0b2f11926c647674283c660b03add4f2ec9079c5e475fc7\":\"e0365b6af3286dd5f0b2f11926c647674283c660b03add4f2ec9079c5e475fc7\",\"artifact:5410:e74d06dc789316a6fbcbd237de08eedeedb9b3aa843c02f08df98e8b9c0655dd\":\"e74d06dc789316a6fbcbd237de08eedeedb9b3aa843c02f08df98e8b9c0655dd\"},\"sha256\":\"62d8095a1cc20117f0609a50fae7569ec9030651793f56daeebbdaed961e91e0\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: []\nConfigured areas: [{\"id\":\"g2-exponent\",\"title\":\"Lower the G₂ upper-bound exponent (4.27 toward 2)\",\"lane\":\"g2-exponent\",\"topic_ids\":[],\"domain_md\":\"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"adversarial\",\"title\":\"Break accepted lemmas: counterexample search against validators\",\"lane\":\"adversarial\",\"topic_ids\":[],\"domain_md\":\"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"formalize\",\"title\":\"Lean 4 formalization of accepted lemmas\",\"lane\":\"formalize\",\"topic_ids\":[],\"domain_md\":\"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"measure\",\"title\":\"Extend numbered measurement scripts to larger ranges\",\"lane\":\"measure\",\"topic_ids\":[],\"domain_md\":\"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"infinitude\",\"title\":\"Routes toward infinitude not in the refuted registry\",\"lane\":\"infinitude\",\"topic_ids\":[],\"domain_md\":\"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"finiteness-structure\",\"title\":\"What structure a finite twin count would force (disproof-shaped lane)\",\"lane\":\"finiteness-structure\",\"topic_ids\":[],\"domain_md\":\"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"kk-lower-bound\",\"title\":\"Mapped paper claims and separate obligations\",\"paper\":\"kk-lower-bound\",\"topic_ids\":[\"kk-lower-bound.main\",\"kk-lower-bound.o1\",\"kk-lower-bound.o2\",\"kk-lower-bound.o3\",\"kk-lower-bound.o4\",\"kk-lower-bound.o5\",\"kk-lower-bound.o6\",\"kk-lower-bound.o7\"],\"domain_md\":\"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.\"}]\n\nFor each area return applicability_assessment:{source_return_id:5410,input_sha256:\"c8a08ab4725bd4b2d08816ecba1fa57150eacfbd5ac1f57399b1fd101402bb7d\",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/5413/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"400b0b5ca7236efc83b05732f3544ea0ef6d6ad8ac5565402ade99aa759fd214","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}