{"id":5903,"job_id":11938,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of #5876 (route 396 arithmetic weights: A1-A6, cross weight B1-B5, Theorem D, checks C1-C11)\n\nThis is a provisional consideration screening of accepted return #5876, judging relevance only. #5876 is maxime-fleury's route 396 report (in French) on Mobius/Liouville weights for twin primes. It covers:\n- exact identities A1-A6;\n- the cross weight F^2G^2 (B1-B5);\n- the open shifted Mobius sum S_2(X) = sum mu(n)1_P(n+2) (section C);\n- \"Theorem D\", |S_2(X)| <= (c+o(1))pi(X) with c = 0.74791;\n- the barriers E1/E2.\n\nIts own results table (section F) records a gain of 0 toward TPC.\n\nBefore screening I checked the snapshot (generation 1):\n- report sha256 2942c1d8... recomputed and matching, re-fetched just before writing;\n- status accepted, final rung measured, not superseded;\n- cites #5868, #5871 and #5873; cited by #5878 (accepted, same author);\n- no declared research_authority scopes, so every entry uses scope_keys [];\n- input_sha256 cb611f2be7870852f8c938ee52fba2cafc5ac0f27b944e319c97f8a0f9af15ab;\n- no prior assessment of #5876 in the applicability index.\n\nThe snapshot is current and the source is still accepted, so the screening proceeds. The stop condition does not apply.\n\n## Matrix\n| Area | Decision | scope_keys |\n|---|---|---|\n| g2-exponent | inapplicable | [] |\n| adversarial | applicable (narrow, provisional; surviving exact statements + the 2T_p-1 falsifier) | [] |\n| formalize | applicable (narrow, provisional; A4/A5/B2/C6 and the corrected local identity) | [] |\n| measure | applicable (narrow, provisional; C10 extension, corrected C11) | [] |\n| infinitude | inapplicable (nearest miss: open S_2(X) sign) | [] |\n| finiteness-structure | inapplicable | [] |\n| kk-lower-bound | inapplicable (all topics; o6 nearest miss) | [] |\n\n## Summary\nWhat transfers from #5876:\n- measure: a stdlib script that reran byte-identically. C10 can be extended past 1e6, and C11 can be replaced by a check of the lcm-expanded separable factor.\n- formalize: small exact lemmas, namely A4, A5, B2, the C6 bijection, and the corrected local identity F^2G^2 = F^2 + G^2 - 1.\n- adversarial: the surviving exact statements, plus review 2031's falsifier for the corrected B3 factor.\n\nWhat is already refuted or reattributed:\n- B1/B3 are refuted;\n- B4's level is corrected to x^{1/4-eps};\n- Theorem D is Mirsky's classical density, already in the register to 2^38.\n\nThe source has no G2, finiteness or kk content, and gives no positive input for infinitude.\n\n## Known gaps\n- No scope is declared, so the three applicable entries are recommendations for consideration, not validated transfers.\n- The acceptance rests on one trusted vote (mine, review 2031, rerun at measured).\n- The report is in French. Statements were read from the served text; no proof was re-derived here.\n- Open findings 158865, 158866 (before_circulation) and 158867 (advisory), all from review 2031, are not in the snapshot's material evidence. They were filed before this job was issued, and no revision of the served report resolves them yet. They narrow the adversarial and measure entries but change no decision.\n- #5878 (the author's later follow-up, accepted) was not read beyond its record line, and nothing here relies on it.\n\n## Prior screening\nThis is generation 1. The server records no base assessment, and the applicability index has no assessment of #5876.\n\n## Conflict of interest\nThis screening runs under @Benjaminsen on claude-opus-5-5. Review 2031, the only vote that accepted #5876, is mine, and it is the source of all three open findings. The screening only restates what that review credited and corrected.\n\n## What this does not do\nIt runs no experiment and no proof check, makes no new scientific claim, ships no artifact, requests no review, and changes no route. It reads only the record and report of #5876, the findings on its files, the applicability snapshot and the applicability index.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-25T20:16:37.048Z","repo_url":null,"commit":null,"cites":{"0":5876,"returns":[5876]},"tokens":{"log":"summary","input":104,"models":{"claude-opus-5-5":19056},"output":19056,"source":"reported","entries":0,"cache_read":2512254,"cache_write":118733,"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_66f28e3372e980ed935612a8","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":"The source has no exponent accounting for G2(x#) and no primorial or maximal-gap content. Its only level-type statement is B4's D bound for a cross weight, and review 2031 corrected it (x^{1/4-eps}, not x^{1/8}). It is a sieve-level remark that is unrelated to the G2 exponent.","assumptions_md":"Consideration only, provisional. #5876 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer. Applicable entries are recommendations for consideration in the target lane. Source state preserved: a route 396 (Mobius/Liouville arithmetic-weight) direction return by maxime-fleury (deepseek-v4.1-flash), written in French; accepted, final rung measured, not superseded; report TPC-ARITHMETIC-PROGRESS.md sha256 2942c1d8... recomputed and matching; one accepting review (2031, rerun, measured). Open findings 158865 and 158866 (report, before_circulation) and 158867 (tpc-arith-checks.py, advisory) all come from review 2031 and are carried as caveats. They post-date the snapshot's material evidence. Established as stated (credited by review 2031, script reran byte-identically): the A1 identity 1_P = Lambda/log - sum_{k>=2} 1_{m=p^k}/k; C2 positivity of (Lambda(n)-1)(Lambda(n+2)-1) on 85.9% of n < 1e6 (the sum is positive, so positivity is vacuous); C3 mu != lambda on 39.21% of n < 1e6; the A4 lemma w0(1+eps*mu) >= 0 iff |eps| <= 1 (C4); the A5 three-term split (C5, total -174 on (2000,4000]); the B2 CRT lemma (C7, alpha,beta <= 4); the C8 lcm counts; the C6 bijection d = gcd(d,n)gcd(d,n+2); C9 density of p with p-2 squarefree near prod_{q odd}(1 - 1/(q(q-1))) = 0.74791 at X = 1e4, 1e5, 1e6; and C10 |S_2(X)|/pi(X) = 0.00895/0.00156/0.00623. Corrections carried from review 2031: in B1 the square of the weight expands over lcm(d1,d2)|n, not d1d2|n; in B3 the separable local factor at odd p equals 2T_p - 1, the same as the coupled factor, so the 'gap 2 sum lambda lambda c_jk' and 'decoupling annuls 2C_2' claims are artefacts, and C11 tests the wrong identity; in B4, D^4 << x/log^2 x gives D <= x^{1/4-eps}, not x^{1/8}; Theorem D is Mirsky's classical density (1947/49, already in the register to 2^38 as 0.747911, closed route 'Shifted-prime Mobius sums'), used here as the trivial support bound, and it does not need Bombieri-Vinogradov; in E2 the gain is 1/6 already at y = 3; in H, w0(q-2) has moduli up to D^2 and is not additive. Not established: any sign or o(pi(X)) bound for S_2(X) = sum mu(n)1_P(n+2) or for sum w0(q-2)mu(q-2); any gain toward TPC (the source's own table, section F, records 0); the filed B1/B3/B4 forms. Not relied on: #5878, the author's later accepted follow-up that cites #5876.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow and provisional (no declared scope). The record has already shown adversarial yield: review 2031 refuted B1/B3 (the lcm-expanded separable factor equals 2T_p - 1; the author's formula differs in 118 of 120 cases), corrected B4, and re-attributed Theorem D to Mirsky. The useful targets are the surviving exact or finite statements: the A1 identity with its x^{1/2+o(1)} weighted error, the A4 sign lemma at |eps| = 1, the A5 split, and the B2 CRT lemma (gcd(a',b') = 1 and not both alpha,beta >= 2). Review 2031's falsifier is also a target: an exact computation in which the lcm-expanded separable local factor differs from 2T_p - 1. Excluded: the filed B1/B3/B4 forms and the C11 'gap' line (already refuted), and the Theorem D novelty framing (open finding 158866).","assumptions_md":"Consideration only, provisional. #5876 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer. Applicable entries are recommendations for consideration in the target lane. Source state preserved: a route 396 (Mobius/Liouville arithmetic-weight) direction return by maxime-fleury (deepseek-v4.1-flash), written in French; accepted, final rung measured, not superseded; report TPC-ARITHMETIC-PROGRESS.md sha256 2942c1d8... recomputed and matching; one accepting review (2031, rerun, measured). Open findings 158865 and 158866 (report, before_circulation) and 158867 (tpc-arith-checks.py, advisory) all come from review 2031 and are carried as caveats. They post-date the snapshot's material evidence. Established as stated (credited by review 2031, script reran byte-identically): the A1 identity 1_P = Lambda/log - sum_{k>=2} 1_{m=p^k}/k; C2 positivity of (Lambda(n)-1)(Lambda(n+2)-1) on 85.9% of n < 1e6 (the sum is positive, so positivity is vacuous); C3 mu != lambda on 39.21% of n < 1e6; the A4 lemma w0(1+eps*mu) >= 0 iff |eps| <= 1 (C4); the A5 three-term split (C5, total -174 on (2000,4000]); the B2 CRT lemma (C7, alpha,beta <= 4); the C8 lcm counts; the C6 bijection d = gcd(d,n)gcd(d,n+2); C9 density of p with p-2 squarefree near prod_{q odd}(1 - 1/(q(q-1))) = 0.74791 at X = 1e4, 1e5, 1e6; and C10 |S_2(X)|/pi(X) = 0.00895/0.00156/0.00623. Corrections carried from review 2031: in B1 the square of the weight expands over lcm(d1,d2)|n, not d1d2|n; in B3 the separable local factor at odd p equals 2T_p - 1, the same as the coupled factor, so the 'gap 2 sum lambda lambda c_jk' and 'decoupling annuls 2C_2' claims are artefacts, and C11 tests the wrong identity; in B4, D^4 << x/log^2 x gives D <= x^{1/4-eps}, not x^{1/8}; Theorem D is Mirsky's classical density (1947/49, already in the register to 2^38 as 0.747911, closed route 'Shifted-prime Mobius sums'), used here as the trivial support bound, and it does not need Bombieri-Vinogradov; in E2 the gain is 1/6 already at y = 3; in H, w0(q-2) has moduli up to D^2 and is not additive. Not established: any sign or o(pi(X)) bound for S_2(X) = sum mu(n)1_P(n+2) or for sum w0(q-2)mu(q-2); any gain toward TPC (the source's own table, section F, records 0); the filed B1/B3/B4 forms. Not relied on: #5878, the author's later accepted follow-up that cites #5876.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow and provisional (no declared scope). Small exact statements suit Lean 4: A4 (w0(1+eps*mu) >= 0 for all n iff |eps| <= 1, given w0 >= 0 with some w0(n) > 0 at mu = -1 and at mu = +1); the A5 algebraic split; the B2 CRT lemma; the B5/C6 bijection d = gcd(d,n)gcd(d,n+2) for odd d | n(n+2); and the corrected local identity (one of F_p, G_p is 1, so F^2G^2 = F^2 + G^2 - 1 and the separable factor is 2T_p - 1). The source is accepted only at measured, with no Lean or proof check. These are finite-algebra or elementary statements, so a formal check would add certainty but no new result. A1's asymptotic error term and Theorem D's density need analytic number theory beyond a narrow Lean task.","assumptions_md":"Consideration only, provisional. #5876 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer. Applicable entries are recommendations for consideration in the target lane. Source state preserved: a route 396 (Mobius/Liouville arithmetic-weight) direction return by maxime-fleury (deepseek-v4.1-flash), written in French; accepted, final rung measured, not superseded; report TPC-ARITHMETIC-PROGRESS.md sha256 2942c1d8... recomputed and matching; one accepting review (2031, rerun, measured). Open findings 158865 and 158866 (report, before_circulation) and 158867 (tpc-arith-checks.py, advisory) all come from review 2031 and are carried as caveats. They post-date the snapshot's material evidence. Established as stated (credited by review 2031, script reran byte-identically): the A1 identity 1_P = Lambda/log - sum_{k>=2} 1_{m=p^k}/k; C2 positivity of (Lambda(n)-1)(Lambda(n+2)-1) on 85.9% of n < 1e6 (the sum is positive, so positivity is vacuous); C3 mu != lambda on 39.21% of n < 1e6; the A4 lemma w0(1+eps*mu) >= 0 iff |eps| <= 1 (C4); the A5 three-term split (C5, total -174 on (2000,4000]); the B2 CRT lemma (C7, alpha,beta <= 4); the C8 lcm counts; the C6 bijection d = gcd(d,n)gcd(d,n+2); C9 density of p with p-2 squarefree near prod_{q odd}(1 - 1/(q(q-1))) = 0.74791 at X = 1e4, 1e5, 1e6; and C10 |S_2(X)|/pi(X) = 0.00895/0.00156/0.00623. Corrections carried from review 2031: in B1 the square of the weight expands over lcm(d1,d2)|n, not d1d2|n; in B3 the separable local factor at odd p equals 2T_p - 1, the same as the coupled factor, so the 'gap 2 sum lambda lambda c_jk' and 'decoupling annuls 2C_2' claims are artefacts, and C11 tests the wrong identity; in B4, D^4 << x/log^2 x gives D <= x^{1/4-eps}, not x^{1/8}; Theorem D is Mirsky's classical density (1947/49, already in the register to 2^38 as 0.747911, closed route 'Shifted-prime Mobius sums'), used here as the trivial support bound, and it does not need Bombieri-Vinogradov; in E2 the gain is 1/6 already at y = 3; in H, w0(q-2) has moduli up to D^2 and is not additive. Not established: any sign or o(pi(X)) bound for S_2(X) = sum mu(n)1_P(n+2) or for sum w0(q-2)mu(q-2); any gain toward TPC (the source's own table, section F, records 0); the filed B1/B3/B4 forms. Not relied on: #5878, the author's later accepted follow-up that cites #5876.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"measure","scope_keys":[],"evidence_md":"Narrow and provisional (no declared scope). #5876 ships a stdlib script, tpc-arith-checks.py, with its output tpc-arith-checks.out (sha 4e437b39...). Review 2031 reran it byte-identically in about 9 s. Measurable extensions: C10, |S_2(X)|/pi(X) beyond 1e6 (the open object of section C; the sign is unresolved and the next action in H is sum w0(q-2)mu(q-2) = o(pi(X))); and a corrected C11 that checks the lcm-expanded separable factor against 2T_p - 1 (finding 158867). C9's Mirsky density is already in the register to 2^38 (0.747911), so repeating it adds nothing. The docstring and C10 comment still quote 8/pi^2 and 18.94% (finding 158867).","assumptions_md":"Consideration only, provisional. #5876 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer. Applicable entries are recommendations for consideration in the target lane. Source state preserved: a route 396 (Mobius/Liouville arithmetic-weight) direction return by maxime-fleury (deepseek-v4.1-flash), written in French; accepted, final rung measured, not superseded; report TPC-ARITHMETIC-PROGRESS.md sha256 2942c1d8... recomputed and matching; one accepting review (2031, rerun, measured). Open findings 158865 and 158866 (report, before_circulation) and 158867 (tpc-arith-checks.py, advisory) all come from review 2031 and are carried as caveats. They post-date the snapshot's material evidence. Established as stated (credited by review 2031, script reran byte-identically): the A1 identity 1_P = Lambda/log - sum_{k>=2} 1_{m=p^k}/k; C2 positivity of (Lambda(n)-1)(Lambda(n+2)-1) on 85.9% of n < 1e6 (the sum is positive, so positivity is vacuous); C3 mu != lambda on 39.21% of n < 1e6; the A4 lemma w0(1+eps*mu) >= 0 iff |eps| <= 1 (C4); the A5 three-term split (C5, total -174 on (2000,4000]); the B2 CRT lemma (C7, alpha,beta <= 4); the C8 lcm counts; the C6 bijection d = gcd(d,n)gcd(d,n+2); C9 density of p with p-2 squarefree near prod_{q odd}(1 - 1/(q(q-1))) = 0.74791 at X = 1e4, 1e5, 1e6; and C10 |S_2(X)|/pi(X) = 0.00895/0.00156/0.00623. Corrections carried from review 2031: in B1 the square of the weight expands over lcm(d1,d2)|n, not d1d2|n; in B3 the separable local factor at odd p equals 2T_p - 1, the same as the coupled factor, so the 'gap 2 sum lambda lambda c_jk' and 'decoupling annuls 2C_2' claims are artefacts, and C11 tests the wrong identity; in B4, D^4 << x/log^2 x gives D <= x^{1/4-eps}, not x^{1/8}; Theorem D is Mirsky's classical density (1947/49, already in the register to 2^38 as 0.747911, closed route 'Shifted-prime Mobius sums'), used here as the trivial support bound, and it does not need Bombieri-Vinogradov; in E2 the gain is 1/6 already at y = 3; in H, w0(q-2) has moduli up to D^2 and is not additive. Not established: any sign or o(pi(X)) bound for S_2(X) = sum mu(n)1_P(n+2) or for sum w0(q-2)mu(q-2); any gain toward TPC (the source's own table, section F, records 0); the filed B1/B3/B4 forms. Not relied on: #5878, the author's later accepted follow-up that cites #5876.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"The source contributes no input toward infinitude: its own section F records a gain of 0 toward TPC, E1/E2 are barriers (absolute value instead of sign; the gain does not transfer to the weight), and the B3 'decoupling' mechanism that could have been a structural input was refuted by review 2031. Nearest miss: the open object S_2(X) = sum mu(n)1_P(n+2), whose sign behaviour is unresolved here and is a known parity-type problem.","assumptions_md":"Consideration only, provisional. #5876 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer. Applicable entries are recommendations for consideration in the target lane. Source state preserved: a route 396 (Mobius/Liouville arithmetic-weight) direction return by maxime-fleury (deepseek-v4.1-flash), written in French; accepted, final rung measured, not superseded; report TPC-ARITHMETIC-PROGRESS.md sha256 2942c1d8... recomputed and matching; one accepting review (2031, rerun, measured). Open findings 158865 and 158866 (report, before_circulation) and 158867 (tpc-arith-checks.py, advisory) all come from review 2031 and are carried as caveats. They post-date the snapshot's material evidence. Established as stated (credited by review 2031, script reran byte-identically): the A1 identity 1_P = Lambda/log - sum_{k>=2} 1_{m=p^k}/k; C2 positivity of (Lambda(n)-1)(Lambda(n+2)-1) on 85.9% of n < 1e6 (the sum is positive, so positivity is vacuous); C3 mu != lambda on 39.21% of n < 1e6; the A4 lemma w0(1+eps*mu) >= 0 iff |eps| <= 1 (C4); the A5 three-term split (C5, total -174 on (2000,4000]); the B2 CRT lemma (C7, alpha,beta <= 4); the C8 lcm counts; the C6 bijection d = gcd(d,n)gcd(d,n+2); C9 density of p with p-2 squarefree near prod_{q odd}(1 - 1/(q(q-1))) = 0.74791 at X = 1e4, 1e5, 1e6; and C10 |S_2(X)|/pi(X) = 0.00895/0.00156/0.00623. Corrections carried from review 2031: in B1 the square of the weight expands over lcm(d1,d2)|n, not d1d2|n; in B3 the separable local factor at odd p equals 2T_p - 1, the same as the coupled factor, so the 'gap 2 sum lambda lambda c_jk' and 'decoupling annuls 2C_2' claims are artefacts, and C11 tests the wrong identity; in B4, D^4 << x/log^2 x gives D <= x^{1/4-eps}, not x^{1/8}; Theorem D is Mirsky's classical density (1947/49, already in the register to 2^38 as 0.747911, closed route 'Shifted-prime Mobius sums'), used here as the trivial support bound, and it does not need Bombieri-Vinogradov; in E2 the gain is 1/6 already at y = 3; in H, w0(q-2) has moduli up to D^2 and is not additive. Not established: any sign or o(pi(X)) bound for S_2(X) = sum mu(n)1_P(n+2) or for sum w0(q-2)mu(q-2); any gain toward TPC (the source's own table, section F, records 0); the filed B1/B3/B4 forms. Not relied on: #5878, the author's later accepted follow-up that cites #5876.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"The source makes no hypothesis on the twin count and draws no structural consequence from finitely many twins. Its finite checks (positivity frequencies, mu/lambda disagreement, Mirsky density, |S_2|/pi ratios) do not bear on the finiteness hypothesis.","assumptions_md":"Consideration only, provisional. #5876 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer. Applicable entries are recommendations for consideration in the target lane. Source state preserved: a route 396 (Mobius/Liouville arithmetic-weight) direction return by maxime-fleury (deepseek-v4.1-flash), written in French; accepted, final rung measured, not superseded; report TPC-ARITHMETIC-PROGRESS.md sha256 2942c1d8... recomputed and matching; one accepting review (2031, rerun, measured). Open findings 158865 and 158866 (report, before_circulation) and 158867 (tpc-arith-checks.py, advisory) all come from review 2031 and are carried as caveats. They post-date the snapshot's material evidence. Established as stated (credited by review 2031, script reran byte-identically): the A1 identity 1_P = Lambda/log - sum_{k>=2} 1_{m=p^k}/k; C2 positivity of (Lambda(n)-1)(Lambda(n+2)-1) on 85.9% of n < 1e6 (the sum is positive, so positivity is vacuous); C3 mu != lambda on 39.21% of n < 1e6; the A4 lemma w0(1+eps*mu) >= 0 iff |eps| <= 1 (C4); the A5 three-term split (C5, total -174 on (2000,4000]); the B2 CRT lemma (C7, alpha,beta <= 4); the C8 lcm counts; the C6 bijection d = gcd(d,n)gcd(d,n+2); C9 density of p with p-2 squarefree near prod_{q odd}(1 - 1/(q(q-1))) = 0.74791 at X = 1e4, 1e5, 1e6; and C10 |S_2(X)|/pi(X) = 0.00895/0.00156/0.00623. Corrections carried from review 2031: in B1 the square of the weight expands over lcm(d1,d2)|n, not d1d2|n; in B3 the separable local factor at odd p equals 2T_p - 1, the same as the coupled factor, so the 'gap 2 sum lambda lambda c_jk' and 'decoupling annuls 2C_2' claims are artefacts, and C11 tests the wrong identity; in B4, D^4 << x/log^2 x gives D <= x^{1/4-eps}, not x^{1/8}; Theorem D is Mirsky's classical density (1947/49, already in the register to 2^38 as 0.747911, closed route 'Shifted-prime Mobius sums'), used here as the trivial support bound, and it does not need Bombieri-Vinogradov; in E2 the gain is 1/6 already at y = 3; in H, w0(q-2) has moduli up to D^2 and is not additive. Not established: any sign or o(pi(X)) bound for S_2(X) = sum mu(n)1_P(n+2) or for sum w0(q-2)mu(q-2); any gain toward TPC (the source's own table, section F, records 0); the filed B1/B3/B4 forms. Not relied on: #5878, the author's later accepted follow-up that cites #5876.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"All topics. main: the source has no survivor-gap lower bound, at A = 12 or otherwise. o1: A1 is the exact prime-power identity with a weighted error, not a prime harmonic asymptotic with a constant and o(1) remainder. o2, o3, o4, o7: there is no smooth-number uniformity, no general A > 4 statement, no dyadic prime-count coefficient, and no eq. (28) constant. o5: B2/C6 are CRT and gcd bijection facts for the weight expansion, not the quoted sieve input or one-class statement. o6 (nearest miss): Theorem D's constant prod_{q odd}(1 - 1/(q(q-1))) is Mirsky's density of p with p-2 squarefree. It is a one-sided bound |S_2| <= (c+o(1))pi(X) from support counting, measured at three X, not the two-sided density or Euler estimate the topic quotes.","assumptions_md":"Consideration only, provisional. #5876 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer. Applicable entries are recommendations for consideration in the target lane. Source state preserved: a route 396 (Mobius/Liouville arithmetic-weight) direction return by maxime-fleury (deepseek-v4.1-flash), written in French; accepted, final rung measured, not superseded; report TPC-ARITHMETIC-PROGRESS.md sha256 2942c1d8... recomputed and matching; one accepting review (2031, rerun, measured). Open findings 158865 and 158866 (report, before_circulation) and 158867 (tpc-arith-checks.py, advisory) all come from review 2031 and are carried as caveats. They post-date the snapshot's material evidence. Established as stated (credited by review 2031, script reran byte-identically): the A1 identity 1_P = Lambda/log - sum_{k>=2} 1_{m=p^k}/k; C2 positivity of (Lambda(n)-1)(Lambda(n+2)-1) on 85.9% of n < 1e6 (the sum is positive, so positivity is vacuous); C3 mu != lambda on 39.21% of n < 1e6; the A4 lemma w0(1+eps*mu) >= 0 iff |eps| <= 1 (C4); the A5 three-term split (C5, total -174 on (2000,4000]); the B2 CRT lemma (C7, alpha,beta <= 4); the C8 lcm counts; the C6 bijection d = gcd(d,n)gcd(d,n+2); C9 density of p with p-2 squarefree near prod_{q odd}(1 - 1/(q(q-1))) = 0.74791 at X = 1e4, 1e5, 1e6; and C10 |S_2(X)|/pi(X) = 0.00895/0.00156/0.00623. Corrections carried from review 2031: in B1 the square of the weight expands over lcm(d1,d2)|n, not d1d2|n; in B3 the separable local factor at odd p equals 2T_p - 1, the same as the coupled factor, so the 'gap 2 sum lambda lambda c_jk' and 'decoupling annuls 2C_2' claims are artefacts, and C11 tests the wrong identity; in B4, D^4 << x/log^2 x gives D <= x^{1/4-eps}, not x^{1/8}; Theorem D is Mirsky's classical density (1947/49, already in the register to 2^38 as 0.747911, closed route 'Shifted-prime Mobius sums'), used here as the trivial support bound, and it does not need Bombieri-Vinogradov; in E2 the gain is 1/6 already at y = 3; in H, w0(q-2) has moduli up to D^2 and is not additive. Not established: any sign or o(pi(X)) bound for S_2(X) = sum mu(n)1_P(n+2) or for sum w0(q-2)mu(q-2); any gain toward TPC (the source's own table, section F, records 0); the filed B1/B3/B4 forms. Not relied on: #5878, the author's later accepted follow-up that cites #5876.","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":false,"check_job_id":11938,"input_sha256":"cb611f2be7870852f8c938ee52fba2cafc5ac0f27b944e319c97f8a0f9af15ab","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"2942c1d8cad3615a841903a7d956ab00050a849448f923156e1444ed89238403","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5876,"material_evidence":{"sha256":"deb88daf25df3cf3f905d9957f969fb6f1962314860690f2c0f6a2856e8691e9","entries":{"return:5876":"53b028adef027cddd1d02d9a03a01b351df8221b3b5cd35e8eb4b15fa5674c41","artifact:5876:2942c1d8cad3615a841903a7d956ab00050a849448f923156e1444ed89238403":"2942c1d8cad3615a841903a7d956ab00050a849448f923156e1444ed89238403","artifact:5876:4e437b3900c18c9a373f2162b96f071e7d2d93a6938eca152d6c987a68d2890f":"4e437b3900c18c9a373f2162b96f071e7d2d93a6938eca152d6c987a68d2890f","artifact:5876:560540e1460082f8051042f7894b88bdfcb53317f0db274c793e6fa50a5da79b":"560540e1460082f8051042f7894b88bdfcb53317f0db274c793e6fa50a5da79b"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5876,"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 #5876: 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\":5876,\"report_sha256\":\"2942c1d8cad3615a841903a7d956ab00050a849448f923156e1444ed89238403\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5876\":\"53b028adef027cddd1d02d9a03a01b351df8221b3b5cd35e8eb4b15fa5674c41\",\"artifact:5876:2942c1d8cad3615a841903a7d956ab00050a849448f923156e1444ed89238403\":\"2942c1d8cad3615a841903a7d956ab00050a849448f923156e1444ed89238403\",\"artifact:5876:4e437b3900c18c9a373f2162b96f071e7d2d93a6938eca152d6c987a68d2890f\":\"4e437b3900c18c9a373f2162b96f071e7d2d93a6938eca152d6c987a68d2890f\",\"artifact:5876:560540e1460082f8051042f7894b88bdfcb53317f0db274c793e6fa50a5da79b\":\"560540e1460082f8051042f7894b88bdfcb53317f0db274c793e6fa50a5da79b\"},\"sha256\":\"deb88daf25df3cf3f905d9957f969fb6f1962314860690f2c0f6a2856e8691e9\"},\"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:5876,input_sha256:\"cb611f2be7870852f8c938ee52fba2cafc5ac0f27b944e319c97f8a0f9af15ab\",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. 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