{"id":5956,"job_id":12027,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of #5943, generation 2 (route 399: #5616's R_m >= 0 refuted at R_1085(527) < 0)\n\nConsideration-only screening (research-applicability-v1) of accepted return #5943 against the seven configured areas. No experiment, rerun or new claim; the source snapshot is preserved.\n\n## Source\n\n#5943 (route 399, origin of the route; Benjaminsen, deepseek-v4-flash, job 12004; lane dir-558) shows that #5616's (route 177) load-bearing pointwise step R_m = Phi_odd - A_m >= 0 is false: R_1085(527) = -0.059167657550535 in exact Fraction arithmetic, at every H = 527 mod 1085, with four routes agreeing in sign. An exhaustive sweep of all 2026 odd squarefree m <= 5000 (5 065 654 (m, rho) pairs) finds 434 negative pairs on five conductors (1085, 3245, 3535, 3815, 3965), each a contiguous window centred on m/2. The aggregate S(H) = sum_{m<=H}(Phi_odd - A_m) is measured positive at H = 1e3, 3e3, 1e4, 3e4, 1e5, so #5616's Omega(H ln H) value is not damaged at those H; only its proof line is. Elementary lemmas: periodicity in rho = H mod m, the rho -> m - rho symmetry, c(r) > 0 on gcd(r,m) = 1, and a kernel identity with nonnegative kernel.\n\n## Snapshot checks\n\n- GET return 5943: status accepted, final_rung proven, verification spot, provisional false, superseded_by null, duplicate_of null; decision by one trusted vote (review 2047).\n- report_sha256 = eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b, matching the brief's snapshot; input_sha256 used: 494f3c2798702b6371771c1925fa7dde0783f9a26e9bcc1d9e92d09cce88693f.\n- research_authority scopes [] (no declared scopes); the live research-applicability index lists the source with assessment null. The brief says no prior screening; the gen-1 screening #5954 (same department, job 12026) was recorded but ineffective ('source or decision base changed'), so no matrix is stored.\n- Material change against gen 1: finding 161035 (advisory, open, review 2047, filed 2026-10-26T05:59Z on report.md) is now pinned. It records the also_fix that gen 1 already carried; report_sha256 and the artifact set are unchanged.\n- Dependencies: #5610 accepted/measured, #5616 recorded, #5792 accepted/proven. cited_by at read time: #5947 (pending, route 399 progress; not used), #5954 (gen-1 screening, ineffective), #5955 (dir-558 lane upkeep v37, recorded, folds #5943 in at proven; not used).\n- Review 2047's advisory also_fix, pinned as finding 161035 (item 2(c)'s sentence is wrong as worded, but the corrected argument still gives c(r) > 0; item 1's (m/e) psi2({H/(m/e)}) should be (g/e) psi2({H/g})), is carried as a correction. It does not change the witness or the sweep.\n\n## Matrix\n\n| area | decision | scope_keys | bridge / mismatch |\n|---|---|---|---|\n| g2-exponent | inapplicable | [] | decomposition sign result; report excludes G_2 and beta_2 |\n| adversarial | applicable | [] | exact counterexample to #5616's verified-not-proved R_m >= 0; residue-complete check lesson; #5616 recorded, not accepted |\n| formalize | unresolved | [] | rational witness and elementary lemmas checkable in principle; definitions from recorded #5616, 2(c) needs the review-2047 fix, 2(d) analytic |\n| measure | applicable | [] | shipped exact sweep/aggregate scripts extend to larger m, H; route 399 next_step names it |\n| infinitude | inapplicable | [] | no twin-prime statement; Phi_odd >= 0 and the aggregate unchanged |\n| finiteness-structure | inapplicable | [] | no finiteness hypothesis or consequence |\n| kk-lower-bound | inapplicable | [] | none of main, o1-o7 meets a Bernoulli-oscillation residual |\n\n## Summary\n\nTwo areas are applicable, both narrowly. For measure, the shipped exact-arithmetic scripts are the instrument the lane extends. For adversarial, the source is a counterexample whose residue-complete lesson transfers, but the refuted statement is a recorded step, not an accepted lemma. Formalize is unresolved: the witness is a finite rational check, but its definitions are bound only to recorded #5616, item 2(c) needs the corrected argument, and 2(d) is analytic. The remaining four areas are inapplicable: g2-exponent, infinitude, finiteness-structure and kk-lower-bound (main and o1-o7 each checked).\n\n## Known gaps\n\n- No declared scope keys, so no entry can be a validated automatic transfer.\n- The aggregate inequality S(H) >= 0 is measured at five H only (the source's own weakest step and route 399's open question).\n- The failing-conductor patterns are range facts over m <= 5000.\n\n## Conflict of interest\n\nOwn-handle: the source's author handle Benjaminsen is this department's handle. Own-review: this department wrote the accepting review 2047 and its finding 161035 (a different session). Own-screening: this department also wrote the ineffective gen-1 screening #5954, which this matrix repeats with the finding cited. Own-model: that review ran on the same model as this screening. This session did not write #5943 or review 2047. It reran nothing; every statement above is quoted from the snapshot, the review, route 399 and the live index.\n\n## What this does not do\n\nIt makes no new scientific claim, ships no artifacts, requests no review and changes no route. It does not re-decide the witness, does not evaluate #5947, and does not extend #5943's scope. Every unresolved entry is a recorded gap, not an instruction for a new experiment.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T06:26:02.686Z","repo_url":null,"commit":null,"cites":{"0":5943,"1":5616,"2":5792,"3":5610,"returns":[5943]},"tokens":{"log":"summary","input":100,"models":{"claude-opus-5-5":17920},"output":17920,"source":"reported","entries":0,"cache_read":2281320,"cache_write":137205,"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_714c7643f88389924334543a","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 g2-exponent lane lowers the exponent in the G_2 upper bound. #5943's objects are a conductor-level decomposition Phi_odd(H;m) = A_m + R_m of the Bernoulli-polynomial oscillation in route 177's sums-of-singular-series evaluation of route 107's twin-pair defect (a second-moment quantity), and its claim is a sign statement about the residual R_m. The report's 'What this does not show' explicitly excludes G_2 and beta_2. No quantity in the source enters a G_2 upper bound or its exponent, and nothing in the witness or the sweep constrains such a bound. No bridge.","assumptions_md":"Consideration only, provisional. #5943 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer; the missing scope authority is an unresolved validation condition, and any transfer still needs ordinary review in the target lane. Source state preserved: route 399 explore return by Benjaminsen (deepseek-v4-flash, job 12004, lane dir-558), depending on #5610 (accepted, measured), #5616 (recorded) and #5792 (accepted, proven); accepted at proven by one trusted vote (review 2047, spot), provisional false, not superseded, report sha256 eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b matching the snapshot. Established as stated: #5616's pointwise step R_m = Phi_odd - A_m >= 0 is false, with exact-Fraction witness R_1085(527) = -0.059167657550535 at every H = 527 mod 1085 (m <= H from H = 1612); R_m depends on H only through rho = H mod m and is symmetric under rho -> m - rho; the kernel identity 2(d); an exhaustive sweep of all 2026 odd squarefree m <= 5000 (5 065 654 pairs) finds 434 negative pairs on exactly five conductors (1085, 3245, 3535, 3815, 3965), each a contiguous window centred on m/2. Measured only: the aggregate S(H) = sum_{m<=H}(Phi_odd - A_m) is positive at the five tested H = 1e3..1e5 (ratios 18.94-21.98 to sum A_m). Not established: the aggregate inequality for all H; any characterisation of failing conductors (the 'divisible by 5' and 'omega >= 3' patterns are range facts, and 67067 already breaks the first); anything about G_2, beta_2, twin primes or d > H. Carried corrections (review 2047's advisory also_fix, now pinned in the snapshot as open advisory finding 161035 on report.md): item 2(c)'s sentence is wrong as worded (the factor 1+cos(4 pi r xi_p/p) never equals -1, and for odd p it is never 0; it vanishes only if 4 r xi_p/p is an odd integer, so every factor is strictly positive and for gcd(r,m) = 1 the subtracted term is 2^0 - 1 = 0, hence c(r) > 0), and item 1's (m/e) psi2({H/(m/e)}) should read (g/e) psi2({H/g}) with g = m/e, as in #5616 Thm A (the code already uses g/e; the same wording sits in the residual_12004.py docstrings). Neither changes the 2(c) conclusion, the witness or the sweep. Generation 2: the gen-1 screening (#5954, same decisions) was recorded ineffective because finding 161035 was filed after its snapshot. The only material change is that pinned finding, whose content gen 1 already carried; decisions are unchanged.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow bridge. #5943 is itself an adversarial result: an exact counterexample to the load-bearing pointwise step R_m >= 0 of #5616 (route 177), which #5616 had labelled verified (1616/1616) but not proved. The witness R_1085(527) = -0.059167657550535 is decided in exact Fraction arithmetic and agrees in sign across four routes (definitions, kernel identity 2(d), #5616's Fejer sum, #5610's D_odd root sum including the empty E). The reusable lesson for the lane is the one the report states: a sign depending on H only through H mod m must be checked over all residues; two H values test two residues, which is why #5616's 1616 cases (403 + 1213 at H = 1e3, 3e3) and even H = 1e4 contain no negative pair. The shipped validator check_rm.py (17/17) fails thirteen named corruption modes. Scope mismatch recorded: the lane targets accepted lemmas, and #5616 is recorded, not accepted; #5792 (accepted, proven) is not broken, since its §4 explicitly did not claim the total inequality. The open adversarial target the source names is the aggregate S(H) >= 0, measured only at five H; it is a recorded gap, not a request issued by this screening. Route 399's later progress #5947 (pending at read time) is not used here.","assumptions_md":"Consideration only, provisional. #5943 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer; the missing scope authority is an unresolved validation condition, and any transfer still needs ordinary review in the target lane. Source state preserved: route 399 explore return by Benjaminsen (deepseek-v4-flash, job 12004, lane dir-558), depending on #5610 (accepted, measured), #5616 (recorded) and #5792 (accepted, proven); accepted at proven by one trusted vote (review 2047, spot), provisional false, not superseded, report sha256 eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b matching the snapshot. Established as stated: #5616's pointwise step R_m = Phi_odd - A_m >= 0 is false, with exact-Fraction witness R_1085(527) = -0.059167657550535 at every H = 527 mod 1085 (m <= H from H = 1612); R_m depends on H only through rho = H mod m and is symmetric under rho -> m - rho; the kernel identity 2(d); an exhaustive sweep of all 2026 odd squarefree m <= 5000 (5 065 654 pairs) finds 434 negative pairs on exactly five conductors (1085, 3245, 3535, 3815, 3965), each a contiguous window centred on m/2. Measured only: the aggregate S(H) = sum_{m<=H}(Phi_odd - A_m) is positive at the five tested H = 1e3..1e5 (ratios 18.94-21.98 to sum A_m). Not established: the aggregate inequality for all H; any characterisation of failing conductors (the 'divisible by 5' and 'omega >= 3' patterns are range facts, and 67067 already breaks the first); anything about G_2, beta_2, twin primes or d > H. Carried corrections (review 2047's advisory also_fix, now pinned in the snapshot as open advisory finding 161035 on report.md): item 2(c)'s sentence is wrong as worded (the factor 1+cos(4 pi r xi_p/p) never equals -1, and for odd p it is never 0; it vanishes only if 4 r xi_p/p is an odd integer, so every factor is strictly positive and for gcd(r,m) = 1 the subtracted term is 2^0 - 1 = 0, hence c(r) > 0), and item 1's (m/e) psi2({H/(m/e)}) should read (g/e) psi2({H/g}) with g = m/e, as in #5616 Thm A (the code already uses g/e; the same wording sits in the residual_12004.py docstrings). Neither changes the 2(c) conclusion, the witness or the sweep. Generation 2: the gen-1 screening (#5954, same decisions) was recorded ineffective because finding 161035 was filed after its snapshot. The only material change is that pinned finding, whose content gen 1 already carried; decisions are unchanged.","transfer_eligible":false,"provisional_relevance":true},{"decision":"unresolved","target_id":"formalize","scope_keys":[],"evidence_md":"A bridge exists in principle: #5943 is accepted at proven and its headline is a finite exact rational computation (R_1085(527) < 0 in Fractions) plus elementary lemmas 2(a) periodicity in rho, 2(b) the rho -> m - rho symmetry, 2(c) c(r) > 0 for gcd(r,m) = 1, and 2(d) a trigonometric kernel identity. A rational witness is the kind of statement a Lean 4 decision procedure can check. Unresolved because: (i) the definitions of U_m, Phi_odd and the aligned class A_m come from #5610 (measured) and #5616 (recorded, not accepted), so a statement binding to an accepted definition is missing; (ii) item 2(c)'s published argument is wrong as worded and must be formalized from review 2047's corrected argument (finding 161035, open); (iii) 2(d) uses the two-sided trigamma sum and sin^-2 kernel, which is real-analytic and not a finite check; (iv) no Lean artifact or size estimate exists. These are recorded gaps, not a formalization request.","assumptions_md":"Consideration only, provisional. #5943 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer; the missing scope authority is an unresolved validation condition, and any transfer still needs ordinary review in the target lane. Source state preserved: route 399 explore return by Benjaminsen (deepseek-v4-flash, job 12004, lane dir-558), depending on #5610 (accepted, measured), #5616 (recorded) and #5792 (accepted, proven); accepted at proven by one trusted vote (review 2047, spot), provisional false, not superseded, report sha256 eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b matching the snapshot. Established as stated: #5616's pointwise step R_m = Phi_odd - A_m >= 0 is false, with exact-Fraction witness R_1085(527) = -0.059167657550535 at every H = 527 mod 1085 (m <= H from H = 1612); R_m depends on H only through rho = H mod m and is symmetric under rho -> m - rho; the kernel identity 2(d); an exhaustive sweep of all 2026 odd squarefree m <= 5000 (5 065 654 pairs) finds 434 negative pairs on exactly five conductors (1085, 3245, 3535, 3815, 3965), each a contiguous window centred on m/2. Measured only: the aggregate S(H) = sum_{m<=H}(Phi_odd - A_m) is positive at the five tested H = 1e3..1e5 (ratios 18.94-21.98 to sum A_m). Not established: the aggregate inequality for all H; any characterisation of failing conductors (the 'divisible by 5' and 'omega >= 3' patterns are range facts, and 67067 already breaks the first); anything about G_2, beta_2, twin primes or d > H. Carried corrections (review 2047's advisory also_fix, now pinned in the snapshot as open advisory finding 161035 on report.md): item 2(c)'s sentence is wrong as worded (the factor 1+cos(4 pi r xi_p/p) never equals -1, and for odd p it is never 0; it vanishes only if 4 r xi_p/p is an odd integer, so every factor is strictly positive and for gcd(r,m) = 1 the subtracted term is 2^0 - 1 = 0, hence c(r) > 0), and item 1's (m/e) psi2({H/(m/e)}) should read (g/e) psi2({H/g}) with g = m/e, as in #5616 Thm A (the code already uses g/e; the same wording sits in the residual_12004.py docstrings). Neither changes the 2(c) conclusion, the witness or the sweep. Generation 2: the gen-1 screening (#5954, same decisions) was recorded ineffective because finding 161035 was filed after its snapshot. The only material change is that pinned finding, whose content gen 1 already carried; decisions are unchanged.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"measure","scope_keys":[],"evidence_md":"Direct bridge. The source ships numbered measurement scripts (residual.py, sweep.py, sweep2.py, aggregate.py, counterexample.py, check_rm.py; stdlib Python, exact Fraction for every sign, about 2 min wall) with an exhaustive sweep of all (m, rho) for odd squarefree m <= 5000 and the aggregate S(H) at H = 1e3, 3e3, 1e4, 3e4, 1e5, plus conductor-level checks at H = 1e4, 3e4, 1e5. Extending the same instrument to larger m and H is exactly this lane's domain, and the kernel identity 2(d) gives an O(m) per-conductor screen. Route 399's next_step already names the extension (half-period census R_m((m-1)/2) to m <= 3e5, CRT-collected negative mass against S(H)), and #5947 (pending at read time) reports a sweep to m <= 5e4; this screening neither runs nor evaluates it. Scope kept: the measured S(H) > 0 values are measurements at five H, not a proof of the aggregate inequality, and any extension stays a measurement of the same quantities.","assumptions_md":"Consideration only, provisional. #5943 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer; the missing scope authority is an unresolved validation condition, and any transfer still needs ordinary review in the target lane. Source state preserved: route 399 explore return by Benjaminsen (deepseek-v4-flash, job 12004, lane dir-558), depending on #5610 (accepted, measured), #5616 (recorded) and #5792 (accepted, proven); accepted at proven by one trusted vote (review 2047, spot), provisional false, not superseded, report sha256 eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b matching the snapshot. Established as stated: #5616's pointwise step R_m = Phi_odd - A_m >= 0 is false, with exact-Fraction witness R_1085(527) = -0.059167657550535 at every H = 527 mod 1085 (m <= H from H = 1612); R_m depends on H only through rho = H mod m and is symmetric under rho -> m - rho; the kernel identity 2(d); an exhaustive sweep of all 2026 odd squarefree m <= 5000 (5 065 654 pairs) finds 434 negative pairs on exactly five conductors (1085, 3245, 3535, 3815, 3965), each a contiguous window centred on m/2. Measured only: the aggregate S(H) = sum_{m<=H}(Phi_odd - A_m) is positive at the five tested H = 1e3..1e5 (ratios 18.94-21.98 to sum A_m). Not established: the aggregate inequality for all H; any characterisation of failing conductors (the 'divisible by 5' and 'omega >= 3' patterns are range facts, and 67067 already breaks the first); anything about G_2, beta_2, twin primes or d > H. Carried corrections (review 2047's advisory also_fix, now pinned in the snapshot as open advisory finding 161035 on report.md): item 2(c)'s sentence is wrong as worded (the factor 1+cos(4 pi r xi_p/p) never equals -1, and for odd p it is never 0; it vanishes only if 4 r xi_p/p is an odd integer, so every factor is strictly positive and for gcd(r,m) = 1 the subtracted term is 2^0 - 1 = 0, hence c(r) > 0), and item 1's (m/e) psi2({H/(m/e)}) should read (g/e) psi2({H/g}) with g = m/e, as in #5616 Thm A (the code already uses g/e; the same wording sits in the residual_12004.py docstrings). Neither changes the 2(c) conclusion, the witness or the sweep. Generation 2: the gen-1 screening (#5954, same decisions) was recorded ineffective because finding 161035 was filed after its snapshot. The only material change is that pinned finding, whose content gen 1 already carried; decisions are unchanged.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"The infinitude lane collects routes toward infinitude of twin primes outside the refuted registry. #5943 states nothing about twin primes ('nothing about G_2, beta_2, twin primes, d > H'). Its contribution is a negative fact about a decomposition step inside route 177's evaluation of route 107's twin-pair defect, whose own link to infinitude is labelled conditional on the k-tuple conjecture. Phi_odd itself stays >= 0, and the aggregate conclusion is unchanged at the tested H. No infinitude statement gains or loses support, so there is no bridge.","assumptions_md":"Consideration only, provisional. #5943 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer; the missing scope authority is an unresolved validation condition, and any transfer still needs ordinary review in the target lane. Source state preserved: route 399 explore return by Benjaminsen (deepseek-v4-flash, job 12004, lane dir-558), depending on #5610 (accepted, measured), #5616 (recorded) and #5792 (accepted, proven); accepted at proven by one trusted vote (review 2047, spot), provisional false, not superseded, report sha256 eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b matching the snapshot. Established as stated: #5616's pointwise step R_m = Phi_odd - A_m >= 0 is false, with exact-Fraction witness R_1085(527) = -0.059167657550535 at every H = 527 mod 1085 (m <= H from H = 1612); R_m depends on H only through rho = H mod m and is symmetric under rho -> m - rho; the kernel identity 2(d); an exhaustive sweep of all 2026 odd squarefree m <= 5000 (5 065 654 pairs) finds 434 negative pairs on exactly five conductors (1085, 3245, 3535, 3815, 3965), each a contiguous window centred on m/2. Measured only: the aggregate S(H) = sum_{m<=H}(Phi_odd - A_m) is positive at the five tested H = 1e3..1e5 (ratios 18.94-21.98 to sum A_m). Not established: the aggregate inequality for all H; any characterisation of failing conductors (the 'divisible by 5' and 'omega >= 3' patterns are range facts, and 67067 already breaks the first); anything about G_2, beta_2, twin primes or d > H. Carried corrections (review 2047's advisory also_fix, now pinned in the snapshot as open advisory finding 161035 on report.md): item 2(c)'s sentence is wrong as worded (the factor 1+cos(4 pi r xi_p/p) never equals -1, and for odd p it is never 0; it vanishes only if 4 r xi_p/p is an odd integer, so every factor is strictly positive and for gcd(r,m) = 1 the subtracted term is 2^0 - 1 = 0, hence c(r) > 0), and item 1's (m/e) psi2({H/(m/e)}) should read (g/e) psi2({H/g}) with g = m/e, as in #5616 Thm A (the code already uses g/e; the same wording sits in the residual_12004.py docstrings). Neither changes the 2(c) conclusion, the witness or the sweep. Generation 2: the gen-1 screening (#5954, same decisions) was recorded ineffective because finding 161035 was filed after its snapshot. The only material change is that pinned finding, whose content gen 1 already carried; decisions are unchanged.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"This lane asks what structure a finite twin count would force. #5943 assumes nothing about the twin count and draws no consequence from finiteness. Its objects are the residual R_m over odd squarefree conductors m and residues rho = H mod m, and its negative windows (centred at m/2 on five conductors <= 5000) are properties of an arithmetic function of (m, rho), not structure forced by finitely many twins. No bridge between the hypotheses; recorded for completeness.","assumptions_md":"Consideration only, provisional. #5943 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer; the missing scope authority is an unresolved validation condition, and any transfer still needs ordinary review in the target lane. Source state preserved: route 399 explore return by Benjaminsen (deepseek-v4-flash, job 12004, lane dir-558), depending on #5610 (accepted, measured), #5616 (recorded) and #5792 (accepted, proven); accepted at proven by one trusted vote (review 2047, spot), provisional false, not superseded, report sha256 eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b matching the snapshot. Established as stated: #5616's pointwise step R_m = Phi_odd - A_m >= 0 is false, with exact-Fraction witness R_1085(527) = -0.059167657550535 at every H = 527 mod 1085 (m <= H from H = 1612); R_m depends on H only through rho = H mod m and is symmetric under rho -> m - rho; the kernel identity 2(d); an exhaustive sweep of all 2026 odd squarefree m <= 5000 (5 065 654 pairs) finds 434 negative pairs on exactly five conductors (1085, 3245, 3535, 3815, 3965), each a contiguous window centred on m/2. Measured only: the aggregate S(H) = sum_{m<=H}(Phi_odd - A_m) is positive at the five tested H = 1e3..1e5 (ratios 18.94-21.98 to sum A_m). Not established: the aggregate inequality for all H; any characterisation of failing conductors (the 'divisible by 5' and 'omega >= 3' patterns are range facts, and 67067 already breaks the first); anything about G_2, beta_2, twin primes or d > H. Carried corrections (review 2047's advisory also_fix, now pinned in the snapshot as open advisory finding 161035 on report.md): item 2(c)'s sentence is wrong as worded (the factor 1+cos(4 pi r xi_p/p) never equals -1, and for odd p it is never 0; it vanishes only if 4 r xi_p/p is an odd integer, so every factor is strictly positive and for gcd(r,m) = 1 the subtracted term is 2^0 - 1 = 0, hence c(r) > 0), and item 1's (m/e) psi2({H/(m/e)}) should read (g/e) psi2({H/g}) with g = m/e, as in #5616 Thm A (the code already uses g/e; the same wording sits in the residual_12004.py docstrings). Neither changes the 2(c) conclusion, the witness or the sweep. Generation 2: the gen-1 screening (#5954, same decisions) was recorded ineffective because finding 161035 was filed after its snapshot. The only material change is that pinned finding, whose content gen 1 already carried; decisions are unchanged.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Checked against kk-lower-bound.main and o1-o7 separately. main (survivor-gap lower bound, full cutoff, sufficient A = 12): #5943 has no sieve survivors, gaps or A parameter. o1 (prime harmonic asymptotic): no sum over 1/p appears. o2 (uniform smooth-number asymptotic): none. o3 (every fixed A > 4): none. o4 (dyadic prime-count asymptotic, coefficient 1/2): none. o5 (quoted general sieve input; finite dimension-4, CRT and covering results): #5943 uses no sieve input, and its CRT appears only as conductor factorisation inside #5610's root sum, not as a covering or dimension-4 result. o6 (two-sided density and Euler estimates): the products prod_{p|m}[1 + cos(4 pi r xi_p/p)] are finite local factors of one conductor, not density or Euler-product estimates. o7 (exact constant in equation (28)): no relation. The source's quantities and quantifiers do not meet any of the eight designated statements.","assumptions_md":"Consideration only, provisional. #5943 declares no research_authority scopes (scopes []), so every entry uses scope_keys [] and no entry is a validated automatic transfer; the missing scope authority is an unresolved validation condition, and any transfer still needs ordinary review in the target lane. Source state preserved: route 399 explore return by Benjaminsen (deepseek-v4-flash, job 12004, lane dir-558), depending on #5610 (accepted, measured), #5616 (recorded) and #5792 (accepted, proven); accepted at proven by one trusted vote (review 2047, spot), provisional false, not superseded, report sha256 eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b matching the snapshot. Established as stated: #5616's pointwise step R_m = Phi_odd - A_m >= 0 is false, with exact-Fraction witness R_1085(527) = -0.059167657550535 at every H = 527 mod 1085 (m <= H from H = 1612); R_m depends on H only through rho = H mod m and is symmetric under rho -> m - rho; the kernel identity 2(d); an exhaustive sweep of all 2026 odd squarefree m <= 5000 (5 065 654 pairs) finds 434 negative pairs on exactly five conductors (1085, 3245, 3535, 3815, 3965), each a contiguous window centred on m/2. Measured only: the aggregate S(H) = sum_{m<=H}(Phi_odd - A_m) is positive at the five tested H = 1e3..1e5 (ratios 18.94-21.98 to sum A_m). Not established: the aggregate inequality for all H; any characterisation of failing conductors (the 'divisible by 5' and 'omega >= 3' patterns are range facts, and 67067 already breaks the first); anything about G_2, beta_2, twin primes or d > H. Carried corrections (review 2047's advisory also_fix, now pinned in the snapshot as open advisory finding 161035 on report.md): item 2(c)'s sentence is wrong as worded (the factor 1+cos(4 pi r xi_p/p) never equals -1, and for odd p it is never 0; it vanishes only if 4 r xi_p/p is an odd integer, so every factor is strictly positive and for gcd(r,m) = 1 the subtracted term is 2^0 - 1 = 0, hence c(r) > 0), and item 1's (m/e) psi2({H/(m/e)}) should read (g/e) psi2({H/g}) with g = m/e, as in #5616 Thm A (the code already uses g/e; the same wording sits in the residual_12004.py docstrings). Neither changes the 2(c) conclusion, the witness or the sweep. Generation 2: the gen-1 screening (#5954, same decisions) was recorded ineffective because finding 161035 was filed after its snapshot. The only material change is that pinned finding, whose content gen 1 already carried; decisions are unchanged.","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":12027,"input_sha256":"494f3c2798702b6371771c1925fa7dde0783f9a26e9bcc1d9e92d09cce88693f","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5943,"material_evidence":{"sha256":"b2bf678fe220751f361a03b9e9eaaf2a6258ae4efc7bb2ab0355ab1f24967082","entries":{"return:5943":"17179ecbd71f8e400c9a954eeff3dce0b4e8da73ce789f777a59e47167cbb6b9","finding:161035":"0422cf4a2427daef4ae560f4c7d961a1ab7108af51eb9470227ee854deade57a","artifact:5943:0e5eefd149b4398806eb73cc4834b91cd8eda7cf62477c542cecc828d2abe293":"0e5eefd149b4398806eb73cc4834b91cd8eda7cf62477c542cecc828d2abe293","artifact:5943:2a111a0fd39b6e6d6210a07043adc5dac9276f24d95c528a4ea259745c290454":"2a111a0fd39b6e6d6210a07043adc5dac9276f24d95c528a4ea259745c290454","artifact:5943:361deccc62f0c9a3f421e09a70652ee537591bac011452b09d3948e71f85bd35":"361deccc62f0c9a3f421e09a70652ee537591bac011452b09d3948e71f85bd35","artifact:5943:408875c1ab48236430c8199098e4176f473305c91e29ebb6bcae2427018dae6a":"408875c1ab48236430c8199098e4176f473305c91e29ebb6bcae2427018dae6a","artifact:5943:45f6ea416a793ba7c99d9f6e753297b682fca57a8d783ad4241e6c8974daff9e":"45f6ea416a793ba7c99d9f6e753297b682fca57a8d783ad4241e6c8974daff9e","artifact:5943:4d54f88336be55716290ce46525cda006417ce326c71d35529f8a93194846515":"4d54f88336be55716290ce46525cda006417ce326c71d35529f8a93194846515","artifact:5943:5077a32d521b2bbdffe3f418472cb96eeb39e59e1a35ece23f2ed8f2cbd078c8":"5077a32d521b2bbdffe3f418472cb96eeb39e59e1a35ece23f2ed8f2cbd078c8","artifact:5943:6bddd47f1d421d23c4ea1315178b0d42239ffb9cfd09314c8ffbed8244427ee3":"6bddd47f1d421d23c4ea1315178b0d42239ffb9cfd09314c8ffbed8244427ee3","artifact:5943:6c07229faccecb259c4ac3efc66a20c70a83126ea32bab21ed4e8de32c3660a3":"6c07229faccecb259c4ac3efc66a20c70a83126ea32bab21ed4e8de32c3660a3","artifact:5943:8ff193ad56d3c732d54a329009479b69f007aba5a9a1f6e746c830663f2da407":"8ff193ad56d3c732d54a329009479b69f007aba5a9a1f6e746c830663f2da407","artifact:5943:c011a07ef84341f1848b5d0d666f4d4d81460fe481b81575ec4da84dd236b332":"c011a07ef84341f1848b5d0d666f4d4d81460fe481b81575ec4da84dd236b332","artifact:5943:ca4863d57a388b8ec72da39f04a3bbc9e816a7a44c0b68469f2045c665f850cf":"ca4863d57a388b8ec72da39f04a3bbc9e816a7a44c0b68469f2045c665f850cf","artifact:5943:e6293ec4907450c53fe3e545d39130f9ec945ff53785d68481b16b7ecf8ecfdc":"e6293ec4907450c53fe3e545d39130f9ec945ff53785d68481b16b7ecf8ecfdc","artifact:5943:e9a6a586006f2e5be6cf08ec5e028556f337d44a278ea3b3797e1e8d1a8a1672":"e9a6a586006f2e5be6cf08ec5e028556f337d44a278ea3b3797e1e8d1a8a1672","artifact:5943:eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b":"eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5943,"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 #5943: 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\":5943,\"report_sha256\":\"eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5943\":\"17179ecbd71f8e400c9a954eeff3dce0b4e8da73ce789f777a59e47167cbb6b9\",\"finding:161035\":\"0422cf4a2427daef4ae560f4c7d961a1ab7108af51eb9470227ee854deade57a\",\"artifact:5943:0e5eefd149b4398806eb73cc4834b91cd8eda7cf62477c542cecc828d2abe293\":\"0e5eefd149b4398806eb73cc4834b91cd8eda7cf62477c542cecc828d2abe293\",\"artifact:5943:2a111a0fd39b6e6d6210a07043adc5dac9276f24d95c528a4ea259745c290454\":\"2a111a0fd39b6e6d6210a07043adc5dac9276f24d95c528a4ea259745c290454\",\"artifact:5943:361deccc62f0c9a3f421e09a70652ee537591bac011452b09d3948e71f85bd35\":\"361deccc62f0c9a3f421e09a70652ee537591bac011452b09d3948e71f85bd35\",\"artifact:5943:408875c1ab48236430c8199098e4176f473305c91e29ebb6bcae2427018dae6a\":\"408875c1ab48236430c8199098e4176f473305c91e29ebb6bcae2427018dae6a\",\"artifact:5943:45f6ea416a793ba7c99d9f6e753297b682fca57a8d783ad4241e6c8974daff9e\":\"45f6ea416a793ba7c99d9f6e753297b682fca57a8d783ad4241e6c8974daff9e\",\"artifact:5943:4d54f88336be55716290ce46525cda006417ce326c71d35529f8a93194846515\":\"4d54f88336be55716290ce46525cda006417ce326c71d35529f8a93194846515\",\"artifact:5943:5077a32d521b2bbdffe3f418472cb96eeb39e59e1a35ece23f2ed8f2cbd078c8\":\"5077a32d521b2bbdffe3f418472cb96eeb39e59e1a35ece23f2ed8f2cbd078c8\",\"artifact:5943:6bddd47f1d421d23c4ea1315178b0d42239ffb9cfd09314c8ffbed8244427ee3\":\"6bddd47f1d421d23c4ea1315178b0d42239ffb9cfd09314c8ffbed8244427ee3\",\"artifact:5943:6c07229faccecb259c4ac3efc66a20c70a83126ea32bab21ed4e8de32c3660a3\":\"6c07229faccecb259c4ac3efc66a20c70a83126ea32bab21ed4e8de32c3660a3\",\"artifact:5943:8ff193ad56d3c732d54a329009479b69f007aba5a9a1f6e746c830663f2da407\":\"8ff193ad56d3c732d54a329009479b69f007aba5a9a1f6e746c830663f2da407\",\"artifact:5943:c011a07ef84341f1848b5d0d666f4d4d81460fe481b81575ec4da84dd236b332\":\"c011a07ef84341f1848b5d0d666f4d4d81460fe481b81575ec4da84dd236b332\",\"artifact:5943:ca4863d57a388b8ec72da39f04a3bbc9e816a7a44c0b68469f2045c665f850cf\":\"ca4863d57a388b8ec72da39f04a3bbc9e816a7a44c0b68469f2045c665f850cf\",\"artifact:5943:e6293ec4907450c53fe3e545d39130f9ec945ff53785d68481b16b7ecf8ecfdc\":\"e6293ec4907450c53fe3e545d39130f9ec945ff53785d68481b16b7ecf8ecfdc\",\"artifact:5943:e9a6a586006f2e5be6cf08ec5e028556f337d44a278ea3b3797e1e8d1a8a1672\":\"e9a6a586006f2e5be6cf08ec5e028556f337d44a278ea3b3797e1e8d1a8a1672\",\"artifact:5943:eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b\":\"eca896f7390be0a2415404055ccf0ceb98703c7d05c3ef436a781512cce2dc4b\"},\"sha256\":\"b2bf678fe220751f361a03b9e9eaaf2a6258ae4efc7bb2ab0355ab1f24967082\"},\"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:5943,input_sha256:\"494f3c2798702b6371771c1925fa7dde0783f9a26e9bcc1d9e92d09cce88693f\",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/5956/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"3b4bed060277f786b0d5f693e48c4d7df960966c45cee358696f883222dccc8e","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}