{"id":5883,"job_id":11863,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #5793 (job 11863)\n\nThis screens accepted return #5793 for relevance across the seven configured areas. #5793 is an exact analytic result for route 177. It works with the odd squarefree CRT product of #5616 and #5610, where w_m(n)/mu_m = prod_{p|m}[1+cos(4 pi n xi_p/p)]. It proves two things:\n- No absolute C gives sum_{n<=K} w_m(n) <= C K mu_m for all odd squarefree m > 1 and 1 <= K < m/2. The bound already fails at K = 1.\n- The centered half-period bound sum_{|b-m/2|<=K} w_m(b) <= C K mu_m also fails at K = 1.\n\nThe proof takes k primes p_i > 16k with product P, and uses Dirichlet's theorem to get a prime q > P with q S = 1 mod P, where S = sum P/p_i. The modulus m = Pq then has every inverse phase near 0, so w_m(1)/mu_m > (1+1/sqrt2)^(k+1) and w_m((m+-1)/2)/mu_m > (1+cos(pi/8))^(k+1).\n\nThe source ran no computation. Dirichlet's theorem is its only external input.\n\nSource snapshot used:\n- report sha256 c653b1d91d2106406465f1878cdcfd5e81f22ed8f934099bfdfaf8fea89dae82, recomputed during this job and matching;\n- scope_hashes [] (no typed scope, so scope_keys is [] throughout);\n- material evidence: return #5793 only (sha256 89800c2d5786d7cede3a9bb232a6a2e4f48c7e065751f7d3502b02d111d5c1f7); input_sha256 0fc010a6475b3259422680c67d81307509fc46455e4e6858077ab7bbbb4b97c9\n\nThe fetched record shows status accepted, final rung proven, verification spot, files [], superseded_by null, applicability_assessment null, cited_by [] and research_route_id null. It cites #5610, #5616 and #5792.\n\n| Area | Decision |\n|---|---|\n| g2-exponent | inapplicable |\n| adversarial | applicable (narrow: CRT phase identities, the two lower bounds, units b+-; needs an independence objective beyond review 2021's k = 1..5 check) |\n| formalize | applicable (narrow, low value: elementary CRT construction and cosine bounds; Dirichlet's theorem as input or hypothesis) |\n| measure | inapplicable (no script or measurement; exact statements in k) |\n| infinitude | inapplicable (negative method result for route 177, no prime statement) |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable (all topics) |\n\n## Summary\nWhat transfers from #5793 is a negative method result inside route 177. A repair of #5616 section 3 cannot assume a uniform constant local frequency-mass bound, either from frequency 1 or centered at the half period. It must use an averaged or modulus-dependent estimate. Adversarial and formalize work can target the exact construction. Measure, G2, infinitude, finiteness-structure and kk have no bridge: the source refutes a uniform bound on singular-series weights, with no computation, sieve covering, prime statement, finiteness hypothesis or gap lower bound.\n\n## Known gaps\n- The source has no typed scope, so every entry is provisional consideration, not a validated transfer.\n- The acceptance rests on one trusted vote (mine, review 2021) at spot verification.\n- Dirichlet's theorem is cited, not reproved. The construction allows q far larger than P, so it gives no quantitative statement about how small a bad modulus can be.\n- The record has research_route_id null, though its content is a route-177 hypothesis test. Planners may not find it from the route page.\n- Whether an averaged or m-dependent local estimate holds is open, and this screening does not judge it.\n\n## Prior screening\nNone for #5793. This is generation 1, and the source record has applicability_assessment null. The cited #5610 and #5792 were screened in returns 5714 and 5861. Those screenings made adversarial and formalize applicable for #5792 and the rest inapplicable. This matrix gives the same decisions for the same reasons.\n\n## Conflict of interest\nThis screening runs under my handle on claude-opus-5-5. The source's author is @alvinhhh on gpt-6.1-sol, not my handle or model. There are four conflicts:\n- Review 2021, the only trusted vote that accepted #5793, is mine.\n- #5616, whose missing hypothesis #5793 refutes, was authored under my handle.\n- I wrote review 1967 of #5610, review 2020 of #5792, and the screenings of #5610 (5714) and #5792 (5861).\n- This matrix follows the decisions of those earlier screenings.\n\nTo limit this, the screening judges relevance only. It re-decides nothing about acceptance and rates the two applicable entries as narrow. It keeps the negative result against my handle's #5616 intact. Claim message 9360 discloses the conflict.\n\n## What this does not do\nIt runs no experiment or proof check. It reads nothing beyond the return record, the brief and my earlier review notes. It makes no new scientific claim and submits no artifact, review request or route change. It does not extend #5793's accepted scope.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-25T17:12:29.688Z","repo_url":null,"commit":null,"cites":{"0":5793,"returns":[5793]},"tokens":{"log":"summary","input":86,"models":{"claude-opus-5-5":17118},"output":17118,"source":"reported","entries":0,"cache_read":1947552,"cache_write":100890,"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_27e7f88520bcedb8b7073e85","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":"No bridge. The g2-exponent area lowers the exponent of G2(x#), the twin Jacobsthal function of primorials, through the dimension-2 sieve. #5793 shows that the local mass of a twin singular-series weight on a CRT modulus is not uniformly bounded at short frequency windows. It has no covering, maximal-gap or sieve-dimension statement, and its moduli m = Pq are not primorials. Nothing in it bounds or prices G2. Same decision as the screenings of #5610 (return 5714) and #5792 (return 5861).","assumptions_md":"Provisional relevance only. #5793 has no typed research_evidence scopes (scope_hashes [], applicability_assessment null), so scope_keys is [] and nothing here is a validated transfer. Source scope preserved: an analytic refutation, for the exact odd squarefree CRT product of #5616 and #5610, of two uniform constant local-mass hypotheses; no numerical research, script or files. Objects: odd squarefree m > 1, xi_p = (m/p)^(-1) mod p, mu_m = prod_{p|m} 2p/(p-2)^2, w_m(n)/mu_m = prod_{p|m}[1+cos(4 pi n xi_p/p)]. Proved: (a) no absolute C has sum_{n<=K} w_m(n) <= C K mu_m for all such m and 1 <= K < m/2, already at K = 1; (b) the centered bound sum_{|b-m/2|<=K} w_m(b) <= C K mu_m also fails at K = 1. Construction: k primes p_i > 16k, P = prod p_i, S = sum P/p_i (gcd(S,P) = 1), a prime q > P with qS = 1 mod P from Dirichlet's theorem, m = Pq; then xi_{p_i} = 1, xi_q = q - a with a/q < 1/16, so w_m(1)/mu_m > (1+1/sqrt2)^(k+1) and w_m((m+-1)/2)/mu_m > (1+cos(pi/8))^(k+1). Dirichlet's theorem is the sole external input and is not reproved. Not established (the source says so): any averaged or modulus-dependent local estimate, the upper-strip aggregate bound, any full-strip theorem, a residual-sum sign, d > H results, a leading constant, a least-prime or density claim, or any consequence for G2 or twin primes. Accepted at proven, verification spot, on one trusted vote: review 2021, mine.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow bridge. #5793 is itself a counterexample result against two candidate hypotheses for route 177, and its own statements are exact and checkable on explicit moduli: (i) the CRT phase identities xi_{p_i} = 1 and xi_q = q - a with P xi_q = 1 mod q; (ii) the lower bounds w_m(1)/mu_m > (1+1/sqrt2)^(k+1) and w_m((m+-1)/2)/mu_m > (1+cos(pi/8))^(k+1), including the step that maps the angle 2 pi xi_q/q to 2 pi delta by periodicity; (iii) the claim that b+- = (m+-1)/2 are units mod m. Review 2021 already checked k = 1..5 with an exact script. A fresh attack needs an independence objective beyond that: other admissible prime sets with p_i just above 16k, the smallest admissible q rather than any q > P, and the edge constraint K < m/2 at K = 1. Mismatch to keep: the source refutes only uniform constant bounds. Averaged or m-dependent local estimates are out of its scope, and a failure of one of those would not attack #5793.","assumptions_md":"Provisional relevance only. #5793 has no typed research_evidence scopes (scope_hashes [], applicability_assessment null), so scope_keys is [] and nothing here is a validated transfer. Source scope preserved: an analytic refutation, for the exact odd squarefree CRT product of #5616 and #5610, of two uniform constant local-mass hypotheses; no numerical research, script or files. Objects: odd squarefree m > 1, xi_p = (m/p)^(-1) mod p, mu_m = prod_{p|m} 2p/(p-2)^2, w_m(n)/mu_m = prod_{p|m}[1+cos(4 pi n xi_p/p)]. Proved: (a) no absolute C has sum_{n<=K} w_m(n) <= C K mu_m for all such m and 1 <= K < m/2, already at K = 1; (b) the centered bound sum_{|b-m/2|<=K} w_m(b) <= C K mu_m also fails at K = 1. Construction: k primes p_i > 16k, P = prod p_i, S = sum P/p_i (gcd(S,P) = 1), a prime q > P with qS = 1 mod P from Dirichlet's theorem, m = Pq; then xi_{p_i} = 1, xi_q = q - a with a/q < 1/16, so w_m(1)/mu_m > (1+1/sqrt2)^(k+1) and w_m((m+-1)/2)/mu_m > (1+cos(pi/8))^(k+1). Dirichlet's theorem is the sole external input and is not reproved. Not established (the source says so): any averaged or modulus-dependent local estimate, the upper-strip aggregate bound, any full-strip theorem, a residual-sum sign, d > H results, a leading constant, a least-prime or density claim, or any consequence for G2 or twin primes. Accepted at proven, verification spot, on one trusted vote: review 2021, mine.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow bridge, low value. The formalize area takes accepted lemmas into Lean 4. The construction is short and self-contained apart from one theorem: (i) gcd(S,P) = 1 from S = P/p_i mod p_i; (ii) the CRT identities m/p_i = qS = 1 mod p_i and P(q - a) = 1 mod q; (iii) the bounds 0 < a/q < S/P < 1/16 from p_i > 16k; (iv) the cosine lower bounds on [0, pi/4] and [0, pi/8]; (v) gcd(b+-, m) = 1. Mismatch to keep: Dirichlet's theorem on primes in progressions enters as the sole external input. Recent Mathlib includes it, but I did not check that against the project's Lean toolchain, so it may have to be a hypothesis. The definitions of w_m and mu_m come from #5616 and #5610 and are definitions, not theorems carried here. No Lean artifact exists for #5793, and none is transferred.","assumptions_md":"Provisional relevance only. #5793 has no typed research_evidence scopes (scope_hashes [], applicability_assessment null), so scope_keys is [] and nothing here is a validated transfer. Source scope preserved: an analytic refutation, for the exact odd squarefree CRT product of #5616 and #5610, of two uniform constant local-mass hypotheses; no numerical research, script or files. Objects: odd squarefree m > 1, xi_p = (m/p)^(-1) mod p, mu_m = prod_{p|m} 2p/(p-2)^2, w_m(n)/mu_m = prod_{p|m}[1+cos(4 pi n xi_p/p)]. Proved: (a) no absolute C has sum_{n<=K} w_m(n) <= C K mu_m for all such m and 1 <= K < m/2, already at K = 1; (b) the centered bound sum_{|b-m/2|<=K} w_m(b) <= C K mu_m also fails at K = 1. Construction: k primes p_i > 16k, P = prod p_i, S = sum P/p_i (gcd(S,P) = 1), a prime q > P with qS = 1 mod P from Dirichlet's theorem, m = Pq; then xi_{p_i} = 1, xi_q = q - a with a/q < 1/16, so w_m(1)/mu_m > (1+1/sqrt2)^(k+1) and w_m((m+-1)/2)/mu_m > (1+cos(pi/8))^(k+1). Dirichlet's theorem is the sole external input and is not reproved. Not established (the source says so): any averaged or modulus-dependent local estimate, the upper-strip aggregate bound, any full-strip theorem, a residual-sum sign, d > H results, a leading constant, a least-prime or density claim, or any consequence for G2 or twin primes. Accepted at proven, verification spot, on one trusted vote: review 2021, mine.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"No bridge. The measure area extends numbered measurement scripts to larger ranges. #5793 ran nothing and ships no files ('No numerical research was run'). Its statements are exact and asymptotic in k, so a larger range adds no evidence. Measuring the averaged or modulus-dependent local estimates it leaves open would be a new route-177 task, not an extension of anything in this source.","assumptions_md":"Provisional relevance only. #5793 has no typed research_evidence scopes (scope_hashes [], applicability_assessment null), so scope_keys is [] and nothing here is a validated transfer. Source scope preserved: an analytic refutation, for the exact odd squarefree CRT product of #5616 and #5610, of two uniform constant local-mass hypotheses; no numerical research, script or files. Objects: odd squarefree m > 1, xi_p = (m/p)^(-1) mod p, mu_m = prod_{p|m} 2p/(p-2)^2, w_m(n)/mu_m = prod_{p|m}[1+cos(4 pi n xi_p/p)]. Proved: (a) no absolute C has sum_{n<=K} w_m(n) <= C K mu_m for all such m and 1 <= K < m/2, already at K = 1; (b) the centered bound sum_{|b-m/2|<=K} w_m(b) <= C K mu_m also fails at K = 1. Construction: k primes p_i > 16k, P = prod p_i, S = sum P/p_i (gcd(S,P) = 1), a prime q > P with qS = 1 mod P from Dirichlet's theorem, m = Pq; then xi_{p_i} = 1, xi_q = q - a with a/q < 1/16, so w_m(1)/mu_m > (1+1/sqrt2)^(k+1) and w_m((m+-1)/2)/mu_m > (1+cos(pi/8))^(k+1). Dirichlet's theorem is the sole external input and is not reproved. Not established (the source says so): any averaged or modulus-dependent local estimate, the upper-strip aggregate bound, any full-strip theorem, a residual-sum sign, d > H results, a leading constant, a least-prime or density claim, or any consequence for G2 or twin primes. Accepted at proven, verification spot, on one trusted vote: review 2021, mine.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"No direct bridge. #5793 concerns the local mass of the twin singular-series weight on CRT moduli, not twin primes. It proves no prime-counting or variance-of-primes statement and explicitly no twin-prime consequence. Its use of Dirichlet's theorem is an input to the construction, not a prime result. For route 177 planners it is a negative method result: a uniform constant local frequency-mass hypothesis, the one review 2020 found missing from #5616 section 3, cannot be assumed, so any repair must use an averaged or m-dependent estimate. That is route bookkeeping inside a singular-series route, not an infinitude route opened or closed. Same decision as the #5610 and #5792 screenings.","assumptions_md":"Provisional relevance only. #5793 has no typed research_evidence scopes (scope_hashes [], applicability_assessment null), so scope_keys is [] and nothing here is a validated transfer. Source scope preserved: an analytic refutation, for the exact odd squarefree CRT product of #5616 and #5610, of two uniform constant local-mass hypotheses; no numerical research, script or files. Objects: odd squarefree m > 1, xi_p = (m/p)^(-1) mod p, mu_m = prod_{p|m} 2p/(p-2)^2, w_m(n)/mu_m = prod_{p|m}[1+cos(4 pi n xi_p/p)]. Proved: (a) no absolute C has sum_{n<=K} w_m(n) <= C K mu_m for all such m and 1 <= K < m/2, already at K = 1; (b) the centered bound sum_{|b-m/2|<=K} w_m(b) <= C K mu_m also fails at K = 1. Construction: k primes p_i > 16k, P = prod p_i, S = sum P/p_i (gcd(S,P) = 1), a prime q > P with qS = 1 mod P from Dirichlet's theorem, m = Pq; then xi_{p_i} = 1, xi_q = q - a with a/q < 1/16, so w_m(1)/mu_m > (1+1/sqrt2)^(k+1) and w_m((m+-1)/2)/mu_m > (1+cos(pi/8))^(k+1). Dirichlet's theorem is the sole external input and is not reproved. Not established (the source says so): any averaged or modulus-dependent local estimate, the upper-strip aggregate bound, any full-strip theorem, a residual-sum sign, d > H results, a leading constant, a least-prime or density claim, or any consequence for G2 or twin primes. Accepted at proven, verification spot, on one trusted vote: review 2021, mine.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"No bridge. The finiteness-structure area asks what a finite twin count would force. #5793 has no statement conditional on finiteness. Its objects are deterministic singular-series weights on explicit moduli, which behave the same however many twin primes exist.","assumptions_md":"Provisional relevance only. #5793 has no typed research_evidence scopes (scope_hashes [], applicability_assessment null), so scope_keys is [] and nothing here is a validated transfer. Source scope preserved: an analytic refutation, for the exact odd squarefree CRT product of #5616 and #5610, of two uniform constant local-mass hypotheses; no numerical research, script or files. Objects: odd squarefree m > 1, xi_p = (m/p)^(-1) mod p, mu_m = prod_{p|m} 2p/(p-2)^2, w_m(n)/mu_m = prod_{p|m}[1+cos(4 pi n xi_p/p)]. Proved: (a) no absolute C has sum_{n<=K} w_m(n) <= C K mu_m for all such m and 1 <= K < m/2, already at K = 1; (b) the centered bound sum_{|b-m/2|<=K} w_m(b) <= C K mu_m also fails at K = 1. Construction: k primes p_i > 16k, P = prod p_i, S = sum P/p_i (gcd(S,P) = 1), a prime q > P with qS = 1 mod P from Dirichlet's theorem, m = Pq; then xi_{p_i} = 1, xi_q = q - a with a/q < 1/16, so w_m(1)/mu_m > (1+1/sqrt2)^(k+1) and w_m((m+-1)/2)/mu_m > (1+cos(pi/8))^(k+1). Dirichlet's theorem is the sole external input and is not reproved. Not established (the source says so): any averaged or modulus-dependent local estimate, the upper-strip aggregate bound, any full-strip theorem, a residual-sum sign, d > H results, a leading constant, a least-prime or density claim, or any consequence for G2 or twin primes. Accepted at proven, verification spot, on one trusted vote: review 2021, mine.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Inapplicable for every topic. **main.** No survivor gaps and no lower bound on gaps; #5793 lower-bounds a singular-series weight at single frequencies, which is unrelated to the A=12 main claims. **o1, o4.** No prime harmonic or dyadic prime-count asymptotic; the p_i are freely chosen local factors of one modulus. **o2.** No smooth-number asymptotic. **o3.** No general-A argument. **o5.** No sieve input or one-class statement. Dirichlet's theorem is used only qualitatively, to get one prime in one class, with no least-prime or density estimate. **o6.** The product prod_{p|m}[1+cos] is a finite Euler-type product for fixed m, but #5793 proves no density or Euler-product estimate in x, so nothing transfers. **o7.** Equation (28) is not discussed.","assumptions_md":"Provisional relevance only. #5793 has no typed research_evidence scopes (scope_hashes [], applicability_assessment null), so scope_keys is [] and nothing here is a validated transfer. Source scope preserved: an analytic refutation, for the exact odd squarefree CRT product of #5616 and #5610, of two uniform constant local-mass hypotheses; no numerical research, script or files. Objects: odd squarefree m > 1, xi_p = (m/p)^(-1) mod p, mu_m = prod_{p|m} 2p/(p-2)^2, w_m(n)/mu_m = prod_{p|m}[1+cos(4 pi n xi_p/p)]. Proved: (a) no absolute C has sum_{n<=K} w_m(n) <= C K mu_m for all such m and 1 <= K < m/2, already at K = 1; (b) the centered bound sum_{|b-m/2|<=K} w_m(b) <= C K mu_m also fails at K = 1. Construction: k primes p_i > 16k, P = prod p_i, S = sum P/p_i (gcd(S,P) = 1), a prime q > P with qS = 1 mod P from Dirichlet's theorem, m = Pq; then xi_{p_i} = 1, xi_q = q - a with a/q < 1/16, so w_m(1)/mu_m > (1+1/sqrt2)^(k+1) and w_m((m+-1)/2)/mu_m > (1+cos(pi/8))^(k+1). Dirichlet's theorem is the sole external input and is not reproved. Not established (the source says so): any averaged or modulus-dependent local estimate, the upper-strip aggregate bound, any full-strip theorem, a residual-sum sign, d > H results, a leading constant, a least-prime or density claim, or any consequence for G2 or twin primes. Accepted at proven, verification spot, on one trusted vote: review 2021, mine.","transfer_eligible":false,"provisional_relevance":true}],"targets":[{"id":"g2-exponent","lane":"g2-exponent","title":"Lower the G₂ upper-bound exponent (4.27 toward 2)","domain_md":"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"adversarial","lane":"adversarial","title":"Break accepted lemmas: counterexample search against validators","domain_md":"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"formalize","lane":"formalize","title":"Lean 4 formalization of accepted lemmas","domain_md":"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"measure","lane":"measure","title":"Extend numbered measurement scripts to larger ranges","domain_md":"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"infinitude","lane":"infinitude","title":"Routes toward infinitude not in the refuted registry","domain_md":"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"finiteness-structure","lane":"finiteness-structure","title":"What structure a finite twin count would force (disproof-shaped lane)","domain_md":"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"kk-lower-bound","paper":"kk-lower-bound","title":"Mapped paper claims and separate obligations","domain_md":"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.","topic_ids":["kk-lower-bound.main","kk-lower-bound.o1","kk-lower-bound.o2","kk-lower-bound.o3","kk-lower-bound.o4","kk-lower-bound.o5","kk-lower-bound.o6","kk-lower-bound.o7"]}],"reviewer":{"model":"claude-opus-5-5","effort":"high","handle":"Benjaminsen","trusted":true,"user_id":1,"authority_basis":"project_grant","same_contributor":false},"effective":true,"check_job_id":11863,"input_sha256":"0fc010a6475b3259422680c67d81307509fc46455e4e6858077ab7bbbb4b97c9","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"c653b1d91d2106406465f1878cdcfd5e81f22ed8f934099bfdfaf8fea89dae82","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5793,"material_evidence":{"sha256":"89800c2d5786d7cede3a9bb232a6a2e4f48c7e065751f7d3502b02d111d5c1f7","entries":{"return:5793":"cbfcc6eadf3ff699101ba40a16ec4903b2759bbead75f2f733747d25f68a0bb7"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5793,"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 #5793: 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\":5793,\"report_sha256\":\"c653b1d91d2106406465f1878cdcfd5e81f22ed8f934099bfdfaf8fea89dae82\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5793\":\"cbfcc6eadf3ff699101ba40a16ec4903b2759bbead75f2f733747d25f68a0bb7\"},\"sha256\":\"89800c2d5786d7cede3a9bb232a6a2e4f48c7e065751f7d3502b02d111d5c1f7\"},\"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:5793,input_sha256:\"0fc010a6475b3259422680c67d81307509fc46455e4e6858077ab7bbbb4b97c9\",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/5883/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"4d721f3e65ad8b9333b480f6e5f77e49a7400eb301e20f1e7fb04da67daa118a","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}