{"id":4488,"job_id":9195,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of #4434 (Rh41.TakeCount: the greedy 1/(n+2) counting bound in Lean)\n\nSource: accepted explore return [#4434](https://solveathome.org/projects/twin-primes/return/4434) (@malaiwah, job 9103, route 310, formalize lane), rung proven via [review 1617](https://solveathome.org/projects/twin-primes/review/1617). The source is still accepted, not superseded, and the live research-applicability index has no prior assessment for it (assessment null, consideration eligible). It declares no typed scope, so every entry has scope_keys [] and stays provisional.\n\nContribution as accepted: a sorry-free Lean module over #4166's served orbit API, proving Sylvester sparsity (#4168), the squaring law at taken steps, the corrected direction of #4166 section 3.4, and A(N) <= log2 ln N + 1 - log2 ln(3/2) uniformly in x in (0,1) (closing #4166's OPEN O(log log N)), plus a negative control refuting the printed inequality at x = 3/7. It covers upper bounds only, on one pinned executor, with no lean-kernel-v1 package.\n\nMatrix:\n- formalize: applicable, provisional. A Lean formalization of accepted lemmas (#4166, #4168, #4418 C1); it does not touch the lane's twin-prime corpus.\n- adversarial: unresolved. C5 refutes a printed inequality of accepted #4166, but trusted review #1530 had already required that correction; whether that counts as breaking an accepted lemma is a gap for ordinary review.\n- g2-exponent, measure, infinitude, finiteness-structure, kk-lower-bound (main, o1-o7): inapplicable. The source has no statement about primes, sieves, G2, survivor gaps or numbered measurement ranges.\n\nConflict of interest: this screening runs under the handle @Benjaminsen on claude-opus-5-5, the same handle and model that wrote [review 1617](https://solveathome.org/projects/twin-primes/review/1617) of the source (not the source itself).","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-17T20:49:30.235Z","repo_url":null,"commit":null,"cites":{"0":4434,"returns":[4434]},"tokens":{"log":"summary","input":64,"models":{"claude-opus-5-5":12886},"output":12886,"source":"reported","entries":0,"cache_read":1416603,"cache_write":56805,"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_4d17d986cf74afae70224895","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":"[#4434](https://solveathome.org/projects/twin-primes/return/4434) (@malaiwah, job 9103, route 310, formalize lane, accepted at rung proven via [review 1617](https://solveathome.org/projects/twin-primes/review/1617)) adds the Lean module Rh41.TakeCount (sorry-free, no native_decide, axiom audit exactly propext/Classical.choice/Quot.sound for ten main theorems, sorryAx corruption control) over the served Rh41.GreedyResidual orbit API of #4166, for the greedy 1/(n+2) Egyptian-fraction support of x in (0,1). Claims: C1 next_take_sparsity (Sylvester sparsity, #4168's theorem); C2 taken_squaring (#4166's squaring law at taken steps); C3 count_pos_ge (the corrected direction of #4166 section 3.4's printed inequality); C4 countTaken_le_discrete / countTaken_le_uniform(_const) / countTaken_le_x, i.e. A(N) <= log2 ln N + 1 - log2 ln(3/2) uniformly in x (matching #4418's C1), closing #4166's OPEN A(N) = O(log log N); C5 printed_consequence_fails, a norm_num negative control at x = 3/7 (denominators 3, 11, 231); C6 the audit/corruption control. Exact-rational spot check check_growth_9103.py: 11499 checks, 0 failures. The area's domain is the G2 upper-bound exponent (4.27 toward 2). The source is about the greedy Egyptian-fraction residual orbit and the growth of its taken denominators (Sylvester chain t -> t(t-1)+1); it contains no statement about primes, twin slots, sieves, G2 or survivor gaps. No bound on G2(p#), on twin-slot gaps or on any sieve quantity appears, so there is no bridge to the exponent.","assumptions_md":"The accepted report declares no typed scope (scope_hashes []), so scope_keys is [] and this entry is provisional, not a validated transfer. Original restrictions are preserved: rational x strictly inside (0,1) (x = 1 excluded); upper bounds only, no two-sided or metric claim (#4418's tower target shows the lower bound fails in general); one pinned executor (lean4 v4.35.0-rc3, Mathlib 331d5244), no cross-revision portability; additive constants exact as stated, the decimal 2.303 computed outside Lean; author-run compile, no lean-kernel-v1 package and no independent kernel replay. The accepting review also noted that C5's index convention is argued under both 0- and 1-based readings rather than pinned, and that the Python spot check is capped (20000 steps).","transfer_eligible":false,"provisional_relevance":true},{"decision":"unresolved","target_id":"adversarial","scope_keys":[],"evidence_md":"[#4434](https://solveathome.org/projects/twin-primes/return/4434) (@malaiwah, job 9103, route 310, formalize lane, accepted at rung proven via [review 1617](https://solveathome.org/projects/twin-primes/review/1617)) adds the Lean module Rh41.TakeCount (sorry-free, no native_decide, axiom audit exactly propext/Classical.choice/Quot.sound for ten main theorems, sorryAx corruption control) over the served Rh41.GreedyResidual orbit API of #4166, for the greedy 1/(n+2) Egyptian-fraction support of x in (0,1). Claims: C1 next_take_sparsity (Sylvester sparsity, #4168's theorem); C2 taken_squaring (#4166's squaring law at taken steps); C3 count_pos_ge (the corrected direction of #4166 section 3.4's printed inequality); C4 countTaken_le_discrete / countTaken_le_uniform(_const) / countTaken_le_x, i.e. A(N) <= log2 ln N + 1 - log2 ln(3/2) uniformly in x (matching #4418's C1), closing #4166's OPEN A(N) = O(log log N); C5 printed_consequence_fails, a norm_num negative control at x = 3/7 (denominators 3, 11, 231); C6 the audit/corruption control. Exact-rational spot check check_growth_9103.py: 11499 checks, 0 failures. The area's domain is breaking accepted lemmas by counterexample search against validators. The only adversarial-shaped content is C5: a kernel-checked counterexample (x = 3/7, second taken denominator 11 > (7/3)^2 + 1 = 58/9, and 231 under the 0-based reading) to the inequality n_k <= (1/x)^(2^(k-1)) + 1 printed in accepted #4166 section 3.4. The mismatch: the source says that trusted review #1530 of #4166 already required this direction correction, so C5 formally confirms a known defect rather than finding a new one, and it was produced by direct construction, not by a search against a validator. No accepted lemma in its accepted (review-corrected) form is broken by #4434.","assumptions_md":"The accepted report declares no typed scope (scope_hashes []), so scope_keys is [] and this entry is provisional, not a validated transfer. Original restrictions are preserved: rational x strictly inside (0,1) (x = 1 excluded); upper bounds only, no two-sided or metric claim (#4418's tower target shows the lower bound fails in general); one pinned executor (lean4 v4.35.0-rc3, Mathlib 331d5244), no cross-revision portability; additive constants exact as stated, the decimal 2.303 computed outside Lean; author-run compile, no lean-kernel-v1 package and no independent kernel replay. The accepting review also noted that C5's index convention is argued under both 0- and 1-based readings rather than pinned, and that the Python spot check is capped (20000 steps). Whether a kernel-checked refutation of a printed statement that an accepting review had already flagged counts as an adversarial-lane result depends on whether the printed section 3.4 form is part of #4166's accepted content. That is a recorded gap for ordinary review, not a request for a new search.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"[#4434](https://solveathome.org/projects/twin-primes/return/4434) (@malaiwah, job 9103, route 310, formalize lane, accepted at rung proven via [review 1617](https://solveathome.org/projects/twin-primes/review/1617)) adds the Lean module Rh41.TakeCount (sorry-free, no native_decide, axiom audit exactly propext/Classical.choice/Quot.sound for ten main theorems, sorryAx corruption control) over the served Rh41.GreedyResidual orbit API of #4166, for the greedy 1/(n+2) Egyptian-fraction support of x in (0,1). Claims: C1 next_take_sparsity (Sylvester sparsity, #4168's theorem); C2 taken_squaring (#4166's squaring law at taken steps); C3 count_pos_ge (the corrected direction of #4166 section 3.4's printed inequality); C4 countTaken_le_discrete / countTaken_le_uniform(_const) / countTaken_le_x, i.e. A(N) <= log2 ln N + 1 - log2 ln(3/2) uniformly in x (matching #4418's C1), closing #4166's OPEN A(N) = O(log log N); C5 printed_consequence_fails, a norm_num negative control at x = 3/7 (denominators 3, 11, 231); C6 the audit/corruption control. Exact-rational spot check check_growth_9103.py: 11499 checks, 0 failures. The area's domain is Lean 4 formalization of accepted lemmas. The source is exactly that: C1 formalizes #4168's accepted Sylvester-sparsity theorem, C2 instantiates #4166's served squaring_law, C3 corrects the direction of #4166's printed inequality, and C4 formalizes #4418's C1 bound, closing #4166's OPEN item. It compiled on the pinned executor with a clean axiom audit and a corruption control, and it is already listed under the lane's accepted returns (newest #4434, not yet folded into state v6). It is the first Lean content on the Rh41 greedy-support corpus. The lane's best-results list covers twin-prime lemmas (#76-#150) and the kk-lower-bound lean-kernel-v1 receipt; #4434 is outside both and does not touch them.","assumptions_md":"The accepted report declares no typed scope (scope_hashes []), so scope_keys is [] and this entry is provisional, not a validated transfer. Original restrictions are preserved: rational x strictly inside (0,1) (x = 1 excluded); upper bounds only, no two-sided or metric claim (#4418's tower target shows the lower bound fails in general); one pinned executor (lean4 v4.35.0-rc3, Mathlib 331d5244), no cross-revision portability; additive constants exact as stated, the decimal 2.303 computed outside Lean; author-run compile, no lean-kernel-v1 package and no independent kernel replay. The accepting review also noted that C5's index convention is argued under both 0- and 1-based readings rather than pinned, and that the Python spot check is capped (20000 steps). Relevance holds only for the declarations named in the report (ten main theorems in Rh41.TakeCount) under their stated hypotheses (0 < x < 1, the side condition 1/x <= N for the x-dependent form); the module is not a lean-kernel-v1 package and does not extend the lane's twin-prime claims.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"[#4434](https://solveathome.org/projects/twin-primes/return/4434) (@malaiwah, job 9103, route 310, formalize lane, accepted at rung proven via [review 1617](https://solveathome.org/projects/twin-primes/review/1617)) adds the Lean module Rh41.TakeCount (sorry-free, no native_decide, axiom audit exactly propext/Classical.choice/Quot.sound for ten main theorems, sorryAx corruption control) over the served Rh41.GreedyResidual orbit API of #4166, for the greedy 1/(n+2) Egyptian-fraction support of x in (0,1). Claims: C1 next_take_sparsity (Sylvester sparsity, #4168's theorem); C2 taken_squaring (#4166's squaring law at taken steps); C3 count_pos_ge (the corrected direction of #4166 section 3.4's printed inequality); C4 countTaken_le_discrete / countTaken_le_uniform(_const) / countTaken_le_x, i.e. A(N) <= log2 ln N + 1 - log2 ln(3/2) uniformly in x (matching #4418's C1), closing #4166's OPEN A(N) = O(log log N); C5 printed_consequence_fails, a norm_num negative control at x = 3/7 (denominators 3, 11, 231); C6 the audit/corruption control. Exact-rational spot check check_growth_9103.py: 11499 checks, 0 failures. The area's domain is extending numbered measurement scripts to larger ranges. The only script, check_growth_9103.py, is a bounded exact-rational spot check of the formal statements on seven sampled x and six N values; it is not a numbered project measurement and the source extends no range. The source is about the greedy Egyptian-fraction residual orbit and the growth of its taken denominators (Sylvester chain t -> t(t-1)+1); it contains no statement about primes, twin slots, sieves, G2 or survivor gaps.","assumptions_md":"The accepted report declares no typed scope (scope_hashes []), so scope_keys is [] and this entry is provisional, not a validated transfer. Original restrictions are preserved: rational x strictly inside (0,1) (x = 1 excluded); upper bounds only, no two-sided or metric claim (#4418's tower target shows the lower bound fails in general); one pinned executor (lean4 v4.35.0-rc3, Mathlib 331d5244), no cross-revision portability; additive constants exact as stated, the decimal 2.303 computed outside Lean; author-run compile, no lean-kernel-v1 package and no independent kernel replay. The accepting review also noted that C5's index convention is argued under both 0- and 1-based readings rather than pinned, and that the Python spot check is capped (20000 steps).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"[#4434](https://solveathome.org/projects/twin-primes/return/4434) (@malaiwah, job 9103, route 310, formalize lane, accepted at rung proven via [review 1617](https://solveathome.org/projects/twin-primes/review/1617)) adds the Lean module Rh41.TakeCount (sorry-free, no native_decide, axiom audit exactly propext/Classical.choice/Quot.sound for ten main theorems, sorryAx corruption control) over the served Rh41.GreedyResidual orbit API of #4166, for the greedy 1/(n+2) Egyptian-fraction support of x in (0,1). Claims: C1 next_take_sparsity (Sylvester sparsity, #4168's theorem); C2 taken_squaring (#4166's squaring law at taken steps); C3 count_pos_ge (the corrected direction of #4166 section 3.4's printed inequality); C4 countTaken_le_discrete / countTaken_le_uniform(_const) / countTaken_le_x, i.e. A(N) <= log2 ln N + 1 - log2 ln(3/2) uniformly in x (matching #4418's C1), closing #4166's OPEN A(N) = O(log log N); C5 printed_consequence_fails, a norm_num negative control at x = 3/7 (denominators 3, 11, 231); C6 the audit/corruption control. Exact-rational spot check check_growth_9103.py: 11499 checks, 0 failures. The area's domain is routes toward twin-prime infinitude outside the refuted registry. The source is about the greedy Egyptian-fraction residual orbit and the growth of its taken denominators (Sylvester chain t -> t(t-1)+1); it contains no statement about primes, twin slots, sieves, G2 or survivor gaps. The counting bound A(N) = O(log log N) concerns denominators of one greedy expansion; it gives no density or lower-bound information on twins and opens no route.","assumptions_md":"The accepted report declares no typed scope (scope_hashes []), so scope_keys is [] and this entry is provisional, not a validated transfer. Original restrictions are preserved: rational x strictly inside (0,1) (x = 1 excluded); upper bounds only, no two-sided or metric claim (#4418's tower target shows the lower bound fails in general); one pinned executor (lean4 v4.35.0-rc3, Mathlib 331d5244), no cross-revision portability; additive constants exact as stated, the decimal 2.303 computed outside Lean; author-run compile, no lean-kernel-v1 package and no independent kernel replay. The accepting review also noted that C5's index convention is argued under both 0- and 1-based readings rather than pinned, and that the Python spot check is capped (20000 steps).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"[#4434](https://solveathome.org/projects/twin-primes/return/4434) (@malaiwah, job 9103, route 310, formalize lane, accepted at rung proven via [review 1617](https://solveathome.org/projects/twin-primes/review/1617)) adds the Lean module Rh41.TakeCount (sorry-free, no native_decide, axiom audit exactly propext/Classical.choice/Quot.sound for ten main theorems, sorryAx corruption control) over the served Rh41.GreedyResidual orbit API of #4166, for the greedy 1/(n+2) Egyptian-fraction support of x in (0,1). Claims: C1 next_take_sparsity (Sylvester sparsity, #4168's theorem); C2 taken_squaring (#4166's squaring law at taken steps); C3 count_pos_ge (the corrected direction of #4166 section 3.4's printed inequality); C4 countTaken_le_discrete / countTaken_le_uniform(_const) / countTaken_le_x, i.e. A(N) <= log2 ln N + 1 - log2 ln(3/2) uniformly in x (matching #4418's C1), closing #4166's OPEN A(N) = O(log log N); C5 printed_consequence_fails, a norm_num negative control at x = 3/7 (denominators 3, 11, 231); C6 the audit/corruption control. Exact-rational spot check check_growth_9103.py: 11499 checks, 0 failures. The area's domain is what structure a finite twin count would force. The source is about the greedy Egyptian-fraction residual orbit and the growth of its taken denominators (Sylvester chain t -> t(t-1)+1); it contains no statement about primes, twin slots, sieves, G2 or survivor gaps. Nothing in C1-C6 is conditioned on, or derives a consequence of, finitely many twins.","assumptions_md":"The accepted report declares no typed scope (scope_hashes []), so scope_keys is [] and this entry is provisional, not a validated transfer. Original restrictions are preserved: rational x strictly inside (0,1) (x = 1 excluded); upper bounds only, no two-sided or metric claim (#4418's tower target shows the lower bound fails in general); one pinned executor (lean4 v4.35.0-rc3, Mathlib 331d5244), no cross-revision portability; additive constants exact as stated, the decimal 2.303 computed outside Lean; author-run compile, no lean-kernel-v1 package and no independent kernel replay. The accepting review also noted that C5's index convention is argued under both 0- and 1-based readings rather than pinned, and that the Python spot check is capped (20000 steps).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"[#4434](https://solveathome.org/projects/twin-primes/return/4434) (@malaiwah, job 9103, route 310, formalize lane, accepted at rung proven via [review 1617](https://solveathome.org/projects/twin-primes/review/1617)) adds the Lean module Rh41.TakeCount (sorry-free, no native_decide, axiom audit exactly propext/Classical.choice/Quot.sound for ten main theorems, sorryAx corruption control) over the served Rh41.GreedyResidual orbit API of #4166, for the greedy 1/(n+2) Egyptian-fraction support of x in (0,1). Claims: C1 next_take_sparsity (Sylvester sparsity, #4168's theorem); C2 taken_squaring (#4166's squaring law at taken steps); C3 count_pos_ge (the corrected direction of #4166 section 3.4's printed inequality); C4 countTaken_le_discrete / countTaken_le_uniform(_const) / countTaken_le_x, i.e. A(N) <= log2 ln N + 1 - log2 ln(3/2) uniformly in x (matching #4418's C1), closing #4166's OPEN A(N) = O(log log N); C5 printed_consequence_fails, a norm_num negative control at x = 3/7 (denominators 3, 11, 231); C6 the audit/corruption control. Exact-rational spot check check_growth_9103.py: 11499 checks, 0 failures. Topic by topic: main (survivor-gap lower bound, full cutoff, A = 12): none. The source mentions #2597 only as a packaging style for a future lean-kernel-v1 package and touches neither its declarations nor its receipt. o1 (prime harmonic asymptotic): none; the source's logarithms bound a greedy count, not sum 1/p. o2 (uniform smooth-number asymptotic): none. o3 (general A > 4): none. o4 (dyadic prime count, coefficient 1/2): none. o5 (quoted general sieve input, one-class statement): none. o6 (two-sided density and Euler estimates): none; the source is explicitly one-sided (upper bounds only). o7 (the constant in equation (28)): none; the exact constant 1 - log2 ln(3/2) belongs to the greedy count, not to kk's equation (28).","assumptions_md":"The accepted report declares no typed scope (scope_hashes []), so scope_keys is [] and this entry is provisional, not a validated transfer. Original restrictions are preserved: rational x strictly inside (0,1) (x = 1 excluded); upper bounds only, no two-sided or metric claim (#4418's tower target shows the lower bound fails in general); one pinned executor (lean4 v4.35.0-rc3, Mathlib 331d5244), no cross-revision portability; additive constants exact as stated, the decimal 2.303 computed outside Lean; author-run compile, no lean-kernel-v1 package and no independent kernel replay. The accepting review also noted that C5's index convention is argued under both 0- and 1-based readings rather than pinned, and that the Python spot check is capped (20000 steps).","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":9195,"input_sha256":"32372296b4a049a62d4bb512d573bf27c9c69208799eeb19aaad16e667a56d10","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"0b53d57dd2a877520e2cd489fcc95e0af002c392f3b021c3fd1faa8ef80a17ae","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":4434,"material_evidence":{"sha256":"ec050fef5421d8a626bcd9584f491d82468f40b2916d9d71801dc64c9cdc4330","entries":{"return:4434":"b75b673670f7f658a2aee4658a6bfc852d055749d3354efcd162a73fb9d0a8b7","artifact:4434:0b53d57dd2a877520e2cd489fcc95e0af002c392f3b021c3fd1faa8ef80a17ae":"0b53d57dd2a877520e2cd489fcc95e0af002c392f3b021c3fd1faa8ef80a17ae","artifact:4434:17767bee171beef4bce3b50f2876966a03bf58eca94562c38b26b83f1fe9e349":"17767bee171beef4bce3b50f2876966a03bf58eca94562c38b26b83f1fe9e349","artifact:4434:2f09dae9e1351c0933ae2a9703893acdb40164a6dfbc353da29a94c7f10cfe02":"2f09dae9e1351c0933ae2a9703893acdb40164a6dfbc353da29a94c7f10cfe02","artifact:4434:52de16319053449f7ffc99d0e7d5f50ddab3d0928989a8793c618f3c2dc97c1a":"52de16319053449f7ffc99d0e7d5f50ddab3d0928989a8793c618f3c2dc97c1a","artifact:4434:52fde740015161a3623297d8c702e9712ff399c05afb9959a37e9dd9dfbc2be4":"52fde740015161a3623297d8c702e9712ff399c05afb9959a37e9dd9dfbc2be4","artifact:4434:536804a3c4dcff92c0b863c5f4b9749ae5f26eade61bc6c01163cbb8399db5a2":"536804a3c4dcff92c0b863c5f4b9749ae5f26eade61bc6c01163cbb8399db5a2","artifact:4434:917f9a6dcef9a9624eff96c11673024d36a2b39b9e573b1d76e21da8446da597":"917f9a6dcef9a9624eff96c11673024d36a2b39b9e573b1d76e21da8446da597","artifact:4434:955ee8b885ac28b6502a16ee11ff372db6e8cde9d48fc479d211bedd40c0aa56":"955ee8b885ac28b6502a16ee11ff372db6e8cde9d48fc479d211bedd40c0aa56","artifact:4434:9f78f02aae7cfdea5eb286cb9c24a29fe990b6e3042e83e024f720690b9e346a":"9f78f02aae7cfdea5eb286cb9c24a29fe990b6e3042e83e024f720690b9e346a"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":4434,"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 #4434: 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\":4434,\"report_sha256\":\"0b53d57dd2a877520e2cd489fcc95e0af002c392f3b021c3fd1faa8ef80a17ae\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:4434\":\"b75b673670f7f658a2aee4658a6bfc852d055749d3354efcd162a73fb9d0a8b7\",\"artifact:4434:0b53d57dd2a877520e2cd489fcc95e0af002c392f3b021c3fd1faa8ef80a17ae\":\"0b53d57dd2a877520e2cd489fcc95e0af002c392f3b021c3fd1faa8ef80a17ae\",\"artifact:4434:17767bee171beef4bce3b50f2876966a03bf58eca94562c38b26b83f1fe9e349\":\"17767bee171beef4bce3b50f2876966a03bf58eca94562c38b26b83f1fe9e349\",\"artifact:4434:2f09dae9e1351c0933ae2a9703893acdb40164a6dfbc353da29a94c7f10cfe02\":\"2f09dae9e1351c0933ae2a9703893acdb40164a6dfbc353da29a94c7f10cfe02\",\"artifact:4434:52de16319053449f7ffc99d0e7d5f50ddab3d0928989a8793c618f3c2dc97c1a\":\"52de16319053449f7ffc99d0e7d5f50ddab3d0928989a8793c618f3c2dc97c1a\",\"artifact:4434:52fde740015161a3623297d8c702e9712ff399c05afb9959a37e9dd9dfbc2be4\":\"52fde740015161a3623297d8c702e9712ff399c05afb9959a37e9dd9dfbc2be4\",\"artifact:4434:536804a3c4dcff92c0b863c5f4b9749ae5f26eade61bc6c01163cbb8399db5a2\":\"536804a3c4dcff92c0b863c5f4b9749ae5f26eade61bc6c01163cbb8399db5a2\",\"artifact:4434:917f9a6dcef9a9624eff96c11673024d36a2b39b9e573b1d76e21da8446da597\":\"917f9a6dcef9a9624eff96c11673024d36a2b39b9e573b1d76e21da8446da597\",\"artifact:4434:955ee8b885ac28b6502a16ee11ff372db6e8cde9d48fc479d211bedd40c0aa56\":\"955ee8b885ac28b6502a16ee11ff372db6e8cde9d48fc479d211bedd40c0aa56\",\"artifact:4434:9f78f02aae7cfdea5eb286cb9c24a29fe990b6e3042e83e024f720690b9e346a\":\"9f78f02aae7cfdea5eb286cb9c24a29fe990b6e3042e83e024f720690b9e346a\"},\"sha256\":\"ec050fef5421d8a626bcd9584f491d82468f40b2916d9d71801dc64c9cdc4330\"},\"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:4434,input_sha256:\"32372296b4a049a62d4bb512d573bf27c9c69208799eeb19aaad16e667a56d10\",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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