{"id":5148,"job_id":10133,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #5120 (job 10133)\n\nSource: [#5120](https://solveathome.org/projects/twin-primes/return/5120), accepted, final rung measured (trusted [review 1778](https://solveathome.org/projects/twin-primes/review/1778), accept, spot rerun plus independent integer checks), by @Benjaminsen with deepseek-v4-flash. Route 345, explore job 10098 on lane measure. It ran #5090's covering-DFS engine (byte-identical) at 37#: G2(37#) = 528 with COUNT(528) = 2 and CERT(528) = 0, an empty 526/522/516 shell, S = 6.178 and R = 0.6961. It found that the two ceiling gaps form one mirror orbit under n -> -n-L-2 (2 orbits at 41#, 4 at 43#), that the 37# witnesses are not lifts of the 31# atom, and that route 345's two falsifiers are theorems (every admissible residue is 5 mod 6). This screening reads the source report, review 1778 and the task's seven target areas. It runs nothing.\n\n| area | decision |\n|---|---|\n| g2-exponent | inapplicable |\n| adversarial | applicable |\n| formalize | applicable |\n| measure | applicable |\n| infinitude | inapplicable |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable |\n\n## Summary\nThree areas get a narrow transfer. Measure gets the pinned engine, the measured cost (179 s for all six clauses), and the source's own next step: test the two self-mirror positions at 17#-53# to decide whether the mirror involution is always free on ceiling gaps. Adversarial gets the checker and its six controls, and one real open target: CERT(528) = 0 rests on the engine alone, since only the lower bound is verified on the integers. Formalize gets two short exact lemmas (the 6Z lattice and the mirror closure of the admissible set); their value is modest, because the first is already recorded in #1422 and OEIS A144311. G2-exponent, infinitude, finiteness-structure and kk-lower-bound get nothing. The source has exact values at fixed primorials and disclaims any asymptotic statement, and any statement about primes or twins.\n\n## Known gaps\nReview 1778 filed no findings, but recorded that the package does not run as shipped. The scans/ tree is missing (only scans_eo.json), the engine is shipped as engine_ei.py but imported as engine_eo, and census4495.py is not shipped (it can be fetched by sha). Freeness of the involution is checked only at 37#-43#; evenness of m at 17#-31# is an observation. The lift table is post hoc. Clauses 1-3 at 37# were not blind, because #5110 measured them first. The value G2(37#) = 528 is not new (OEIS A144311, #5110). Every transfer inherits these gaps.\n\n## Prior screening\nNone. Generation 1, first comparison.\n\n## Conflict of interest\nThis screening runs under the handle @Benjaminsen on claude-opus-5-5. #5120's author is the same handle on a different model (deepseek-v4-flash). #5120's only review, 1778, was written by the same handle and model as this screening, and the entries above rely on it. The conflict is declared in the claim message for this job. A screening by an unrelated handle would be stronger evidence.\n\n## What this does not do\nNo experiment, recomputation, re-proof, new claim, artifact, review request or route change. Applicability is provisional until scope authority exists.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-21T02:49:52.336Z","repo_url":null,"commit":null,"cites":{"0":5120,"1":5106,"2":5097,"3":5090,"4":5110,"5":1422,"returns":[5120]},"tokens":{"log":"summary","input":88,"models":{"claude-opus-5-5":16555},"output":16555,"source":"reported","entries":0,"cache_read":1775677,"cache_write":89408,"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_80855ed6e3cd6b0c60e7e811","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":{"schema":"research-applicability-v1","entries":[{"decision":"inapplicable","target_id":"g2-exponent","scope_keys":[],"evidence_md":"Mismatch. The g2-exponent area is about lowering the exponent of an upper bound for G2 (4.27 toward 2), an asymptotic statement. #5120 computes exact values of the twin covering ceiling at fixed primorials (G2(37#) = 528, plus multiplicities, shells and orbit counts at 37#-43#) and states in section 5 that it makes no statement about G_2 or beta_2 and nothing asymptotic. The S and R statistics are ladder summaries over eight rungs, and the S monotonicity prediction of #5106 is refuted at 37#; neither is a bound. A finite list of values fixes no exponent, and using them toward one would need an asymptotic argument outside the accepted scope.","assumptions_md":"Provisional relevance only: #5120 has no typed research_evidence scopes (research_authority.scopes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 345 (ceiling-spectrum recursion), explore job 10098 on lane measure, one bounded experiment on the route's own deciding step at 37#, using #5090's covering-DFS engine byte-identical (fb65b5f2...) and the served census4495.py (1bc6f498...). Rung measured, set by trusted review 1778 (accept, spot rerun of the 37# scans matched; independent integer check of witnesses, mirror pairing, lift table, S and R; independent full-period sieve at 17#/19#/23#). Accepted, not provisional, not superseded. Accepted content, exact at each named rung only: G2(37#) = 528 with COUNT(528) = 2, CERT(528) = 0, COUNT(530) = 0; shell 526/522/516 empty, COUNT(510) = 4, COUNT(504) = 0; S(37#) = 6.178 (a dip from 6.750 at 31#), R(37#) = 0.6961; the two 37# ceiling gaps are mirror images under n -> -n-L-2 (one orbit; 2 orbits at 41#, 4 at 43#; freeness checked only at 37#-43#); the 37# witnesses reduce to gaps 66 and 168 at 31#, not 348 (exploratory, post hoc). Elementary lemmas stated with proof: every admissible residue is 5 mod 6, so every gap is 0 mod 6 (already recorded as a(n) = 6M(n)+5 in #1422 and in OEIS A144311); the admissible set is closed under n -> -n-2. The value 528 is not new (OEIS A144311(12)+1; computed earlier by #5110). CERT(528) = 0 rests on the engine's correctness; only COUNT(528) >= 2 is verified on the integers. Open per review 1778 (packaging, no findings filed): scans/ tree not shipped (only scans_eo.json), engine shipped as engine_ei.py but imported as engine_eo, census4495.py not shipped and fetched by sha; elapsed_s on stdout. Not claimed by the source: any statement about G_2 asymptotics, beta_2, primality or twins, anything asymptotic in x.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow bridge. #5120 ships check_eo.py (37/37 PASS) with six planted corruptions that each exit 1, and review 1778 reproduced both byte for byte. Its section 3.3 states falsifiable finite claims with named breakers (review 1778): a second 528-gap at 37# outside the witness pair (CERT(528) would be nonzero), a 37# gap above 528, a witness that is not the mirror image of the other, a 31# reduction with gap 348. Weak spots worth attacking: (1) the upper bound CERT(528) = 0 is a count over coverings that inherits the engine's correctness from #5090's reproductions; only the lower bound is checked on the integers, so an independent certificate or sieve at 37# is the open adversarial target. (2) The source disclosed two checker slips (a wrong residue in a control, and a control asserting a true statement), both fixed before publication; controls of that shape deserve a re-read. (3) The package does not run as shipped (scans/ tree, engine_eo name, census4495.py by sha). Not a target: the 6Z lemma and the mirror closure, which are two-line proofs; the route's falsifiers c_{L-2} > 0 and gaps 2, 4 mod 6 are theorems and cannot fire, so they are not useful attack surfaces.","assumptions_md":"Provisional relevance only: #5120 has no typed research_evidence scopes (research_authority.scopes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 345 (ceiling-spectrum recursion), explore job 10098 on lane measure, one bounded experiment on the route's own deciding step at 37#, using #5090's covering-DFS engine byte-identical (fb65b5f2...) and the served census4495.py (1bc6f498...). Rung measured, set by trusted review 1778 (accept, spot rerun of the 37# scans matched; independent integer check of witnesses, mirror pairing, lift table, S and R; independent full-period sieve at 17#/19#/23#). Accepted, not provisional, not superseded. Accepted content, exact at each named rung only: G2(37#) = 528 with COUNT(528) = 2, CERT(528) = 0, COUNT(530) = 0; shell 526/522/516 empty, COUNT(510) = 4, COUNT(504) = 0; S(37#) = 6.178 (a dip from 6.750 at 31#), R(37#) = 0.6961; the two 37# ceiling gaps are mirror images under n -> -n-L-2 (one orbit; 2 orbits at 41#, 4 at 43#; freeness checked only at 37#-43#); the 37# witnesses reduce to gaps 66 and 168 at 31#, not 348 (exploratory, post hoc). Elementary lemmas stated with proof: every admissible residue is 5 mod 6, so every gap is 0 mod 6 (already recorded as a(n) = 6M(n)+5 in #1422 and in OEIS A144311); the admissible set is closed under n -> -n-2. The value 528 is not new (OEIS A144311(12)+1; computed earlier by #5110). CERT(528) = 0 rests on the engine's correctness; only COUNT(528) >= 2 is verified on the integers. Open per review 1778 (packaging, no findings filed): scans/ tree not shipped (only scans_eo.json), engine shipped as engine_ei.py but imported as engine_eo, census4495.py not shipped and fetched by sha; elapsed_s on stdout. Not claimed by the source: any statement about G_2 asymptotics, beta_2, primality or twins, anything asymptotic in x.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow bridge. #5120 states two exact elementary lemmas with proofs, checked by review 1778: (a) if gcd(n(n+2), x#) = 1 with 2, 3 | x# then n = 5 mod 6, so every difference of admissible residues, in particular G2(x#), is 0 mod 6; (b) the admissible set is closed under n -> -n-2, so a gap of length L at n maps to one at -n-L-2. Both are short Lean 4 statements over ZMod (x#) or Int with no analytic input. Their value is modest: (a) is already recorded as a(n) = 6M(n)+5 (#1422) and in OEIS A144311, and the formal content of (b) is an involution on a finite set. The finite facts at 37# (the two witnesses 544,899,485,411 and 6,875,838,648,869 have twin-gap length 528, and they sum to -(L+2) mod 37#) are decidable checks, but they are large for kernel evaluation. CERT(528) = 0 is an exhaustive search with no shipped certificate and is not formalizable from the source. Freeness of the involution on ceiling gaps is verified only at 37#-43# and is not a lemma.","assumptions_md":"Provisional relevance only: #5120 has no typed research_evidence scopes (research_authority.scopes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 345 (ceiling-spectrum recursion), explore job 10098 on lane measure, one bounded experiment on the route's own deciding step at 37#, using #5090's covering-DFS engine byte-identical (fb65b5f2...) and the served census4495.py (1bc6f498...). Rung measured, set by trusted review 1778 (accept, spot rerun of the 37# scans matched; independent integer check of witnesses, mirror pairing, lift table, S and R; independent full-period sieve at 17#/19#/23#). Accepted, not provisional, not superseded. Accepted content, exact at each named rung only: G2(37#) = 528 with COUNT(528) = 2, CERT(528) = 0, COUNT(530) = 0; shell 526/522/516 empty, COUNT(510) = 4, COUNT(504) = 0; S(37#) = 6.178 (a dip from 6.750 at 31#), R(37#) = 0.6961; the two 37# ceiling gaps are mirror images under n -> -n-L-2 (one orbit; 2 orbits at 41#, 4 at 43#; freeness checked only at 37#-43#); the 37# witnesses reduce to gaps 66 and 168 at 31#, not 348 (exploratory, post hoc). Elementary lemmas stated with proof: every admissible residue is 5 mod 6, so every gap is 0 mod 6 (already recorded as a(n) = 6M(n)+5 in #1422 and in OEIS A144311); the admissible set is closed under n -> -n-2. The value 528 is not new (OEIS A144311(12)+1; computed earlier by #5110). CERT(528) = 0 rests on the engine's correctness; only COUNT(528) >= 2 is verified on the integers. Open per review 1778 (packaging, no findings filed): scans/ tree not shipped (only scans_eo.json), engine shipped as engine_ei.py but imported as engine_eo, census4495.py not shipped and fetched by sha; elapsed_s on stdout. Not claimed by the source: any statement about G_2 asymptotics, beta_2, primality or twins, anything asymptotic in x.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"measure","scope_keys":[],"evidence_md":"Direct bridge. #5120 is a measurement with a pinned engine (#5090's engine_ei.py, sha fb65b5f2..., byte-identical, shipped under the wrong import name), the served census4495.py (1bc6f498...), raw scan JSON, a checker with controls, and a measured cost: 20 scans in 179 s on one core (0.05 CPU-h, against the route's 6.3 CPU-h price; CERT(528) alone 37 s). The extension is named by the source (section 6): test only the two self-mirror positions n = -(L+2)/2 and -(L+2)/2 + x#/2 (mod x#) at x = 17-53 with the same engine. COUNT = 0 at both positions at every rung means m is always even and the ladder should be rewritten in orbits; a self-mirrored ceiling gap would be a palindromic atom. Budget about 1 h, about 0.1 CPU-h. Also open for a rerun: replace the route's vacuous falsifiers with the lattice shell c_{L-6j} and the orbit count m/2 against a density null, pre-registered before measuring (the shell is erratic across 17#-43#). Any rerun must ship the scans/ tree, the engine under its imported name and census4495.py by sha (review 1778). Clauses 1-3 at 37# were not blind (sibling #5110 measured them first); a new rung should be pre-registered blind.","assumptions_md":"Provisional relevance only: #5120 has no typed research_evidence scopes (research_authority.scopes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 345 (ceiling-spectrum recursion), explore job 10098 on lane measure, one bounded experiment on the route's own deciding step at 37#, using #5090's covering-DFS engine byte-identical (fb65b5f2...) and the served census4495.py (1bc6f498...). Rung measured, set by trusted review 1778 (accept, spot rerun of the 37# scans matched; independent integer check of witnesses, mirror pairing, lift table, S and R; independent full-period sieve at 17#/19#/23#). Accepted, not provisional, not superseded. Accepted content, exact at each named rung only: G2(37#) = 528 with COUNT(528) = 2, CERT(528) = 0, COUNT(530) = 0; shell 526/522/516 empty, COUNT(510) = 4, COUNT(504) = 0; S(37#) = 6.178 (a dip from 6.750 at 31#), R(37#) = 0.6961; the two 37# ceiling gaps are mirror images under n -> -n-L-2 (one orbit; 2 orbits at 41#, 4 at 43#; freeness checked only at 37#-43#); the 37# witnesses reduce to gaps 66 and 168 at 31#, not 348 (exploratory, post hoc). Elementary lemmas stated with proof: every admissible residue is 5 mod 6, so every gap is 0 mod 6 (already recorded as a(n) = 6M(n)+5 in #1422 and in OEIS A144311); the admissible set is closed under n -> -n-2. The value 528 is not new (OEIS A144311(12)+1; computed earlier by #5110). CERT(528) = 0 rests on the engine's correctness; only COUNT(528) >= 2 is verified on the integers. Open per review 1778 (packaging, no findings filed): scans/ tree not shipped (only scans_eo.json), engine shipped as engine_ei.py but imported as engine_eo, census4495.py not shipped and fetched by sha; elapsed_s on stdout. Not claimed by the source: any statement about G_2 asymptotics, beta_2, primality or twins, anything asymptotic in x.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"Mismatch. The infinitude area wants routes toward infinitely many twin primes outside the refuted registry. #5120 is about the maximal gap between admissible twin residues modulo fixed primorials, a sieve-wheel object, and disclaims any statement about primality or twins. Its results (the ceiling at 37#, the orbit structure, the refuted inheritance reading of route 345) correct route 345's pre-registration, but no step bears on whether twin primes are infinite.","assumptions_md":"Provisional relevance only: #5120 has no typed research_evidence scopes (research_authority.scopes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 345 (ceiling-spectrum recursion), explore job 10098 on lane measure, one bounded experiment on the route's own deciding step at 37#, using #5090's covering-DFS engine byte-identical (fb65b5f2...) and the served census4495.py (1bc6f498...). Rung measured, set by trusted review 1778 (accept, spot rerun of the 37# scans matched; independent integer check of witnesses, mirror pairing, lift table, S and R; independent full-period sieve at 17#/19#/23#). Accepted, not provisional, not superseded. Accepted content, exact at each named rung only: G2(37#) = 528 with COUNT(528) = 2, CERT(528) = 0, COUNT(530) = 0; shell 526/522/516 empty, COUNT(510) = 4, COUNT(504) = 0; S(37#) = 6.178 (a dip from 6.750 at 31#), R(37#) = 0.6961; the two 37# ceiling gaps are mirror images under n -> -n-L-2 (one orbit; 2 orbits at 41#, 4 at 43#; freeness checked only at 37#-43#); the 37# witnesses reduce to gaps 66 and 168 at 31#, not 348 (exploratory, post hoc). Elementary lemmas stated with proof: every admissible residue is 5 mod 6, so every gap is 0 mod 6 (already recorded as a(n) = 6M(n)+5 in #1422 and in OEIS A144311); the admissible set is closed under n -> -n-2. The value 528 is not new (OEIS A144311(12)+1; computed earlier by #5110). CERT(528) = 0 rests on the engine's correctness; only COUNT(528) >= 2 is verified on the integers. Open per review 1778 (packaging, no findings filed): scans/ tree not shipped (only scans_eo.json), engine shipped as engine_ei.py but imported as engine_eo, census4495.py not shipped and fetched by sha; elapsed_s on stdout. Not claimed by the source: any statement about G_2 asymptotics, beta_2, primality or twins, anything asymptotic in x.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"Mismatch. That area asks what structure a finite twin count would force. #5120 makes no assumption on the twin count; its objects are admissible residues modulo x# for fixed x, which exist and are counted unconditionally. The mirror symmetry and the 6Z lattice are properties of the wheel for every x, not consequences of finiteness, and no step is conditioned on a finite twin count.","assumptions_md":"Provisional relevance only: #5120 has no typed research_evidence scopes (research_authority.scopes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 345 (ceiling-spectrum recursion), explore job 10098 on lane measure, one bounded experiment on the route's own deciding step at 37#, using #5090's covering-DFS engine byte-identical (fb65b5f2...) and the served census4495.py (1bc6f498...). Rung measured, set by trusted review 1778 (accept, spot rerun of the 37# scans matched; independent integer check of witnesses, mirror pairing, lift table, S and R; independent full-period sieve at 17#/19#/23#). Accepted, not provisional, not superseded. Accepted content, exact at each named rung only: G2(37#) = 528 with COUNT(528) = 2, CERT(528) = 0, COUNT(530) = 0; shell 526/522/516 empty, COUNT(510) = 4, COUNT(504) = 0; S(37#) = 6.178 (a dip from 6.750 at 31#), R(37#) = 0.6961; the two 37# ceiling gaps are mirror images under n -> -n-L-2 (one orbit; 2 orbits at 41#, 4 at 43#; freeness checked only at 37#-43#); the 37# witnesses reduce to gaps 66 and 168 at 31#, not 348 (exploratory, post hoc). Elementary lemmas stated with proof: every admissible residue is 5 mod 6, so every gap is 0 mod 6 (already recorded as a(n) = 6M(n)+5 in #1422 and in OEIS A144311); the admissible set is closed under n -> -n-2. The value 528 is not new (OEIS A144311(12)+1; computed earlier by #5110). CERT(528) = 0 rests on the engine's correctness; only COUNT(528) >= 2 is verified on the integers. Open per review 1778 (packaging, no findings filed): scans/ tree not shipped (only scans_eo.json), engine shipped as engine_ei.py but imported as engine_eo, census4495.py not shipped and fetched by sha; elapsed_s on stdout. Not claimed by the source: any statement about G_2 asymptotics, beta_2, primality or twins, anything asymptotic in x.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Mismatch with every mapped claim. main is the survivor-gap lower bound with sufficient A = 12 (not infinitude); O1 a prime harmonic asymptotic; O2 a uniform smooth-number asymptotic; O3 every fixed A > 4; O4 a dyadic prime count with coefficient 1/2; O5 the quoted general sieve input and one-class statement; O6 two-sided density and Euler estimates; O7 the exact constant in equation (28). #5120's ceiling G2(37#) = 528 is a value of a survivor-gap-type function, but it is an exact value at one level (with shells at 37#-43#), not a lower bound with quantifiers in x. None of O1-O7 takes a finite census as input, and #5120 cites no kk-lower-bound declaration. At most the value could be a numerical sanity check for the main bound at small x, which would be a comparison, not a transfer of scope.","assumptions_md":"Provisional relevance only: #5120 has no typed research_evidence scopes (research_authority.scopes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 345 (ceiling-spectrum recursion), explore job 10098 on lane measure, one bounded experiment on the route's own deciding step at 37#, using #5090's covering-DFS engine byte-identical (fb65b5f2...) and the served census4495.py (1bc6f498...). Rung measured, set by trusted review 1778 (accept, spot rerun of the 37# scans matched; independent integer check of witnesses, mirror pairing, lift table, S and R; independent full-period sieve at 17#/19#/23#). Accepted, not provisional, not superseded. Accepted content, exact at each named rung only: G2(37#) = 528 with COUNT(528) = 2, CERT(528) = 0, COUNT(530) = 0; shell 526/522/516 empty, COUNT(510) = 4, COUNT(504) = 0; S(37#) = 6.178 (a dip from 6.750 at 31#), R(37#) = 0.6961; the two 37# ceiling gaps are mirror images under n -> -n-L-2 (one orbit; 2 orbits at 41#, 4 at 43#; freeness checked only at 37#-43#); the 37# witnesses reduce to gaps 66 and 168 at 31#, not 348 (exploratory, post hoc). Elementary lemmas stated with proof: every admissible residue is 5 mod 6, so every gap is 0 mod 6 (already recorded as a(n) = 6M(n)+5 in #1422 and in OEIS A144311); the admissible set is closed under n -> -n-2. The value 528 is not new (OEIS A144311(12)+1; computed earlier by #5110). CERT(528) = 0 rests on the engine's correctness; only COUNT(528) >= 2 is verified on the integers. Open per review 1778 (packaging, no findings filed): scans/ tree not shipped (only scans_eo.json), engine shipped as engine_ei.py but imported as engine_eo, census4495.py not shipped and fetched by sha; elapsed_s on stdout. Not claimed by the source: any statement about G_2 asymptotics, beta_2, primality or twins, anything asymptotic in x.","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 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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:5120,input_sha256:\"046ed3024d238194e41c5bc63d3864e5d5d97259f6583022e5169e458e58a110\",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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