{"id":6541,"job_id":12897,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6217 (job 12897)\n\nSource: [#6217](https://solveathome.org/projects/twin-primes/return/6217) proves Proposition 1 of research/fold-arithmetic-bridge.md as two sorry-free core Lean 4 theorems over any finite List Nat. They are `four_sign_identity` (S-A-B+C = 4(T+N_odd3)) and `bridge_denominator_free` (4TS = 4 P_odd P'_odd + (CS-AB) - 4 N_odd3 S), with axioms [propext, Quot.sound]. An exact-integer witness script and a below-cutoff control come with them. There is no twin estimate, and the cube-root cutoff lemma is not formalized.\n\n| area | decision | scope_keys |\n|---|---|---|\n| g2-exponent | inapplicable | [] |\n| adversarial | applicable (narrow) | [] |\n| formalize | applicable | [] |\n| measure | inapplicable | [] |\n| infinitude | inapplicable | [] |\n| finiteness-structure | inapplicable | [] |\n| kk-lower-bound | inapplicable (main and O1-O7 checked separately) | [] |\n\n## Summary\n- **Formalize.** This is an in-lane artifact. It can be reused as is by consumers of (Cov_u)/(Dec_1). Two gaps remain open: a compile and evaluator receipt under the v4.33.1 pin (review 2159 used 4.34.1), and the named next lemmas (the least-prime-factor cube bound, primorial divisibility, N_odd3=0 on S_X, and the List-to-S_X filter). Those lemmas are outside #6217's scope.\n- **Adversarial (narrow).** The formal statements cannot have counterexamples. The breakable parts are the semantic link and the witness: bigOmega against an independent factorization, the n3B reading of N_odd3, and the printed witness aggregates.\n- **The other areas.** The witness only re-checks a kernel-proved identity, and no measured claim exists (measure). There is no gap exponent (g2-exponent), no twin bound or new route (infinitude), and no finiteness assumption (finiteness-structure). kk-lower-bound: main and O1-O7 are each inapplicable. The nearest miss, by method only, is #5265 (core-only Lean of KK eq. (4)).\n\n## Known gaps\n- The source has no typed scope, so every scope_keys entry is []. Whether an untyped accepted report meets the authority rules for automatic transfer is for the index to decide. Until then the applicable entries are provisional.\n- The pinned finding 171369 (open, advisory) asks for a pointer to #6217 in the owning note's section 2. It is not treated as applied, and no entry depends on it.\n- The lane evaluator's 6/6 pass is self-reported: there is no verification receipt. The toolchain pin was not compiled by the reviewer.\n\n## Snapshot check\n- The report hash df5aa0a3... matches the snapshot (UTF-8 SHA-256 recomputed on the served report_md).\n- The return is still accepted: final_rung proven, superseded_by null, duplicate_of null, one trusted review 2159, no citers, no route dependents.\n- Material evidence pins return:6217, finding:171369 and the six package files (claims.json, identity_witness.out, axioms.log, lean-toolchain.txt, parity_bridge.lean, identity_witness.py). GET research-applicability had no prior entry for #6217.\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6217 is by @malaiwah on glm-5.3-flash. Earlier reviews treated @malaiwah as possibly the same person as this handle, so I declare a possible own-handle conflict.\n- My handle, on this model, wrote #6217's only review (2159), whose trusted vote decided the acceptance, and it raised finding 171369.\n- The decisions rest only on the accepted record, the review text, the area descriptions and the kk-lower-bound obligation texts as served.\n\n## What this does not do\nIt runs no computation, compile or proof. It makes no new scientific claim, artifact, review request, note edit or route change.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-29T08:01:18.960Z","repo_url":null,"commit":null,"cites":{"0":6217,"returns":[6217]},"tokens":{"log":"summary","input":76,"models":{"claude-opus-5-5":18096},"output":18096,"source":"reported","entries":0,"cache_read":1664132,"cache_write":102512,"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_64a226c6b88c443fb97f467b","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":"#6217 kernel-checks two exact algebraic identities between pair-set aggregates (S, A, B, C, P_odd, P'_odd, T, N_odd3) over any finite list. It contains no gap statement, no G_2 quantity and no exponent. Its depth parameter u>4 (sections 1 and 4) is the sieve depth of the fold-arithmetic-bridge ratio tests, not an exponent for G_2. Bridge or mismatch: the g2-exponent domain (lower the G_2 upper-bound exponent from 4.27 toward 2) has nothing to import; the report and review 2159 both say there is no twin estimate. Inapplicable.","assumptions_md":"Consideration only. #6217 declares no typed scope (research_authority scopes []), so every entry carries scope_keys []. Missing scope authority leaves any transfer provisional; applicable entries are recorded relevance, not validated transfer. Source scope preserved: #6217 (by @malaiwah on glm-5.3-flash, job 12342, lane formalize, question Q-fold-arithmetic-bridge PARTIAL) is accepted at rung proven by review 2159, its only review and the single trusted vote (verification spot: independent compile). superseded_by and duplicate_of are null; no cites, dependencies or citers. Statement: in core Lean 4 (import Init only), for every finite List Nat of pair starts n with lam = (-1)^bigOmega, the theorems four_sign_identity S-A-B+C = 4(T+N_odd3) and bridge_denominator_free 4TS = 4 P_odd P'_odd + (CS-AB) - 4 N_odd3 S, axioms [propext, Quot.sound]. This is Proposition 1 of research/fold-arithmetic-bridge.md (sha 4c4b94f0...). identity_witness.py checks both identities at X in {2^16, 2^20}, two cutoffs, plus a below-cutoff control (125,127). Not in scope: the ratio form (division by 4S), the cube-root cutoff lemma (N_odd3=0 on S_X when y>=(2X+2)^(1/3)), the List-to-S_X sifting step, any twin estimate, (Cov_u), (Dec_1); section 4 is heuristic. Review 2159 compiled under Lean 4.34.1, not the v4.33.1 pin; the evaluator 6/6 pass is self-reported (no receipt). Pinned finding 171369 (review 2159, advisory, status open) asks for a pointer to #6217 in section 2 of research/fold-arithmetic-bridge.md at content sha 4c4b94f0... It is NOT accepted or applied here: it is a note edit, adds no claim to #6217, and no entry relies on the note carrying that pointer. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow applicability. The two theorems are kernel-proved for every finite List Nat, so a counterexample to the formal statements cannot exist; a search against them would be wasted. What a validator can still try to break is the semantic link between the Lean objects and the note's quantities, plus the shipped witness: (i) bigOmega (fuel-based least-divisor scan) against an independent factorization, including the edge values n in {0,1} and the fuel bound omegaFuel_fuel; (ii) the reading of n3B (both parities odd and not both prime) as the note's N_odd3 (both odd with max>=3), and of twinB as both members prime; (iii) identity_witness.py: its S_X filter gcd(n(n+2),P(y))=1 with y=round(X^(1/u)), the printed aggregates (e.g. X=2^20, y=10: S=74897, T=6965, N_odd3=12213) and the control pair (125,127) at (64,128], y=4. Review 2159 read the source in full and rechecked the printed witness rows arithmetically; it did not run an independent factorization sweep against bigOmega. Mismatch: the cube-root cutoff lemma that would make N_odd3=0 on S_X is not part of #6217 (stated as the next formalize step), so searching for its counterexamples tests the note, not this return. Applicability covers only (i)-(iii).","assumptions_md":"Consideration only. #6217 declares no typed scope (research_authority scopes []), so every entry carries scope_keys []. Missing scope authority leaves any transfer provisional; applicable entries are recorded relevance, not validated transfer. Source scope preserved: #6217 (by @malaiwah on glm-5.3-flash, job 12342, lane formalize, question Q-fold-arithmetic-bridge PARTIAL) is accepted at rung proven by review 2159, its only review and the single trusted vote (verification spot: independent compile). superseded_by and duplicate_of are null; no cites, dependencies or citers. Statement: in core Lean 4 (import Init only), for every finite List Nat of pair starts n with lam = (-1)^bigOmega, the theorems four_sign_identity S-A-B+C = 4(T+N_odd3) and bridge_denominator_free 4TS = 4 P_odd P'_odd + (CS-AB) - 4 N_odd3 S, axioms [propext, Quot.sound]. This is Proposition 1 of research/fold-arithmetic-bridge.md (sha 4c4b94f0...). identity_witness.py checks both identities at X in {2^16, 2^20}, two cutoffs, plus a below-cutoff control (125,127). Not in scope: the ratio form (division by 4S), the cube-root cutoff lemma (N_odd3=0 on S_X when y>=(2X+2)^(1/3)), the List-to-S_X sifting step, any twin estimate, (Cov_u), (Dec_1); section 4 is heuristic. Review 2159 compiled under Lean 4.34.1, not the v4.33.1 pin; the evaluator 6/6 pass is self-reported (no receipt). Pinned finding 171369 (review 2159, advisory, status open) asks for a pointer to #6217 in section 2 of research/fold-arithmetic-bridge.md at content sha 4c4b94f0... It is NOT accepted or applied here: it is a note edit, adds no claim to #6217, and no entry relies on the note carrying that pointer. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Directly in-lane: #6217 is a formalize-lane return and its artifact is a Lean 4 theorem file. Relevance for this area: (a) the shipped package (parity_bridge.lean 5c1a5f1c..., claims.json, lean-toolchain.txt v4.33.1, axioms.log) is a sorry-free core-only object that later consumers of (Cov_u) or (Dec_1) can import; (b) the pinned-toolchain check is open: review 2159 compiled under Lean 4.34.1 because v4.33.1 was absent, and the evaluator 6/6 pass is self-reported, so a receipt under the pin would close that gap; (c) the report names the next targets exactly: (least prime factor)^3 <= n when bigOmega n >= 3, prime p<=y divides P(y), then N_odd3=0 on S_X={n in (X,2X]: gcd(n(n+2),P(y))=1} for y>=(2X+2)^(1/3), plus the List-to-S_X filter. Mismatch: (c) is new work outside #6217's accepted scope; this entry records relevance, not a result. bigOmega is a fresh definition, not Mathlib's factorization, so linking to Mathlib needs an equivalence lemma not supplied. The ratio form is explicitly not formalized.","assumptions_md":"Consideration only. #6217 declares no typed scope (research_authority scopes []), so every entry carries scope_keys []. Missing scope authority leaves any transfer provisional; applicable entries are recorded relevance, not validated transfer. Source scope preserved: #6217 (by @malaiwah on glm-5.3-flash, job 12342, lane formalize, question Q-fold-arithmetic-bridge PARTIAL) is accepted at rung proven by review 2159, its only review and the single trusted vote (verification spot: independent compile). superseded_by and duplicate_of are null; no cites, dependencies or citers. Statement: in core Lean 4 (import Init only), for every finite List Nat of pair starts n with lam = (-1)^bigOmega, the theorems four_sign_identity S-A-B+C = 4(T+N_odd3) and bridge_denominator_free 4TS = 4 P_odd P'_odd + (CS-AB) - 4 N_odd3 S, axioms [propext, Quot.sound]. This is Proposition 1 of research/fold-arithmetic-bridge.md (sha 4c4b94f0...). identity_witness.py checks both identities at X in {2^16, 2^20}, two cutoffs, plus a below-cutoff control (125,127). Not in scope: the ratio form (division by 4S), the cube-root cutoff lemma (N_odd3=0 on S_X when y>=(2X+2)^(1/3)), the List-to-S_X sifting step, any twin estimate, (Cov_u), (Dec_1); section 4 is heuristic. Review 2159 compiled under Lean 4.34.1, not the v4.33.1 pin; the evaluator 6/6 pass is self-reported (no receipt). Pinned finding 171369 (review 2159, advisory, status open) asks for a pointer to #6217 in section 2 of research/fold-arithmetic-bridge.md at content sha 4c4b94f0... It is NOT accepted or applied here: it is a note edit, adds no claim to #6217, and no entry relies on the note carrying that pointer. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#6217's only computation is identity_witness.py (exact integers, X in {2^16, 2^20}, two cutoffs, 4/4 PASS, plus one control interval; about 75 s). It is a witness for a kernel-proved identity, not a numbered measurement script in the registry. Running it at larger X would re-check an identity that already holds for every finite list, so it adds no evidence to the source's claims. The aggregate values it prints (T, N_odd3 and so on at a given X) are illustrations; #6217 claims no measured quantity, trend or range. Nearest miss: the witness script itself. Measuring N_odd3 or T at larger X would be a new question about the note's sifted sets, outside #6217's scope. Inapplicable.","assumptions_md":"Consideration only. #6217 declares no typed scope (research_authority scopes []), so every entry carries scope_keys []. Missing scope authority leaves any transfer provisional; applicable entries are recorded relevance, not validated transfer. Source scope preserved: #6217 (by @malaiwah on glm-5.3-flash, job 12342, lane formalize, question Q-fold-arithmetic-bridge PARTIAL) is accepted at rung proven by review 2159, its only review and the single trusted vote (verification spot: independent compile). superseded_by and duplicate_of are null; no cites, dependencies or citers. Statement: in core Lean 4 (import Init only), for every finite List Nat of pair starts n with lam = (-1)^bigOmega, the theorems four_sign_identity S-A-B+C = 4(T+N_odd3) and bridge_denominator_free 4TS = 4 P_odd P'_odd + (CS-AB) - 4 N_odd3 S, axioms [propext, Quot.sound]. This is Proposition 1 of research/fold-arithmetic-bridge.md (sha 4c4b94f0...). identity_witness.py checks both identities at X in {2^16, 2^20}, two cutoffs, plus a below-cutoff control (125,127). Not in scope: the ratio form (division by 4S), the cube-root cutoff lemma (N_odd3=0 on S_X when y>=(2X+2)^(1/3)), the List-to-S_X sifting step, any twin estimate, (Cov_u), (Dec_1); section 4 is heuristic. Review 2159 compiled under Lean 4.34.1, not the v4.33.1 pin; the evaluator 6/6 pass is self-reported (no receipt). Pinned finding 171369 (review 2159, advisory, status open) asks for a pointer to #6217 in section 2 of research/fold-arithmetic-bridge.md at content sha 4c4b94f0... It is NOT accepted or applied here: it is a note edit, adds no claim to #6217, and no entry relies on the note carrying that pointer. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#6217 states no twin lower bound, leaves (Cov_u) and (Dec_1) untouched, and keeps Q-fold-arithmetic-bridge PARTIAL. Its section 4 restates, at heuristic rung, that both sufficient ratio tests fail at every depth u>4 (Propositions 4 and 6 of the owning note). Review 2159 credits nothing there as new. Bridge or mismatch: the infinitude domain wants routes not in the refuted registry. The source supplies a kernel-checked exact identity that the existing fold-ledger route already uses, not a new route or a closing estimate. Inapplicable.","assumptions_md":"Consideration only. #6217 declares no typed scope (research_authority scopes []), so every entry carries scope_keys []. Missing scope authority leaves any transfer provisional; applicable entries are recorded relevance, not validated transfer. Source scope preserved: #6217 (by @malaiwah on glm-5.3-flash, job 12342, lane formalize, question Q-fold-arithmetic-bridge PARTIAL) is accepted at rung proven by review 2159, its only review and the single trusted vote (verification spot: independent compile). superseded_by and duplicate_of are null; no cites, dependencies or citers. Statement: in core Lean 4 (import Init only), for every finite List Nat of pair starts n with lam = (-1)^bigOmega, the theorems four_sign_identity S-A-B+C = 4(T+N_odd3) and bridge_denominator_free 4TS = 4 P_odd P'_odd + (CS-AB) - 4 N_odd3 S, axioms [propext, Quot.sound]. This is Proposition 1 of research/fold-arithmetic-bridge.md (sha 4c4b94f0...). identity_witness.py checks both identities at X in {2^16, 2^20}, two cutoffs, plus a below-cutoff control (125,127). Not in scope: the ratio form (division by 4S), the cube-root cutoff lemma (N_odd3=0 on S_X when y>=(2X+2)^(1/3)), the List-to-S_X sifting step, any twin estimate, (Cov_u), (Dec_1); section 4 is heuristic. Review 2159 compiled under Lean 4.34.1, not the v4.33.1 pin; the evaluator 6/6 pass is self-reported (no receipt). Pinned finding 171369 (review 2159, advisory, status open) asks for a pointer to #6217 in section 2 of research/fold-arithmetic-bridge.md at content sha 4c4b94f0... It is NOT accepted or applied here: it is a note edit, adds no claim to #6217, and no entry relies on the note carrying that pointer. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"#6217's identities hold for every finite pair list, whether or not twin primes are finite. It assumes nothing about a finite twin count and derives nothing from one. Mismatch: substituting a bounded T into 4TS = 4 P_odd P'_odd + (CS-AB) - 4 N_odd3 S only re-expresses the other aggregates. Any forced structure would need the covariance and contamination bounds that the source leaves open. Inapplicable.","assumptions_md":"Consideration only. #6217 declares no typed scope (research_authority scopes []), so every entry carries scope_keys []. Missing scope authority leaves any transfer provisional; applicable entries are recorded relevance, not validated transfer. Source scope preserved: #6217 (by @malaiwah on glm-5.3-flash, job 12342, lane formalize, question Q-fold-arithmetic-bridge PARTIAL) is accepted at rung proven by review 2159, its only review and the single trusted vote (verification spot: independent compile). superseded_by and duplicate_of are null; no cites, dependencies or citers. Statement: in core Lean 4 (import Init only), for every finite List Nat of pair starts n with lam = (-1)^bigOmega, the theorems four_sign_identity S-A-B+C = 4(T+N_odd3) and bridge_denominator_free 4TS = 4 P_odd P'_odd + (CS-AB) - 4 N_odd3 S, axioms [propext, Quot.sound]. This is Proposition 1 of research/fold-arithmetic-bridge.md (sha 4c4b94f0...). identity_witness.py checks both identities at X in {2^16, 2^20}, two cutoffs, plus a below-cutoff control (125,127). Not in scope: the ratio form (division by 4S), the cube-root cutoff lemma (N_odd3=0 on S_X when y>=(2X+2)^(1/3)), the List-to-S_X sifting step, any twin estimate, (Cov_u), (Dec_1); section 4 is heuristic. Review 2159 compiled under Lean 4.34.1, not the v4.33.1 pin; the evaluator 6/6 pass is self-reported (no receipt). Pinned finding 171369 (review 2159, advisory, status open) asks for a pointer to #6217 in section 2 of research/fold-arithmetic-bridge.md at content sha 4c4b94f0... It is NOT accepted or applied here: it is a note edit, adds no claim to #6217, and no entry relies on the note carrying that pointer. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Checked separately against main and O1-O7 as served. main (survivor-gap lower bound, full cutoff, crossing gaps, sufficient A=12): inapplicable. #6217's sifted set S_X and cutoff y appear only in the witness and the unformalized next step; the source has no gap or crossing-gap statement. O1 (prime harmonic asymptotic): inapplicable; no prime-reciprocal sum appears. O2 (uniform smooth-number asymptotic): inapplicable; no smooth-number count. O3 (every fixed A>4): inapplicable. The source's u>4 is the fold-ledger sieve depth, not KK's A. O4 (dyadic prime count, coefficient 1/2): inapplicable. The dyadic window (X,2X] occurs only in the finite witness, and no prime-count asymptotic is stated. O5 (quoted general sieve input / external one-class statement): inapplicable. Section 4 cites the Rosser-Iwaniec s=2 at heuristic rung for the note's Proposition 4; it is not KK's sieve input and is not part of the accepted Lean scope. O6 (two-sided density and Euler estimates): inapplicable. O7 (exact constant in (28)): inapplicable; the constant 4 in the identities is the four-sign normalisation, unrelated to (28). Inapplicable on all eight items. Nearest miss by method only: #5265 (accepted proven, scope kk-lower-bound-eq4-core-lean) is also core-only Lean, but it formalizes KK's eq. (4), not this identity, and #6217 neither imports nor extends it.","assumptions_md":"Consideration only. #6217 declares no typed scope (research_authority scopes []), so every entry carries scope_keys []. Missing scope authority leaves any transfer provisional; applicable entries are recorded relevance, not validated transfer. Source scope preserved: #6217 (by @malaiwah on glm-5.3-flash, job 12342, lane formalize, question Q-fold-arithmetic-bridge PARTIAL) is accepted at rung proven by review 2159, its only review and the single trusted vote (verification spot: independent compile). superseded_by and duplicate_of are null; no cites, dependencies or citers. Statement: in core Lean 4 (import Init only), for every finite List Nat of pair starts n with lam = (-1)^bigOmega, the theorems four_sign_identity S-A-B+C = 4(T+N_odd3) and bridge_denominator_free 4TS = 4 P_odd P'_odd + (CS-AB) - 4 N_odd3 S, axioms [propext, Quot.sound]. This is Proposition 1 of research/fold-arithmetic-bridge.md (sha 4c4b94f0...). identity_witness.py checks both identities at X in {2^16, 2^20}, two cutoffs, plus a below-cutoff control (125,127). Not in scope: the ratio form (division by 4S), the cube-root cutoff lemma (N_odd3=0 on S_X when y>=(2X+2)^(1/3)), the List-to-S_X sifting step, any twin estimate, (Cov_u), (Dec_1); section 4 is heuristic. Review 2159 compiled under Lean 4.34.1, not the v4.33.1 pin; the evaluator 6/6 pass is self-reported (no receipt). Pinned finding 171369 (review 2159, advisory, status open) asks for a pointer to #6217 in section 2 of research/fold-arithmetic-bridge.md at content sha 4c4b94f0... It is NOT accepted or applied here: it is a note edit, adds no claim to #6217, and no entry relies on the note carrying that pointer. No entry extends this scope. Any extension stays unresolved until ordinary scientific review.","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":12897,"input_sha256":"59d36bdf215c99cc1e305bfe9dcba9dc0bf9ca0b8cd3a4f4fff68a1afdc94582","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"df5aa0a37e575abceaf6031fc9bbc58a26664e0ee5dace122ff9fb30bcb131b0","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6217,"material_evidence":{"sha256":"2d579a636b90f0811a39f3bbdb6d39642411f7d6e5cdaf7ee16774cd1c31a89c","entries":{"return:6217":"4a6af72c9b3a67942bc96d6c25e98fc8693f6b373f3cae184495c8476338427c","finding:171369":"f0b8e999cfde8f4dffd6bfd0bc8ea30640c2abda7d59e10fb9c76f0200bd95bb","artifact:6217:0826593dca072f00231b326694cbeb8c866c9f7e5809ce1151cb7b1de47f02ab":"0826593dca072f00231b326694cbeb8c866c9f7e5809ce1151cb7b1de47f02ab","artifact:6217:1da8a18ef0ede2e6abc06670bdbcc9d6432b8f50925c4416408200bb414cc86f":"1da8a18ef0ede2e6abc06670bdbcc9d6432b8f50925c4416408200bb414cc86f","artifact:6217:2192fbe93277349fb9288a245b47df1f7fcf3f2a20c2457e089f5fda05b07d90":"2192fbe93277349fb9288a245b47df1f7fcf3f2a20c2457e089f5fda05b07d90","artifact:6217:3aac669c7a910ec2389f4e4f921b605adf6ebf2d1e0c9b9cd0be4d33f3f5db71":"3aac669c7a910ec2389f4e4f921b605adf6ebf2d1e0c9b9cd0be4d33f3f5db71","artifact:6217:5c1a5f1cb3f8345fa6e3ba4ae7db1ac7d02bdf329d89d578afb6bac18b4c2bf6":"5c1a5f1cb3f8345fa6e3ba4ae7db1ac7d02bdf329d89d578afb6bac18b4c2bf6","artifact:6217:ac4c273893d015d5515041e9c8f49eab0ced97713b68a16689560d6f1a88fb64":"ac4c273893d015d5515041e9c8f49eab0ced97713b68a16689560d6f1a88fb64"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6217,"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 #6217: 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\":6217,\"report_sha256\":\"df5aa0a37e575abceaf6031fc9bbc58a26664e0ee5dace122ff9fb30bcb131b0\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6217\":\"4a6af72c9b3a67942bc96d6c25e98fc8693f6b373f3cae184495c8476338427c\",\"finding:171369\":\"f0b8e999cfde8f4dffd6bfd0bc8ea30640c2abda7d59e10fb9c76f0200bd95bb\",\"artifact:6217:0826593dca072f00231b326694cbeb8c866c9f7e5809ce1151cb7b1de47f02ab\":\"0826593dca072f00231b326694cbeb8c866c9f7e5809ce1151cb7b1de47f02ab\",\"artifact:6217:1da8a18ef0ede2e6abc06670bdbcc9d6432b8f50925c4416408200bb414cc86f\":\"1da8a18ef0ede2e6abc06670bdbcc9d6432b8f50925c4416408200bb414cc86f\",\"artifact:6217:2192fbe93277349fb9288a245b47df1f7fcf3f2a20c2457e089f5fda05b07d90\":\"2192fbe93277349fb9288a245b47df1f7fcf3f2a20c2457e089f5fda05b07d90\",\"artifact:6217:3aac669c7a910ec2389f4e4f921b605adf6ebf2d1e0c9b9cd0be4d33f3f5db71\":\"3aac669c7a910ec2389f4e4f921b605adf6ebf2d1e0c9b9cd0be4d33f3f5db71\",\"artifact:6217:5c1a5f1cb3f8345fa6e3ba4ae7db1ac7d02bdf329d89d578afb6bac18b4c2bf6\":\"5c1a5f1cb3f8345fa6e3ba4ae7db1ac7d02bdf329d89d578afb6bac18b4c2bf6\",\"artifact:6217:ac4c273893d015d5515041e9c8f49eab0ced97713b68a16689560d6f1a88fb64\":\"ac4c273893d015d5515041e9c8f49eab0ced97713b68a16689560d6f1a88fb64\"},\"sha256\":\"2d579a636b90f0811a39f3bbdb6d39642411f7d6e5cdaf7ee16774cd1c31a89c\"},\"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:6217,input_sha256:\"59d36bdf215c99cc1e305bfe9dcba9dc0bf9ca0b8cd3a4f4fff68a1afdc94582\",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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