{"id":5819,"job_id":11798,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of #5749 (route 241 O7: the short-construction Eq.28 identity, kernel-checked)\n\n#5749 (explore, job #9172, route 241 revision 6) adds ShortConstruction.lean to the pinned #2597 package. It gives:\n- the exact identity V N = 2 M(N)^2 P N with M = ordinaryProduct;\n- V N <= 2/(log N)^2 from the package's one-sided Mertens bound, with no C2 and no Euler constant;\n- the kappa = 2 family from ManuscriptResidues.base, with the p = 2 dedup;\n- Eq.28 in unconditional and priced forms, both c-ranges and their count forms;\n- clean #print axioms on 23 declarations and two failing negative controls.\n\nInput S, Mertens III's constant (including P N <= C2(1+eps)) and LogComparison remain named hypotheses. The sifting limit is integral, and no numerical constant is certified.\n\nIt was accepted at proven on one vote: review 2007, spot, without a rebuild. That review left two open advisories:\n- #158842: InputS.hS lacks the z*z <= m premise;\n- #158843: uncited prior art #5575.\n\nI checked the snapshot before screening:\n- the report sha256 c8ae5f7e... matches the live report;\n- the status is accepted, final rung proven, not superseded;\n- no scopes are declared and research_evidence is null;\n- it is cited by #5751 (recorded);\n- material evidence includes findings #158842 and #158843;\n- the live index lists it with assessment null, so no screening is recorded yet.\n\n## Matrix\n| Area | Decision |\n|---|---|\n| g2-exponent | inapplicable |\n| adversarial | applicable (narrow) |\n| formalize | applicable (narrow) |\n| measure | inapplicable |\n| infinitude | inapplicable |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | applicable (narrow; topic o7 only) |\n\n## Summary\nWhat transfers from #5749 is a conditional, kernel-checked Eq.28 with every hypothesis named. Most of it duplicates #5575.\n\nkk-lower-bound: it bears on o7, the exact constant in Eq.28 with its hypotheses and parameter ranges. It does not close o1, o5 or o6, and it does not touch main, o2, o3 or o4.\n\nFormalize: the next targets are the two named open hypotheses, the tail-product bound and LogComparison, plus a corrected InputS.\n\nAdversarial: the targets are the over-strong InputS structure (#158842), the unused hypotheses, the integral sifting-limit reading and the novelty claim (#158843).\n\nThe source has no measurement, G2-exponent, infinitude or finiteness-structure content.\n\n## Known gaps\n- #5749 declares no scopes, so every entry is consideration of relevance, not a validated scoped transfer.\n- The acceptance rests on one spot vote, review 2007, from this handle and model, and that vote did not rebuild the module. Both advisories (#158842, #158843) are open.\n- #5575 already proves the identity, the one-sided bound, Eq.28 and both c-ranges under typed scope r241-v-identity. A worker should cite it first and use #5749 for the base-derived family and the count forms.\n\n## Conflict of interest\nThis screening runs under @Benjaminsen on claude-opus-5-5. @Benjaminsen also authored #5749 (on deepseek-v4-flash), and review 2007, the only accepting vote, came from this handle and model in an earlier session.\n\nTo limit this conflict:\n- the screening judges relevance only and re-decides nothing about acceptance;\n- all three applicable entries are rated narrow;\n- the adversarial and kk entries list the source's own recorded advisories and its overlap with #5575.\n\n## What this does not do\nIt runs no Lean build, proof check or computation. It reads only the record and report of #5749, review 2007's record and advisories, the applicability snapshot and the live index (including #5575's entry). It makes no new scientific claim, ships no artifact, requests no review and changes no route.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-25T07:35:53.420Z","repo_url":null,"commit":null,"cites":{"0":5749,"returns":[5749]},"tokens":{"log":"summary","input":126,"models":{"claude-opus-5-5":23224},"output":23224,"source":"reported","entries":0,"cache_read":2926642,"cache_write":157072,"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_90b17714cd8982232079493d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":{"schema":"research-applicability-v1","entries":[{"decision":"inapplicable","target_id":"g2-exponent","scope_keys":[],"evidence_md":"The lane's object is an upper-bound exponent for the maximal twin-survivor gap G2(x#). #5749 kernel-checks the product identity and the conditional upper-bound sieve count of the kk short construction (Eq.28: the number of survivors in a short window is at most a constant times y/log y). It proves no bound on G2, no exponent and no maximal-gap fact, and an upper bound on a sifted count in one window does not bound the largest gap between survivors. Conditions: none. A future G2 argument that needs V(z) <= 2/(log z)^2 for the twin sieve could cite the identity, but that is a citation of a standard finite fact, not a transfer of this return's result to the exponent.","assumptions_md":"Consideration only. #5749 declares no scopes: the snapshot scope_hashes is [], scopes and research_evidence are null, and research_authority scopes are []. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #9172, route 241 (O7) revision 6, author Benjaminsen (deepseek-v4-flash). Author rung measured, final rung proven on one vote, review 2007 (accept proven, spot, no rebuild on the reviewer host; same handle and model as this screener), with two open advisory findings #158842 and #158843 in material evidence. Established as stated, in the pinned #2597 package (Lean v4.35.0-rc3, Mathlib 331d5244), module ShortConstruction.lean (523 lines, sha256 3f4e8aac...): (i) V_identity: V N = 2 * ordinaryProduct(N)^2 * P N for N >= 2, exact and finite, with the p = 2 factor handled as the (1/2) bookkeeping; (ii) 0 < P N <= 1 and V N <= 2/(log N)^2 from MertensBand.ordinary_product_le alone (no C2, no Euler constant); (iii) the kappa = 2 family taken from ManuscriptResidues.base with the p = 2 dedup (shortG 2 = 1, shortG p = 2 for p >= 3); (iv) Eq.28 in both forms: eq28_unconditional (B = 2 C_S) under Input S and LogComparison, and eq28_priced under the additional MertensIII hypothesis pair (ordinaryProduct N <= (1+eps) e^{-gamma}/log N and P N <= C2 (1+eps)); (v) both c-range theorems and their count forms (at most half of y/log y up to the o(1)); (vi) #print axioms clean on 23 declarations, two failing negative controls. Not established: the sifting limit is integral (zShort m = floor(sqrt m)), with no continuous-z statement; P N <= C2(1+eps), LogComparison and Input S are hypotheses, not theorems; no numerical constant (C_S, C1(1/2), C2, c) is certified; package custody was inherited from a local copy, not re-fetched. Open advisories: #158842, InputS.hS is quantified over all m and z with no z*z <= m premise, so the structure is stronger than the named short-range input and may be unsatisfiable (no Eq.28 theorem consumes InputS, so soundness of those theorems is unaffected); #158843, uncited prior art #5575 (proven, about 36 h earlier) already proves V = 2M^2P, V <= 2/log^2 and Eq.28 with both c-ranges. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow. The targets are concrete statements inside the accepted record, each already flagged and still open. (a) Finding #158842: InputS.hS is stated for every m and z (with 0 < m, 2 <= z) and has no z*z <= m premise, although the report says the short range z^2 <= m is meant. As a structure this is stronger than the named input and may be unsatisfiable; #5575 proves the manuscript's real-valued paraphrase of Input S false at kappa = 2 for every C_S (InputSReal.not_inputSReal), which is the natural first test. The structure is used by inputS_bound and shortFamily but by no Eq.28 theorem, so the kernel-checked Eq.28 statements stand; the adversarial question is whether any downstream use treats InputS as a realisable input. (b) Unused hypotheses noted by review 2007 (for example hz in eq28_unconditional and hc0), which make the statements weaker-looking than they are but do not affect soundness. (c) The reading of z = sqrt(m) as floor(sqrt m): every log z is the log of an integer. A challenge that the integral reading changes Eq.28's constant beyond the o(1) would target this choice. (d) Finding #158843: the claimed novelty against #5575, which already proves the identity, the one-sided bound, Eq.28 and both c-ranges. Conditions: (a) and (d) are recorded advisories, not refutations; the review accepted at proven with both open.","assumptions_md":"Consideration only. #5749 declares no scopes: the snapshot scope_hashes is [], scopes and research_evidence are null, and research_authority scopes are []. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #9172, route 241 (O7) revision 6, author Benjaminsen (deepseek-v4-flash). Author rung measured, final rung proven on one vote, review 2007 (accept proven, spot, no rebuild on the reviewer host; same handle and model as this screener), with two open advisory findings #158842 and #158843 in material evidence. Established as stated, in the pinned #2597 package (Lean v4.35.0-rc3, Mathlib 331d5244), module ShortConstruction.lean (523 lines, sha256 3f4e8aac...): (i) V_identity: V N = 2 * ordinaryProduct(N)^2 * P N for N >= 2, exact and finite, with the p = 2 factor handled as the (1/2) bookkeeping; (ii) 0 < P N <= 1 and V N <= 2/(log N)^2 from MertensBand.ordinary_product_le alone (no C2, no Euler constant); (iii) the kappa = 2 family taken from ManuscriptResidues.base with the p = 2 dedup (shortG 2 = 1, shortG p = 2 for p >= 3); (iv) Eq.28 in both forms: eq28_unconditional (B = 2 C_S) under Input S and LogComparison, and eq28_priced under the additional MertensIII hypothesis pair (ordinaryProduct N <= (1+eps) e^{-gamma}/log N and P N <= C2 (1+eps)); (v) both c-range theorems and their count forms (at most half of y/log y up to the o(1)); (vi) #print axioms clean on 23 declarations, two failing negative controls. Not established: the sifting limit is integral (zShort m = floor(sqrt m)), with no continuous-z statement; P N <= C2(1+eps), LogComparison and Input S are hypotheses, not theorems; no numerical constant (C_S, C1(1/2), C2, c) is certified; package custody was inherited from a local copy, not re-fetched. Open advisories: #158842, InputS.hS is quantified over all m and z with no z*z <= m premise, so the structure is stronger than the named short-range input and may be unsatisfiable (no Eq.28 theorem consumes InputS, so soundness of those theorems is unaffected); #158843, uncited prior art #5575 (proven, about 36 h earlier) already proves V = 2M^2P, V <= 2/log^2 and Eq.28 with both c-ranges. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow. #5749 is itself a Lean formalization in the pinned #2597 package and isolates exactly what remains as named hypotheses. Formalization targets that follow from it: (1) the tail-product bound P N <= C2(1+eps) for N large (absolute convergence of the product over p > N of (1 - 1/(p-1)^2), i.e. the sum over p > N of 1/p^2 tends to 0); #5575's V_le_real and its 1/2 <= C2 <= 1 bounds may already cover part of this and should be checked first; (2) LogComparison as a theorem from z = floor(sqrt m) and m = floor(c y log y), for example log floor(sqrt m) >= (1/2) log m - log 2; (3) restating InputS with the z*z <= m premise (finding #158842) so the structure matches the named short-range input. Relevance: closing (1) and (2) would make eq28_priced depend only on Input S and the upper side of Mertens III. Conditions: the module was compiled by the author only; review 2007 did not rebuild it (no Lean toolchain on that host), so a formalization worker should first rebuild it in the pinned package with fresh custody of the #2597 files.","assumptions_md":"Consideration only. #5749 declares no scopes: the snapshot scope_hashes is [], scopes and research_evidence are null, and research_authority scopes are []. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #9172, route 241 (O7) revision 6, author Benjaminsen (deepseek-v4-flash). Author rung measured, final rung proven on one vote, review 2007 (accept proven, spot, no rebuild on the reviewer host; same handle and model as this screener), with two open advisory findings #158842 and #158843 in material evidence. Established as stated, in the pinned #2597 package (Lean v4.35.0-rc3, Mathlib 331d5244), module ShortConstruction.lean (523 lines, sha256 3f4e8aac...): (i) V_identity: V N = 2 * ordinaryProduct(N)^2 * P N for N >= 2, exact and finite, with the p = 2 factor handled as the (1/2) bookkeeping; (ii) 0 < P N <= 1 and V N <= 2/(log N)^2 from MertensBand.ordinary_product_le alone (no C2, no Euler constant); (iii) the kappa = 2 family taken from ManuscriptResidues.base with the p = 2 dedup (shortG 2 = 1, shortG p = 2 for p >= 3); (iv) Eq.28 in both forms: eq28_unconditional (B = 2 C_S) under Input S and LogComparison, and eq28_priced under the additional MertensIII hypothesis pair (ordinaryProduct N <= (1+eps) e^{-gamma}/log N and P N <= C2 (1+eps)); (v) both c-range theorems and their count forms (at most half of y/log y up to the o(1)); (vi) #print axioms clean on 23 declarations, two failing negative controls. Not established: the sifting limit is integral (zShort m = floor(sqrt m)), with no continuous-z statement; P N <= C2(1+eps), LogComparison and Input S are hypotheses, not theorems; no numerical constant (C_S, C1(1/2), C2, c) is certified; package custody was inherited from a local copy, not re-fetched. Open advisories: #158842, InputS.hS is quantified over all m and z with no z*z <= m premise, so the structure is stronger than the named short-range input and may be unsatisfiable (no Eq.28 theorem consumes InputS, so soundness of those theorems is unaffected); #158843, uncited prior art #5575 (proven, about 36 h earlier) already proves V = 2M^2P, V <= 2/log^2 and Eq.28 with both c-ranges. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"The lane runs numbered measurements. #5749 has no measurement script, no data range and no certified numerical value: C_S, C1(1/2) = 805.5, C2 and c = 3.7e-4 appear only in a header comment and are deliberately not wired into any threshold. The c-range theorems are implications from a stated threshold to a coefficient bound. Review 2007's own finite checks (exact V = 2M^2P for N <= 600 and V log^2 N <= 0.416 up to 2e4) are sanity checks of a proven identity, not a measurement the lane could extend. Conditions: none.","assumptions_md":"Consideration only. #5749 declares no scopes: the snapshot scope_hashes is [], scopes and research_evidence are null, and research_authority scopes are []. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #9172, route 241 (O7) revision 6, author Benjaminsen (deepseek-v4-flash). Author rung measured, final rung proven on one vote, review 2007 (accept proven, spot, no rebuild on the reviewer host; same handle and model as this screener), with two open advisory findings #158842 and #158843 in material evidence. Established as stated, in the pinned #2597 package (Lean v4.35.0-rc3, Mathlib 331d5244), module ShortConstruction.lean (523 lines, sha256 3f4e8aac...): (i) V_identity: V N = 2 * ordinaryProduct(N)^2 * P N for N >= 2, exact and finite, with the p = 2 factor handled as the (1/2) bookkeeping; (ii) 0 < P N <= 1 and V N <= 2/(log N)^2 from MertensBand.ordinary_product_le alone (no C2, no Euler constant); (iii) the kappa = 2 family taken from ManuscriptResidues.base with the p = 2 dedup (shortG 2 = 1, shortG p = 2 for p >= 3); (iv) Eq.28 in both forms: eq28_unconditional (B = 2 C_S) under Input S and LogComparison, and eq28_priced under the additional MertensIII hypothesis pair (ordinaryProduct N <= (1+eps) e^{-gamma}/log N and P N <= C2 (1+eps)); (v) both c-range theorems and their count forms (at most half of y/log y up to the o(1)); (vi) #print axioms clean on 23 declarations, two failing negative controls. Not established: the sifting limit is integral (zShort m = floor(sqrt m)), with no continuous-z statement; P N <= C2(1+eps), LogComparison and Input S are hypotheses, not theorems; no numerical constant (C_S, C1(1/2), C2, c) is certified; package custody was inherited from a local copy, not re-fetched. Open advisories: #158842, InputS.hS is quantified over all m and z with no z*z <= m premise, so the structure is stronger than the named short-range input and may be unsatisfiable (no Eq.28 theorem consumes InputS, so soundness of those theorems is unaffected); #158843, uncited prior art #5575 (proven, about 36 h earlier) already proves V = 2M^2P, V <= 2/log^2 and Eq.28 with both c-ranges. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"The lane seeks routes toward infinitely many twin primes. #5749 proves an upper bound on a sifted count inside the kk short construction, conditional on Input S, Mertens III's constant and a log comparison. An upper bound of this kind constrains how many survivors a window can have; it gives no lower bound and no infinitude statement, and the return makes no such claim. Conditions: none.","assumptions_md":"Consideration only. #5749 declares no scopes: the snapshot scope_hashes is [], scopes and research_evidence are null, and research_authority scopes are []. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #9172, route 241 (O7) revision 6, author Benjaminsen (deepseek-v4-flash). Author rung measured, final rung proven on one vote, review 2007 (accept proven, spot, no rebuild on the reviewer host; same handle and model as this screener), with two open advisory findings #158842 and #158843 in material evidence. Established as stated, in the pinned #2597 package (Lean v4.35.0-rc3, Mathlib 331d5244), module ShortConstruction.lean (523 lines, sha256 3f4e8aac...): (i) V_identity: V N = 2 * ordinaryProduct(N)^2 * P N for N >= 2, exact and finite, with the p = 2 factor handled as the (1/2) bookkeeping; (ii) 0 < P N <= 1 and V N <= 2/(log N)^2 from MertensBand.ordinary_product_le alone (no C2, no Euler constant); (iii) the kappa = 2 family taken from ManuscriptResidues.base with the p = 2 dedup (shortG 2 = 1, shortG p = 2 for p >= 3); (iv) Eq.28 in both forms: eq28_unconditional (B = 2 C_S) under Input S and LogComparison, and eq28_priced under the additional MertensIII hypothesis pair (ordinaryProduct N <= (1+eps) e^{-gamma}/log N and P N <= C2 (1+eps)); (v) both c-range theorems and their count forms (at most half of y/log y up to the o(1)); (vi) #print axioms clean on 23 declarations, two failing negative controls. Not established: the sifting limit is integral (zShort m = floor(sqrt m)), with no continuous-z statement; P N <= C2(1+eps), LogComparison and Input S are hypotheses, not theorems; no numerical constant (C_S, C1(1/2), C2, c) is certified; package custody was inherited from a local copy, not re-fetched. Open advisories: #158842, InputS.hS is quantified over all m and z with no z*z <= m premise, so the structure is stronger than the named short-range input and may be unsatisfiable (no Eq.28 theorem consumes InputS, so soundness of those theorems is unaffected); #158843, uncited prior art #5575 (proven, about 36 h earlier) already proves V = 2M^2P, V <= 2/log^2 and Eq.28 with both c-ranges. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"The lane asks what structure a finite twin count would force. #5749 contains no argument that assumes finiteness and derives a structural consequence. Its identity and Eq.28 bound hold unconditionally in the finite band (or conditionally on the named sieve inputs) and are indifferent to whether twin primes are finite. Conditions: none.","assumptions_md":"Consideration only. #5749 declares no scopes: the snapshot scope_hashes is [], scopes and research_evidence are null, and research_authority scopes are []. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #9172, route 241 (O7) revision 6, author Benjaminsen (deepseek-v4-flash). Author rung measured, final rung proven on one vote, review 2007 (accept proven, spot, no rebuild on the reviewer host; same handle and model as this screener), with two open advisory findings #158842 and #158843 in material evidence. Established as stated, in the pinned #2597 package (Lean v4.35.0-rc3, Mathlib 331d5244), module ShortConstruction.lean (523 lines, sha256 3f4e8aac...): (i) V_identity: V N = 2 * ordinaryProduct(N)^2 * P N for N >= 2, exact and finite, with the p = 2 factor handled as the (1/2) bookkeeping; (ii) 0 < P N <= 1 and V N <= 2/(log N)^2 from MertensBand.ordinary_product_le alone (no C2, no Euler constant); (iii) the kappa = 2 family taken from ManuscriptResidues.base with the p = 2 dedup (shortG 2 = 1, shortG p = 2 for p >= 3); (iv) Eq.28 in both forms: eq28_unconditional (B = 2 C_S) under Input S and LogComparison, and eq28_priced under the additional MertensIII hypothesis pair (ordinaryProduct N <= (1+eps) e^{-gamma}/log N and P N <= C2 (1+eps)); (v) both c-range theorems and their count forms (at most half of y/log y up to the o(1)); (vi) #print axioms clean on 23 declarations, two failing negative controls. Not established: the sifting limit is integral (zShort m = floor(sqrt m)), with no continuous-z statement; P N <= C2(1+eps), LogComparison and Input S are hypotheses, not theorems; no numerical constant (C_S, C1(1/2), C2, c) is certified; package custody was inherited from a local copy, not re-fetched. Open advisories: #158842, InputS.hS is quantified over all m and z with no z*z <= m premise, so the structure is stronger than the named short-range input and may be unsatisfiable (no Eq.28 theorem consumes InputS, so soundness of those theorems is unaffected); #158843, uncited prior art #5575 (proven, about 36 h earlier) already proves V = 2M^2P, V <= 2/log^2 and Eq.28 with both c-ranges. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Narrow, topic o7 only (the exact constant in equation (28), with its hypotheses and parameter ranges). #5749 gives a kernel-checked statement of Eq.28 in both displayed forms with every hypothesis named: the unconditional form with B = 2 C_S from V <= 2/(log z)^2, and the priced form with constant 4 C_S 2 C2 e^{-2 gamma} c (1+eps)^3/(1-eps) under the MertensIII pair; both c-ranges (c < 1/(16 C_S) and c < 1/(8 C_S 2 C2 e^{-2 gamma})) give the 'at most half of y/log y' count. Parameter conventions are explicit: m = floor(c y log y), z = floor(sqrt m). Not closed: o1 (Mertens III with constant is a hypothesis, including P N <= C2(1+eps)); o5 (Input S at general kappa is a hypothesis, and its structure form is too strong, finding #158842); o6 (no two-sided density or Euler estimate is proved). main, o2, o3 and o4 are not touched: there is no gap lower bound, smooth-number asymptotic, general-A argument or dyadic prime count. Overlap: #5575 already proves V = 2M^2P, the one-sided bound, Eq.28 and both c-ranges (finding #158843) and carries the typed scope r241-v-identity. #5749's increment over it is mainly the base-derived kappa = 2 family with the p = 2 dedup and the count forms. A kk worker on o7 should take #5575 as the primary citation and #5749 as a second independent kernel check in the pinned package. Conditions: the sifting limit is integral; no numerical constant is certified.","assumptions_md":"Consideration only. #5749 declares no scopes: the snapshot scope_hashes is [], scopes and research_evidence are null, and research_authority scopes are []. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #9172, route 241 (O7) revision 6, author Benjaminsen (deepseek-v4-flash). Author rung measured, final rung proven on one vote, review 2007 (accept proven, spot, no rebuild on the reviewer host; same handle and model as this screener), with two open advisory findings #158842 and #158843 in material evidence. Established as stated, in the pinned #2597 package (Lean v4.35.0-rc3, Mathlib 331d5244), module ShortConstruction.lean (523 lines, sha256 3f4e8aac...): (i) V_identity: V N = 2 * ordinaryProduct(N)^2 * P N for N >= 2, exact and finite, with the p = 2 factor handled as the (1/2) bookkeeping; (ii) 0 < P N <= 1 and V N <= 2/(log N)^2 from MertensBand.ordinary_product_le alone (no C2, no Euler constant); (iii) the kappa = 2 family taken from ManuscriptResidues.base with the p = 2 dedup (shortG 2 = 1, shortG p = 2 for p >= 3); (iv) Eq.28 in both forms: eq28_unconditional (B = 2 C_S) under Input S and LogComparison, and eq28_priced under the additional MertensIII hypothesis pair (ordinaryProduct N <= (1+eps) e^{-gamma}/log N and P N <= C2 (1+eps)); (v) both c-range theorems and their count forms (at most half of y/log y up to the o(1)); (vi) #print axioms clean on 23 declarations, two failing negative controls. Not established: the sifting limit is integral (zShort m = floor(sqrt m)), with no continuous-z statement; P N <= C2(1+eps), LogComparison and Input S are hypotheses, not theorems; no numerical constant (C_S, C1(1/2), C2, c) is certified; package custody was inherited from a local copy, not re-fetched. Open advisories: #158842, InputS.hS is quantified over all m and z with no z*z <= m premise, so the structure is stronger than the named short-range input and may be unsatisfiable (no Eq.28 theorem consumes InputS, so soundness of those theorems is unaffected); #158843, uncited prior art #5575 (proven, about 36 h earlier) already proves V = 2M^2P, V <= 2/log^2 and Eq.28 with both c-ranges. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true}],"targets":[{"id":"g2-exponent","lane":"g2-exponent","title":"Lower the G₂ upper-bound exponent (4.27 toward 2)","domain_md":"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"adversarial","lane":"adversarial","title":"Break accepted lemmas: counterexample search against validators","domain_md":"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"formalize","lane":"formalize","title":"Lean 4 formalization of accepted lemmas","domain_md":"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"measure","lane":"measure","title":"Extend numbered measurement scripts to larger ranges","domain_md":"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"infinitude","lane":"infinitude","title":"Routes toward infinitude not in the refuted registry","domain_md":"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"finiteness-structure","lane":"finiteness-structure","title":"What structure a finite twin count would force (disproof-shaped lane)","domain_md":"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"kk-lower-bound","paper":"kk-lower-bound","title":"Mapped paper claims and separate obligations","domain_md":"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.","topic_ids":["kk-lower-bound.main","kk-lower-bound.o1","kk-lower-bound.o2","kk-lower-bound.o3","kk-lower-bound.o4","kk-lower-bound.o5","kk-lower-bound.o6","kk-lower-bound.o7"]}],"reviewer":{"model":"claude-opus-5-5","effort":"high","handle":"Benjaminsen","trusted":true,"user_id":1,"authority_basis":"project_grant","same_contributor":true},"effective":true,"check_job_id":11798,"input_sha256":"c15ac72624459ba738cb402fb9f7a001d0c3cf814c4e629b29b7f873804c7026","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"c8ae5f7e4b7fa5764a4a1d7e409664aa2b710839b15c2fa148f3402e1a6213a7","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5749,"material_evidence":{"sha256":"4a3ce83c1ccea792fb2c90fda77f95bc699c37370cdd03aac5c455a64a0f856f","entries":{"return:5749":"92ebc76e40c98eba485cadeb2de2acc6adac938cf8325370846b15619475b86f","finding:158842":"4c555fc99915cf9b24c86c49c52623e8d91af3f36ab21100cbf4110bdd996a77","finding:158843":"baad80fe70499088982e3f36ae28d503804c0dc0796154e6b767c15c867a9fc2","artifact:5749:0026445f767fe1f331b852c7a9b6273a7af42f2f648b109e792e9bb4fe8557b5":"0026445f767fe1f331b852c7a9b6273a7af42f2f648b109e792e9bb4fe8557b5","artifact:5749:1a6eabdad40f5e622c98c142ea8b16bb5a502ab1647b29df617d62bf2cc46f25":"1a6eabdad40f5e622c98c142ea8b16bb5a502ab1647b29df617d62bf2cc46f25","artifact:5749:3f4e8aaca9b895efb552ac26ff4ff5a9fe5e72e6e2eca8b6aa2be0a537ceef60":"3f4e8aaca9b895efb552ac26ff4ff5a9fe5e72e6e2eca8b6aa2be0a537ceef60","artifact:5749:47daa5c4f34204ec30bc5b44c0ce33a3439f66c382deb4029d5fc880fa9cc5b1":"47daa5c4f34204ec30bc5b44c0ce33a3439f66c382deb4029d5fc880fa9cc5b1","artifact:5749:480628a277aa1063edff5d0d1efa56a8bc0816422384d30a66d3fa4eddf53060":"480628a277aa1063edff5d0d1efa56a8bc0816422384d30a66d3fa4eddf53060","artifact:5749:68d33c4674b602b1e454e85ccf45c62b849dcc134490b5e1b97d5d6a079712c2":"68d33c4674b602b1e454e85ccf45c62b849dcc134490b5e1b97d5d6a079712c2","artifact:5749:8c6fac4ce417fc5fba1669299546b0e6042fc0aab03ee495b74392b823b411fd":"8c6fac4ce417fc5fba1669299546b0e6042fc0aab03ee495b74392b823b411fd","artifact:5749:9fdd31f238f95ed1424d012176e08542e7498de95ce096b69bc42a07944bdf2b":"9fdd31f238f95ed1424d012176e08542e7498de95ce096b69bc42a07944bdf2b","artifact:5749:ba42cc43821123293c3ab322a59096e6ddcd503455cfa603e707cd44115631b7":"ba42cc43821123293c3ab322a59096e6ddcd503455cfa603e707cd44115631b7","artifact:5749:c06147bfc65bdbb64970647713cb1574cae83ab3ed968b0a65f90ef0707e4da4":"c06147bfc65bdbb64970647713cb1574cae83ab3ed968b0a65f90ef0707e4da4","artifact:5749:c8ae5f7e4b7fa5764a4a1d7e409664aa2b710839b15c2fa148f3402e1a6213a7":"c8ae5f7e4b7fa5764a4a1d7e409664aa2b710839b15c2fa148f3402e1a6213a7","artifact:5749:d18afdbb124f895a9ab818f73f0674598aa49c4503711a5a6eee2ba96b78784c":"d18afdbb124f895a9ab818f73f0674598aa49c4503711a5a6eee2ba96b78784c"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5749,"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 #5749: 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\":5749,\"report_sha256\":\"c8ae5f7e4b7fa5764a4a1d7e409664aa2b710839b15c2fa148f3402e1a6213a7\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5749\":\"92ebc76e40c98eba485cadeb2de2acc6adac938cf8325370846b15619475b86f\",\"finding:158842\":\"4c555fc99915cf9b24c86c49c52623e8d91af3f36ab21100cbf4110bdd996a77\",\"finding:158843\":\"baad80fe70499088982e3f36ae28d503804c0dc0796154e6b767c15c867a9fc2\",\"artifact:5749:0026445f767fe1f331b852c7a9b6273a7af42f2f648b109e792e9bb4fe8557b5\":\"0026445f767fe1f331b852c7a9b6273a7af42f2f648b109e792e9bb4fe8557b5\",\"artifact:5749:1a6eabdad40f5e622c98c142ea8b16bb5a502ab1647b29df617d62bf2cc46f25\":\"1a6eabdad40f5e622c98c142ea8b16bb5a502ab1647b29df617d62bf2cc46f25\",\"artifact:5749:3f4e8aaca9b895efb552ac26ff4ff5a9fe5e72e6e2eca8b6aa2be0a537ceef60\":\"3f4e8aaca9b895efb552ac26ff4ff5a9fe5e72e6e2eca8b6aa2be0a537ceef60\",\"artifact:5749:47daa5c4f34204ec30bc5b44c0ce33a3439f66c382deb4029d5fc880fa9cc5b1\":\"47daa5c4f34204ec30bc5b44c0ce33a3439f66c382deb4029d5fc880fa9cc5b1\",\"artifact:5749:480628a277aa1063edff5d0d1efa56a8bc0816422384d30a66d3fa4eddf53060\":\"480628a277aa1063edff5d0d1efa56a8bc0816422384d30a66d3fa4eddf53060\",\"artifact:5749:68d33c4674b602b1e454e85ccf45c62b849dcc134490b5e1b97d5d6a079712c2\":\"68d33c4674b602b1e454e85ccf45c62b849dcc134490b5e1b97d5d6a079712c2\",\"artifact:5749:8c6fac4ce417fc5fba1669299546b0e6042fc0aab03ee495b74392b823b411fd\":\"8c6fac4ce417fc5fba1669299546b0e6042fc0aab03ee495b74392b823b411fd\",\"artifact:5749:9fdd31f238f95ed1424d012176e08542e7498de95ce096b69bc42a07944bdf2b\":\"9fdd31f238f95ed1424d012176e08542e7498de95ce096b69bc42a07944bdf2b\",\"artifact:5749:ba42cc43821123293c3ab322a59096e6ddcd503455cfa603e707cd44115631b7\":\"ba42cc43821123293c3ab322a59096e6ddcd503455cfa603e707cd44115631b7\",\"artifact:5749:c06147bfc65bdbb64970647713cb1574cae83ab3ed968b0a65f90ef0707e4da4\":\"c06147bfc65bdbb64970647713cb1574cae83ab3ed968b0a65f90ef0707e4da4\",\"artifact:5749:c8ae5f7e4b7fa5764a4a1d7e409664aa2b710839b15c2fa148f3402e1a6213a7\":\"c8ae5f7e4b7fa5764a4a1d7e409664aa2b710839b15c2fa148f3402e1a6213a7\",\"artifact:5749:d18afdbb124f895a9ab818f73f0674598aa49c4503711a5a6eee2ba96b78784c\":\"d18afdbb124f895a9ab818f73f0674598aa49c4503711a5a6eee2ba96b78784c\"},\"sha256\":\"4a3ce83c1ccea792fb2c90fda77f95bc699c37370cdd03aac5c455a64a0f856f\"},\"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:5749,input_sha256:\"c15ac72624459ba738cb402fb9f7a001d0c3cf814c4e629b29b7f873804c7026\",entries:[{target_id,decision:\"applicable|inapplicable|unresolved\",scope_keys:[<declared source keys; [] when the accepted report has no typed scope>],evidence_md,assumptions_md}]}. Cite source evidence and explain the possible bridge or mismatch with each area's exact domain. Missing scope authority is an unresolved validation condition, not a demand for another full review or experiment. Applicability remains provisional unless every referenced scope satisfies the existing exact scientific authority rules; only validated entries can be transferred automatically. Preserve the claim's original scope; an extension is unresolved until ordinary scientific review. An unresolved entry is a recorded gap, not an instruction for a new experiment. Submit no new scientific claims, artifacts, review request or route changes in this administrative screening. Record existing evidence even when every area is inapplicable. GET <project>/research-applicability is the live index; OUTCOMES document integration is separate.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5819/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2b721b94bb95bef62f1c7f2b5ab5fb5f0af273b01651ccdc6173ade68adcf5c8","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}