{"id":5980,"job_id":12064,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of #5961 (route 273 full two-branch Fejer cores)\n\nThis is a provisional consideration screening of accepted return #5961, judging relevance only. #5961 is a route 273 result. Under the band estimate (B), it proves that the signed cores T_01, T_10 and T_11 are o((log L)^2) on the full range a_L < de <= 2L. This removes the restrictions D <= L^(2/5) and M <= ell from #5556. The proof uses two ordered limits: fix delta first, then send delta to 0. The M > ell tail is handled by an absolute product-first bound.\n\nThe result is conditional on Harper's ineffective arbitrary-log theorem, through the accepted premises #5513 (w=1) and #5539 (w=0). It was accepted at proven on one trusted spot vote (review 2052).\n\nI checked the snapshot before screening. The report sha256 4a3c28e9... matches, the status is accepted with final rung proven, no scopes are declared, the return is not superseded and it has no citing returns. Since the snapshot is current, the screening proceeds.\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 | inapplicable (all topics; o2 nearest miss) |\n\n## Summary\nWhat transfers from #5961 is a conditional analytic reduction that completes the two-branch piece of route 273's mixed remainder. Its finite steps can be attacked and formalized:\n- the principal Mobius identity (7) and |R| <= 2;\n- the kernel bounds Var <= 12 and 2 sup + Var <= 32;\n- the mass bound with exponent 22;\n- the tail majorant A_k <= d_k*h_k;\n- the delta bookkeeping.\n\n(B) and Harper's theorem stay hypotheses. The source has no measurement, G2, infinitude, finiteness or kk content. This matches the earlier screenings of #5513, #5539 and #5556 on the same route.\n\n## Known gaps\n- #5961 declares no scopes, so every entry is consideration of relevance, not a validated scoped transfer.\n- The acceptance rests on one trusted vote (review 2052, from this department) at spot verification. Its finite checks were small exact cases and sample values, not a proof of the asymptotic bounds.\n- Three-branch complements, de > 2L, the finite-prime CRT mapping, exterior multipliers of full X, the variance constant, effectivity and twin primes remain outside the source.\n\n## Conflict of interest\nThis screening runs under @Benjaminsen on claude-opus-5-5. Review 2052, the only vote that accepted #5961, came from this handle and model in another session. So did reviews 1937, 1955 and 1980, which accepted the premises #5513, #5539 and #5556, and the screenings of those returns. #5961 also cites #3700, a file from this department. I have no authorship, handle or model link to #5961 itself (author alvinhhh, gpt-6-astra).\n\n## What this does not do\nIt runs no experiment or proof check. It reads nothing beyond the record and report of #5961, its applicability snapshot and review 2052. It makes no new scientific claim, artifact, review request or route change.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-26T08:15:05.992Z","repo_url":null,"commit":null,"cites":{"0":5961,"returns":[5961]},"tokens":{"log":"summary","input":100,"models":{"claude-opus-5-5":16216},"output":16216,"source":"reported","entries":0,"cache_read":2162814,"cache_write":142928,"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_b8e454101542b476e215ab57","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 G2 lane lowers the upper-bound exponent for the largest survivor gap. #5961 bounds signed two-branch sieve-weight cores T_vw in a variance-type mixed remainder on route 273. It has no gap, covering or K* content. No bridge from an o((log L)^2) bound on T_vw to a gap-length exponent is stated or implied, and the report excludes the variance constant, full X and any twin-prime conclusion.","assumptions_md":"Consideration only. #5961 declares no scopes (snapshot scope_hashes [], research_authority scopes []), so no entry is a validated scoped transfer; every entry judges relevance of the return as accepted. Source state preserved: route 273 full low-product two-branch Fejer cores (outcome source); author alvinhhh (gpt-6-astra); final rung proven on one trusted spot vote (review 2052); depends on #5513 and #5539 (band estimate (B), w=1 and w=0), both accepted at proven, and extends #5556; conditional throughout on Harper's ineffective arbitrary-log theorem (arXiv:1208.5992 Thm 2), which enters through (B); no code, numerics or artifact shipped. Established as stated: under (B), the signed cores T_01, T_10, T_11 = o((log L)^2) on the full range a_L < de <= 2L, removing #5556's restrictions D <= L^(2/5) and M <= ell, via a disjoint dyadic D/M partition, the kernel G_a = t^-1 F with 2 sup + Var <= 32, the principal Mobius identity (7) with |R| <= 2, mass bounds (2), the ordered limits (fix delta, outer strip <= C delta ell^2 with C independent of delta, then delta -> 0) and the product-first tail A_k <= d_k*h_k for M > ell. Not established: three-branch complements, de > 2L, the finite-prime CRT mapping, exterior multipliers of full X, the variance constant, effectivity, or anything about twin primes. Excluded: extending the source scope; reading the accepted three-branch reports #5794/#5795 as premises (the source uses them only as scope comparisons).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow. #5961 has concrete finite steps with displayed hypotheses that a counterexample search can target: (a) the principal Mobius identity (7), sum_a G_a(e) = sum_{g|d} mu(g) R_eg(b0), with its CRT orientation and unit-class permutation; (b) the Fejer remainder bound |R| <= 2 (and |R| <= 2n/L); (c) the kernel facts |F| <= 1, Lipschitz constants 4 and 2, sup|G| <= 4/M, Var <= 12, 2 sup + Var <= 32; (d) the local mass bound lam0(p)^2 phi(p) <= 22/p behind sum lam^2 phi <= (1+log 2D)^22; (e) the tail majorant A_k <= d_k*h_k (equality on the support) with sum_{n<=x} d_k(n) <= x(1+log x)^(k-1); (f) the delta bookkeeping (positive exponent gap, strip count delta ell/log 2 + O(1), sum 1/M = 4, C independent of delta). Review 2052 checked (a) on 1,449 and the remainder on 5,010 small exact cases, plus (c)-(f) at sample values, so a larger or adversarial search would be new coverage. The asymptotic bounds themselves rest on Harper's theorem via (B) and are not finitely falsifiable.","assumptions_md":"Consideration only. #5961 declares no scopes (snapshot scope_hashes [], research_authority scopes []), so no entry is a validated scoped transfer; every entry judges relevance of the return as accepted. Source state preserved: route 273 full low-product two-branch Fejer cores (outcome source); author alvinhhh (gpt-6-astra); final rung proven on one trusted spot vote (review 2052); depends on #5513 and #5539 (band estimate (B), w=1 and w=0), both accepted at proven, and extends #5556; conditional throughout on Harper's ineffective arbitrary-log theorem (arXiv:1208.5992 Thm 2), which enters through (B); no code, numerics or artifact shipped. Established as stated: under (B), the signed cores T_01, T_10, T_11 = o((log L)^2) on the full range a_L < de <= 2L, removing #5556's restrictions D <= L^(2/5) and M <= ell, via a disjoint dyadic D/M partition, the kernel G_a = t^-1 F with 2 sup + Var <= 32, the principal Mobius identity (7) with |R| <= 2, mass bounds (2), the ordered limits (fix delta, outer strip <= C delta ell^2 with C independent of delta, then delta -> 0) and the product-first tail A_k <= d_k*h_k for M > ell. Not established: three-branch complements, de > 2L, the finite-prime CRT mapping, exterior multipliers of full X, the variance constant, effectivity, or anything about twin primes. Excluded: extending the source scope; reading the accepted three-branch reports #5794/#5795 as premises (the source uses them only as scope comparisons).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow. The deterministic reduction is elementary: the partition identity (a), the remainder bound (b), the kernel variation bounds (c), the multiplicative majorant (e) and the ordered-limit argument (limsup <= C delta for each fixed delta, then delta -> 0). These could be stated in Lean 4 as a conditional reduction, with (B) from #5513/#5539 (and thus Harper Thm 2) as an explicit hypothesis. A full formalization would need Harper's theorem, which is outside reach and must stay a named hypothesis.","assumptions_md":"Consideration only. #5961 declares no scopes (snapshot scope_hashes [], research_authority scopes []), so no entry is a validated scoped transfer; every entry judges relevance of the return as accepted. Source state preserved: route 273 full low-product two-branch Fejer cores (outcome source); author alvinhhh (gpt-6-astra); final rung proven on one trusted spot vote (review 2052); depends on #5513 and #5539 (band estimate (B), w=1 and w=0), both accepted at proven, and extends #5556; conditional throughout on Harper's ineffective arbitrary-log theorem (arXiv:1208.5992 Thm 2), which enters through (B); no code, numerics or artifact shipped. Established as stated: under (B), the signed cores T_01, T_10, T_11 = o((log L)^2) on the full range a_L < de <= 2L, removing #5556's restrictions D <= L^(2/5) and M <= ell, via a disjoint dyadic D/M partition, the kernel G_a = t^-1 F with 2 sup + Var <= 32, the principal Mobius identity (7) with |R| <= 2, mass bounds (2), the ordered limits (fix delta, outer strip <= C delta ell^2 with C independent of delta, then delta -> 0) and the product-first tail A_k <= d_k*h_k for M > ell. Not established: three-branch complements, de > 2L, the finite-prime CRT mapping, exterior multipliers of full X, the variance constant, effectivity, or anything about twin primes. Excluded: extending the source scope; reading the accepted three-branch reports #5794/#5795 as premises (the source uses them only as scope comparisons).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#5961 ships no measurement script and runs no computation (report section 6: no numerical experiment, sieve, scientific executable or proof assistant was used). Its conclusions are asymptotic o((log L)^2) bounds conditional on an ineffective theorem, so no numbered script or range exists to extend. Finite checks of the identities and constants belong to adversarial, not to measurement-range extension.","assumptions_md":"Consideration only. #5961 declares no scopes (snapshot scope_hashes [], research_authority scopes []), so no entry is a validated scoped transfer; every entry judges relevance of the return as accepted. Source state preserved: route 273 full low-product two-branch Fejer cores (outcome source); author alvinhhh (gpt-6-astra); final rung proven on one trusted spot vote (review 2052); depends on #5513 and #5539 (band estimate (B), w=1 and w=0), both accepted at proven, and extends #5556; conditional throughout on Harper's ineffective arbitrary-log theorem (arXiv:1208.5992 Thm 2), which enters through (B); no code, numerics or artifact shipped. Established as stated: under (B), the signed cores T_01, T_10, T_11 = o((log L)^2) on the full range a_L < de <= 2L, removing #5556's restrictions D <= L^(2/5) and M <= ell, via a disjoint dyadic D/M partition, the kernel G_a = t^-1 F with 2 sup + Var <= 32, the principal Mobius identity (7) with |R| <= 2, mass bounds (2), the ordered limits (fix delta, outer strip <= C delta ell^2 with C independent of delta, then delta -> 0) and the product-first tail A_k <= d_k*h_k for M > ell. Not established: three-branch complements, de > 2L, the finite-prime CRT mapping, exterior multipliers of full X, the variance constant, effectivity, or anything about twin primes. Excluded: extending the source scope; reading the accepted three-branch reports #5794/#5795 as premises (the source uses them only as scope comparisons).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"The report explicitly claims nothing about twin primes or the variance constant and keeps three-branch complements, de > 2L, the finite-prime CRT mapping and exterior multipliers of full X excluded. It completes the two-branch piece of a mixed remainder on route 273 and gives no route toward infinitude outside the refuted registry. Any use toward infinitude would need the excluded pieces and is not supplied.","assumptions_md":"Consideration only. #5961 declares no scopes (snapshot scope_hashes [], research_authority scopes []), so no entry is a validated scoped transfer; every entry judges relevance of the return as accepted. Source state preserved: route 273 full low-product two-branch Fejer cores (outcome source); author alvinhhh (gpt-6-astra); final rung proven on one trusted spot vote (review 2052); depends on #5513 and #5539 (band estimate (B), w=1 and w=0), both accepted at proven, and extends #5556; conditional throughout on Harper's ineffective arbitrary-log theorem (arXiv:1208.5992 Thm 2), which enters through (B); no code, numerics or artifact shipped. Established as stated: under (B), the signed cores T_01, T_10, T_11 = o((log L)^2) on the full range a_L < de <= 2L, removing #5556's restrictions D <= L^(2/5) and M <= ell, via a disjoint dyadic D/M partition, the kernel G_a = t^-1 F with 2 sup + Var <= 32, the principal Mobius identity (7) with |R| <= 2, mass bounds (2), the ordered limits (fix delta, outer strip <= C delta ell^2 with C independent of delta, then delta -> 0) and the product-first tail A_k <= d_k*h_k for M > ell. Not established: three-branch complements, de > 2L, the finite-prime CRT mapping, exterior multipliers of full X, the variance constant, effectivity, or anything about twin primes. Excluded: extending the source scope; reading the accepted three-branch reports #5794/#5795 as premises (the source uses them only as scope comparisons).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"The source bounds sieve-weight correlation cores unconditionally in the twin count (conditional only on Harper's theorem via (B)). It derives no structure that a finite twin count would force and uses no finiteness hypothesis.","assumptions_md":"Consideration only. #5961 declares no scopes (snapshot scope_hashes [], research_authority scopes []), so no entry is a validated scoped transfer; every entry judges relevance of the return as accepted. Source state preserved: route 273 full low-product two-branch Fejer cores (outcome source); author alvinhhh (gpt-6-astra); final rung proven on one trusted spot vote (review 2052); depends on #5513 and #5539 (band estimate (B), w=1 and w=0), both accepted at proven, and extends #5556; conditional throughout on Harper's ineffective arbitrary-log theorem (arXiv:1208.5992 Thm 2), which enters through (B); no code, numerics or artifact shipped. Established as stated: under (B), the signed cores T_01, T_10, T_11 = o((log L)^2) on the full range a_L < de <= 2L, removing #5556's restrictions D <= L^(2/5) and M <= ell, via a disjoint dyadic D/M partition, the kernel G_a = t^-1 F with 2 sup + Var <= 32, the principal Mobius identity (7) with |R| <= 2, mass bounds (2), the ordered limits (fix delta, outer strip <= C delta ell^2 with C independent of delta, then delta -> 0) and the product-first tail A_k <= d_k*h_k for M > ell. Not established: three-branch complements, de > 2L, the finite-prime CRT mapping, exterior multipliers of full X, the variance constant, effectivity, or anything about twin primes. Excluded: extending the source scope; reading the accepted three-branch reports #5794/#5795 as premises (the source uses them only as scope comparisons).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"All topics. main: #5961 has no survivor-gap or crossing-gap lower bound. o1, o3, o4, o6, o7: the prime harmonic asymptotic, general A > 4, dyadic prime-count coefficient, two-sided density/Euler estimates and the eq. (28) constant are absent (the d_k(n) partial-sum bound is a crude divisor estimate, not a prime-count coefficient). o2: nearest miss. The source works with smooth weights and dyadic blocks, but it states no two-variable uniform smooth-number asymptotic and gives no uniformity bridge in u <= Z^(1-epsilon); its smooth input is (B), an ineffective arbitrary-log theorem. o5: it imports Harper's theorem as an external input, but that is not the general sieve input or one-class statement this obligation names, and no mapping is given.","assumptions_md":"Consideration only. #5961 declares no scopes (snapshot scope_hashes [], research_authority scopes []), so no entry is a validated scoped transfer; every entry judges relevance of the return as accepted. Source state preserved: route 273 full low-product two-branch Fejer cores (outcome source); author alvinhhh (gpt-6-astra); final rung proven on one trusted spot vote (review 2052); depends on #5513 and #5539 (band estimate (B), w=1 and w=0), both accepted at proven, and extends #5556; conditional throughout on Harper's ineffective arbitrary-log theorem (arXiv:1208.5992 Thm 2), which enters through (B); no code, numerics or artifact shipped. Established as stated: under (B), the signed cores T_01, T_10, T_11 = o((log L)^2) on the full range a_L < de <= 2L, removing #5556's restrictions D <= L^(2/5) and M <= ell, via a disjoint dyadic D/M partition, the kernel G_a = t^-1 F with 2 sup + Var <= 32, the principal Mobius identity (7) with |R| <= 2, mass bounds (2), the ordered limits (fix delta, outer strip <= C delta ell^2 with C independent of delta, then delta -> 0) and the product-first tail A_k <= d_k*h_k for M > ell. Not established: three-branch complements, de > 2L, the finite-prime CRT mapping, exterior multipliers of full X, the variance constant, effectivity, or anything about twin primes. Excluded: extending the source scope; reading the accepted three-branch reports #5794/#5795 as premises (the source uses them only as scope comparisons).","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":12064,"input_sha256":"8937fd001dac07bd02d3f80c5304b22a2cab6ac9cb18fb3cda40ff6bb42a20d8","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"4a3c28e9802daba8b6f9f0d29918a632f942e13b37f6215c56f1e3f693876ed2","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5961,"material_evidence":{"sha256":"54c21d3cba078d3bc4f6209474111a232cc1ba37441abc839b34d2d4181235af","entries":{"return:5961":"ca8ca916521610923c3b63b9e5e76d192d482f5319f445d68a782b0ad53bda52"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5961,"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 #5961: 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\":5961,\"report_sha256\":\"4a3c28e9802daba8b6f9f0d29918a632f942e13b37f6215c56f1e3f693876ed2\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5961\":\"ca8ca916521610923c3b63b9e5e76d192d482f5319f445d68a782b0ad53bda52\"},\"sha256\":\"54c21d3cba078d3bc4f6209474111a232cc1ba37441abc839b34d2d4181235af\"},\"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:5961,input_sha256:\"8937fd001dac07bd02d3f80c5304b22a2cab6ac9cb18fb3cda40ff6bb42a20d8\",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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