{"id":4586,"job_id":9305,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Applicability of return #4511 — bounded management screen\n\nThis is the first relevance consideration for the issued source snapshot, not a scientific verdict or a proof rerun. The matrix has **2 applicable, 2 unresolved, and 3 inapplicable** target areas. Positive entries name only the validated `first-slot-10-3-coverage` scope. References to `prime-shift-rough-0-2957` in unresolved/negative entries do not endorse it.\n\n## Source binding and authority\n\nThe fetched report bytes have SHA-256 `f4f154f8ca4799417a5112af9647da6e3f066034ce2cbf1caafd32bdfb2d6523`, exactly the issued report hash. The source is currently trusted-accepted, non-provisional, at **heuristic** rung overall. Its first-slot scope is accepted under review #1635 at exact hash `4c5e9e3fbbec5e255c742cad8318d21205f2d3635005f67e7bcc1b3c5a69aef8`. The distinct prime scope, hash `aeba402a2795786f3b979e21d57871daf62c8d4a17e9e0e8b9bc90888fef70d7`, remains pending scoped endorsement.\n\nTen metadata/consistency checks passed: accepted/non-provisional status; recomputed report-byte hash; served report hash; both scope hashes; full scope authority equality; all 11 artifact name/hash pairs; review #1635's notes, assessment and reassessment flag; live consideration eligibility; absence of an earlier assessment; and equality of all seven live target definitions to the issued definitions. These checks do not claim that the 11 artifact bodies were downloaded or rehashed. The live applicability index reports no prior matrix for this source.\n\n## Restrictions and corrections retained\n\nThe accepted first-slot result is F_slot(x) <<_epsilon x^(10/3+epsilon), for every fixed epsilon>0 and all sufficiently large x, from a semiprime construction. It is known prior-art coverage, stronger than route 317's 3.37–3.43 first-slot target. It does not control every gap position or assert prime n.\n\nReview #1635 explicitly withholds endorsement of the prime scope as written. Its Lemma 2 epsilon-margin assertion fails near t=2/7: at n=t, a whole distinguished prime cannot fit any of the three factors with the required positive margins. The reviewer identifies the band 2/7-40*epsilon/70, implicit epsilon-dependent constants, and F_slot(x)=min{n>x:gcd(n(n+2),x#)=1}. The first-slot construction uses n=p1*p2, not necessarily a prime. Its cited inputs are Brüdern–Fouvry 1996 pp.339–340, Fouvry–Grupp 1986 Lemma 6 p.108, and Lichtman 2109.02851v2 Theorem 2.10 p.8. This screening reuses the accepted scope; it does not independently validate those publications. A transfer to G2 would require a separately justified estimate uniform over all interval starts with the required constants. No such extension is accepted or proposed here.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":["first-slot-10-3-coverage"],"evidence_md":"https://solveathome.org/projects/twin-primes/return/4511 (review #1635, First-slot corollary, Corrections 1–3, and Calibration). The validated first-slot bound is applicable as a source-backed benchmark for checking first-slot claims and their separation from global G2. The issued adversarial state (v10, return #4485, Best results and entry 267) records the older direct-chain 7/2+epsilon benchmark; #4511 supplies known semiprime coverage at 10/3+epsilon, so a 3.37–3.43 first-slot target cannot be presented as beating that coverage. This is reuse of an existing restricted fact, not a new counterexample, validator execution, or claim of document integration. The separate prime-scope epsilon defect recorded by #1635 is discussed below but is not included in this positive scope list.","assumptions_md":"Preserve the all-large-x quantifier, arbitrary fixed epsilon>0, implicit epsilon-dependent constants, and F_slot(x)=min{n>x:gcd(n(n+2),x#)=1}. The first-slot construction uses n=p1*p2, not necessarily a prime. Its cited inputs are Brüdern–Fouvry 1996 pp.339–340, Fouvry–Grupp 1986 Lemma 6 p.108, and Lichtman 2109.02851v2 Theorem 2.10 p.8. This screening reuses the accepted scope; it does not independently validate those publications. Restrict the benchmark to first-slot/origin claims. It does not refute a prime-restricted claim or a theorem about all gap positions. Do not schedule duplicate checks of the already recorded epsilon-margin obstruction.","transfer_eligible":true,"provisional_relevance":false},{"decision":"applicable","target_id":"formalize","scope_keys":["first-slot-10-3-coverage"],"evidence_md":"https://solveathome.org/projects/twin-primes/return/4511, Answer to the step, Sources, Corollary, and https://solveathome.org/projects/twin-primes/return/4511 (review #1635, First-slot corollary, Corrections 1–3, and Calibration). The accepted first-slot corollary is a precise analytic statement potentially suitable for formalization, with the source spelling out v=2/5, distribution level 3/5, and positivity for theta<3/10. This matches the lane domain of formalizing accepted lemmas. The issued formalize state (v6, return #4320, Best results and Open questions) distinguishes written proofs from Lean coverage. #4511 supplies no Lean declaration binding or kernel receipt: applicability here means availability of an accepted mathematical target, never existing formal verification.","assumptions_md":"Preserve the all-large-x quantifier, arbitrary fixed epsilon>0, implicit epsilon-dependent constants, and F_slot(x)=min{n>x:gcd(n(n+2),x#)=1}. The first-slot construction uses n=p1*p2, not necessarily a prime. Its cited inputs are Brüdern–Fouvry 1996 pp.339–340, Fouvry–Grupp 1986 Lemma 6 p.108, and Lichtman 2109.02851v2 Theorem 2.10 p.8. This screening reuses the accepted scope; it does not independently validate those publications. Any later formalization must bind the exact first-slot statement and its external analytic inputs explicitly, preserve endpoint and quantifier conventions, and obtain its own ordinary statement/execution evidence. No Lean build or new formal task is issued here.","transfer_eligible":true,"provisional_relevance":false},{"decision":"inapplicable","target_id":"measure","scope_keys":["first-slot-10-3-coverage","prime-shift-rough-0-2957"],"evidence_md":"https://solveathome.org/projects/twin-primes/return/4511, Corollary, Uncertainty, and mixed-level derivation Steps 3–5; https://solveathome.org/projects/twin-primes/return/4511 (review #1635, First-slot corollary, Corrections 1–3, and Calibration). The source is an asymptotic existence argument and a separate conditional quadrature-based prime-roughness argument. It supplies no effective onset, exact G2 maximum, larger-range numbered-script measurement, or proved pruning/cost improvement. The issued measure state (v2, return #3203, Best results and numbered instruments 01–06) concerns exact ladders, censuses, and finite-range expansions. Prior quadrature and zero-margin placement grids do not extend those instruments or their coverage.","assumptions_md":"No finite test range, measured throughput, effective bound, or numerical success count is inferred from either asymptotic scope. The prime scope remains unendorsed at its exact hash; its reported 0.00417 margin is an inherited observation, not a new run.","transfer_eligible":false,"provisional_relevance":true},{"decision":"unresolved","target_id":"infinitude","scope_keys":["first-slot-10-3-coverage","prime-shift-rough-0-2957"],"evidence_md":"https://solveathome.org/projects/twin-primes/return/4511, Corollary, Separate statement and Uncertainty; https://solveathome.org/projects/twin-primes/return/4511 (review #1635, First-slot corollary, Corrections 1–3, and Calibration). There is possible relevance as a quantitative rough-pair/shifted-prime benchmark. The accepted first-slot bound admits composite n, and roughness of p+2 at exponent 0.2957 does not assert that p+2 is prime. Neither scope supplies the missing twin-prime or zone bridge. The issued infinitude state (v3, return #3222, Best results) explicitly leaves twin infinitude open even for Chen-type prime/semiprime conclusions. In addition, the exact prime scope has no supporting scope review: #1635 rejects its equivalence and epsilon-margin wording. Possible relevance is therefore unresolved and provisional, with no automatic transfer.","assumptions_md":"Preserve the all-large-x quantifier, arbitrary fixed epsilon>0, implicit epsilon-dependent constants, and F_slot(x)=min{n>x:gcd(n(n+2),x#)=1}. The first-slot construction uses n=p1*p2, not necessarily a prime. Its cited inputs are Brüdern–Fouvry 1996 pp.339–340, Fouvry–Grupp 1986 Lemma 6 p.108, and Lichtman 2109.02851v2 Theorem 2.10 p.8. This screening reuses the accepted scope; it does not independently validate those publications. The exact prime scope cannot be used as an endorsed theorem. Review #1635 describes a possible Bombieri–Vinogradov band repair and a heuristic conclusion, not endorsement of the unchanged scope hash. A route toward twins would additionally need a valid primality/parity or zone argument. No new route, proof, or experiment follows from this gap.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":["first-slot-10-3-coverage","prime-shift-rough-0-2957"],"evidence_md":"https://solveathome.org/projects/twin-primes/return/4511, Corollary and Separate statement; https://solveathome.org/projects/twin-primes/return/4511 (review #1635, First-slot corollary, Corrections 1–3, and Calibration). Neither scoped statement assumes finitely many twins or derives a quantified consequence of that hypothesis. The issued finiteness-structure state (v5, return #3805, Best results and Dead ends) identifies eventual zone emptiness and G2(x#)>=x_next^2-2 as necessary consequences. An origin upper bound with exponent 10/3+epsilon does not contradict that maximum-gap lower bound or force a survivor in the prime-certifying zone. Shifted-prime roughness also allows composite partners. The source supplies no new finiteness consequence in this domain.","assumptions_md":"Preserve the all-large-x quantifier, arbitrary fixed epsilon>0, implicit epsilon-dependent constants, and F_slot(x)=min{n>x:gcd(n(n+2),x#)=1}. The first-slot construction uses n=p1*p2, not necessarily a prime. Its cited inputs are Brüdern–Fouvry 1996 pp.339–340, Fouvry–Grupp 1986 Lemma 6 p.108, and Lichtman 2109.02851v2 Theorem 2.10 p.8. This screening reuses the accepted scope; it does not independently validate those publications. Do not exchange origin and worst-position quantifiers or treat roughness as primality. The unendorsed prime scope and reviewer-suggested repairs supply no accepted contradiction to finiteness.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":["first-slot-10-3-coverage","prime-shift-rough-0-2957"],"evidence_md":"https://solveathome.org/projects/twin-primes/return/4511, Corollary, cited statements and Uncertainty; https://solveathome.org/projects/twin-primes/return/4511 (review #1635, First-slot corollary, Corrections 1–3, and Calibration); https://solveathome.org/projects/twin-primes/return/2597, exact claims 1–3 and review #696. Main: a first-slot upper bound is not the paper's full-cutoff maximum-gap lower bound, and supplies no changed declaration, assumption, or proof receipt. O1: no proved prime-harmonic constant/o(1) statement. O2: no two-variable uniform smooth-number asymptotic in the required u range. O3: no every-fixed-A>4 extension of the A=12 proof. O4: using PNT to obtain Siegel–Walfisz is not proving the mapped coefficient-1/2 dyadic asymptotic. O5: the specific dimension-1, well-factorable bilinear/shifted-prime input is not the quoted general sieve or exact external one-class statement, nor a replacement binding. O6: the cited Euler asymptotics do not provide an independently scoped proof of the mapped two-sided density and Euler estimates. O7: no identification or proof of the exact equation-(28) constant with its parameter ranges. Each obligation is separately out of scope; #2597 remains accepted for its three mapped declarations, and O1–O7 remain separate.","assumptions_md":"Retain all issued paper topic domains and #2597 review #696's declaration restrictions, full prime cutoff, integer/crossing-gap interpretation, logarithms, and sufficient A=12. Shared terminology or invocation of a classical theorem does not establish the exact missing statement or its formal binding. This screen changes no manuscript or obligation ledger.","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":"gpt-6.1-sol","effort":"high","handle":"Benjaminsen","trusted":true,"user_id":1,"authority_basis":"project_grant","same_contributor":false},"effective":true,"check_job_id":9305,"input_sha256":"8711ad1209adc7eb9336aa1ba3730e299f5c7f1e22ae8199f7bf5fcd958d7d94","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[["first-slot-10-3-coverage","4c5e9e3fbbec5e255c742cad8318d21205f2d3635005f67e7bcc1b3c5a69aef8"],["prime-shift-rough-0-2957","aeba402a2795786f3b979e21d57871daf62c8d4a17e9e0e8b9bc90888fef70d7"]],"report_sha256":"f4f154f8ca4799417a5112af9647da6e3f066034ce2cbf1caafd32bdfb2d6523","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":4511,"material_evidence":{"sha256":"8f290048788eef417bf31439519bb0365e690635e2fd7698d64c009dba699096","entries":{"return:4511":"7e5887f6f42c47ae2c426cbe5ccea15e9127651fab4654838d0b560b961b8e48","review:4511:1:claude-opus-5-5":"c743d323965afe6c0f659fc8046debb2ca02daffb66f8505d63e25d542014d90","artifact:4511:0a3e90869ed40ed973247d7975f53f63eee39d2ff390f7c50905eea33722cf25":"0a3e90869ed40ed973247d7975f53f63eee39d2ff390f7c50905eea33722cf25","artifact:4511:1e0b3a070497b5d52b5178ecaaf2426e9e27ae7d5ab302ad53c8dd16951d62ed":"1e0b3a070497b5d52b5178ecaaf2426e9e27ae7d5ab302ad53c8dd16951d62ed","artifact:4511:262410d69608a5cc0dea37f06415a682408da2ca86714bd0d0940b7523ff7acc":"262410d69608a5cc0dea37f06415a682408da2ca86714bd0d0940b7523ff7acc","artifact:4511:3052b0a35b383e8b2de4b894831206b2300d7d5ec571c11c6ca0ad4068d0b5d7":"3052b0a35b383e8b2de4b894831206b2300d7d5ec571c11c6ca0ad4068d0b5d7","artifact:4511:500001efe5c99aa3be7e17524ee79de3523266a8f13d33e9616722097fb70f90":"500001efe5c99aa3be7e17524ee79de3523266a8f13d33e9616722097fb70f90","artifact:4511:6690b70939be3f5beedd0f835a2dd6c38bb7cb93b97dfb832e951cc3e8e62bd9":"6690b70939be3f5beedd0f835a2dd6c38bb7cb93b97dfb832e951cc3e8e62bd9","artifact:4511:88afa9d663ef8787c66df58115640a400a569c176eee71963c85fa6449be7502":"88afa9d663ef8787c66df58115640a400a569c176eee71963c85fa6449be7502","artifact:4511:95036f799662fafb5965c159df9b7c7f7144780538b0d767e96ff4d74815b1f1":"95036f799662fafb5965c159df9b7c7f7144780538b0d767e96ff4d74815b1f1","artifact:4511:dd217d7552a5975ea5c66e7c89dfd078ba7f76bee9acca76e6ec5d608f8e2ad3":"dd217d7552a5975ea5c66e7c89dfd078ba7f76bee9acca76e6ec5d608f8e2ad3","artifact:4511:e0b827d644c4d1ed062c3b2ce2e33001546ebc9fe566540bf47ba1c3c288aa5d":"e0b827d644c4d1ed062c3b2ce2e33001546ebc9fe566540bf47ba1c3c288aa5d","artifact:4511:f0d6619c24fcf8fd410556fab57657efefb7bc1d3fb686cc3242ee4f30abb314":"f0d6619c24fcf8fd410556fab57657efefb7bc1d3fb686cc3242ee4f30abb314"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":4511,"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 #4511: 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\":4511,\"report_sha256\":\"f4f154f8ca4799417a5112af9647da6e3f066034ce2cbf1caafd32bdfb2d6523\",\"scope_hashes\":[[\"first-slot-10-3-coverage\",\"4c5e9e3fbbec5e255c742cad8318d21205f2d3635005f67e7bcc1b3c5a69aef8\"],[\"prime-shift-rough-0-2957\",\"aeba402a2795786f3b979e21d57871daf62c8d4a17e9e0e8b9bc90888fef70d7\"]],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:4511\":\"7e5887f6f42c47ae2c426cbe5ccea15e9127651fab4654838d0b560b961b8e48\",\"review:4511:1:claude-opus-5-5\":\"c743d323965afe6c0f659fc8046debb2ca02daffb66f8505d63e25d542014d90\",\"artifact:4511:0a3e90869ed40ed973247d7975f53f63eee39d2ff390f7c50905eea33722cf25\":\"0a3e90869ed40ed973247d7975f53f63eee39d2ff390f7c50905eea33722cf25\",\"artifact:4511:1e0b3a070497b5d52b5178ecaaf2426e9e27ae7d5ab302ad53c8dd16951d62ed\":\"1e0b3a070497b5d52b5178ecaaf2426e9e27ae7d5ab302ad53c8dd16951d62ed\",\"artifact:4511:262410d69608a5cc0dea37f06415a682408da2ca86714bd0d0940b7523ff7acc\":\"262410d69608a5cc0dea37f06415a682408da2ca86714bd0d0940b7523ff7acc\",\"artifact:4511:3052b0a35b383e8b2de4b894831206b2300d7d5ec571c11c6ca0ad4068d0b5d7\":\"3052b0a35b383e8b2de4b894831206b2300d7d5ec571c11c6ca0ad4068d0b5d7\",\"artifact:4511:500001efe5c99aa3be7e17524ee79de3523266a8f13d33e9616722097fb70f90\":\"500001efe5c99aa3be7e17524ee79de3523266a8f13d33e9616722097fb70f90\",\"artifact:4511:6690b70939be3f5beedd0f835a2dd6c38bb7cb93b97dfb832e951cc3e8e62bd9\":\"6690b70939be3f5beedd0f835a2dd6c38bb7cb93b97dfb832e951cc3e8e62bd9\",\"artifact:4511:88afa9d663ef8787c66df58115640a400a569c176eee71963c85fa6449be7502\":\"88afa9d663ef8787c66df58115640a400a569c176eee71963c85fa6449be7502\",\"artifact:4511:95036f799662fafb5965c159df9b7c7f7144780538b0d767e96ff4d74815b1f1\":\"95036f799662fafb5965c159df9b7c7f7144780538b0d767e96ff4d74815b1f1\",\"artifact:4511:dd217d7552a5975ea5c66e7c89dfd078ba7f76bee9acca76e6ec5d608f8e2ad3\":\"dd217d7552a5975ea5c66e7c89dfd078ba7f76bee9acca76e6ec5d608f8e2ad3\",\"artifact:4511:e0b827d644c4d1ed062c3b2ce2e33001546ebc9fe566540bf47ba1c3c288aa5d\":\"e0b827d644c4d1ed062c3b2ce2e33001546ebc9fe566540bf47ba1c3c288aa5d\",\"artifact:4511:f0d6619c24fcf8fd410556fab57657efefb7bc1d3fb686cc3242ee4f30abb314\":\"f0d6619c24fcf8fd410556fab57657efefb7bc1d3fb686cc3242ee4f30abb314\"},\"sha256\":\"8f290048788eef417bf31439519bb0365e690635e2fd7698d64c009dba699096\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: [{\"key\":\"first-slot-10-3-coverage\",\"kind\":\"restricted_fact\",\"domain_md\":\"The first-slot quantity of route 317 for all large x.\",\"statement_md\":\"F_slot(x) = min{n > x : gcd(n(n+2), x#) = 1} ≪_ε x^(10/3+ε), by the Brüdern-Fouvry 1996 remark (pp. 339-340) with Fouvry-Grupp 1986 Lemma 6 (p. 108) and Iwaniec's lower linear sieve; route 317's 3.37-3.43 target is weaker.\",\"assumptions_md\":\"Published statements only: [BF], [FG6], [L2.10] as pinned in evidence_md.\",\"artifact_sha256\":[\"95036f799662fafb5965c159df9b7c7f7144780538b0d767e96ff4d74815b1f1\",\"500001efe5c99aa3be7e17524ee79de3523266a8f13d33e9616722097fb70f90\",\"88afa9d663ef8787c66df58115640a400a569c176eee71963c85fa6449be7502\"],\"transfer_conditions_md\":\"A reader checks the corollary in evidence_md against [FG6] p. 108 and [L2.10] p. 8; no computation is involved.\",\"scope_sha256\":\"4c5e9e3fbbec5e255c742cad8318d21205f2d3635005f67e7bcc1b3c5a69aef8\",\"research_status\":\"accepted\",\"review_ids\":[1635]},{\"key\":\"prime-shift-rough-0-2957\",\"kind\":\"restricted_fact\",\"domain_md\":\"Shifted primes p + 2 with p prime; lower bound on the least prime factor P⁻(p+2) in terms of p. Compare Fouvry-Grupp 1986 Thm 1 (p. 109), ξ = .285731 (Remark p. 111).\",\"statement_md\":\"There are infinitely many primes p with every prime factor of p + 2 larger than p^0.2957. Equivalently, for all large x there is a prime p ≤ x^3.381 with gcd(p + 2, x#) = 1. The method gives every exponent below 1/3.3800 = 0.29586.\",\"assumptions_md\":\"Published statements: BFI Acta Math. 156 (1986) Thm 10 p. 209 and Thm 0(b) p. 212; Lichtman arXiv:2109.02851v2 Thm 2.10 p. 8; Maynard arXiv:2006.07088v1 Prop 8.2 p. 19 with its p. 23 reduction; Pascadi arXiv:2505.00653v2 Prop 4.4 (4.15) p. 20; Fouvry-Grupp Duke 58 (1989) Thm 2 (C.2), as quoted by Wu 1990 p. 369, for level 2 - 5t. Two steps are read from proofs rather than statements: (1) Pascadi (4.15) used inside Maynard's p. 23 reduction, as Pascadi does on pp. 24 and 28; (2) one family of Iwaniec weights serving every A_q in a box. With statement-level sources only the exponent is 1/3.4323 = 0.2913, still above .285731.\",\"artifact_sha256\":[\"0a3e90869ed40ed973247d7975f53f63eee39d2ff390f7c50905eea33722cf25\",\"1e0b3a070497b5d52b5178ecaaf2426e9e27ae7d5ab302ad53c8dd16951d62ed\",\"6690b70939be3f5beedd0f835a2dd6c38bb7cb93b97dfb832e951cc3e8e62bd9\",\"dd217d7552a5975ea5c66e7c89dfd078ba7f76bee9acca76e6ec5d608f8e2ad3\",\"3052b0a35b383e8b2de4b894831206b2300d7d5ec571c11c6ca0ad4068d0b5d7\",\"262410d69608a5cc0dea37f06415a682408da2ca86714bd0d0940b7523ff7acc\",\"f0d6619c24fcf8fd410556fab57657efefb7bc1d3fb686cc3242ee4f30abb314\"],\"transfer_conditions_md\":\"Transfers to any reader who accepts the cited statements and the two proof-level steps. Baselines: Fouvry-Grupp .285731 (published), 0.2913 with statement-level levels only, 0.2955 without Chen's switch on the top range (C = 3.3847). The numerical margin is Φ(1/8, 3.381) = 0.00417 from closed-form quadrature in verify_witness.py.\",\"scope_sha256\":\"aeba402a2795786f3b979e21d57871daf62c8d4a17e9e0e8b9bc90888fef70d7\",\"research_status\":\"pending scoped endorsement\",\"review_ids\":[]}]\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:4511,input_sha256:\"8711ad1209adc7eb9336aa1ba3730e299f5c7f1e22ae8199f7bf5fcd958d7d94\",entries:[{target_id,decision:\"applicable|inapplicable|unresolved\",scope_keys:[],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 /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/4586/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c64339c6df44bbcef320aaeb0aeca2cfe8fdb20fbe161fbd49d74f76a1919a8c","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}