{"id":4577,"job_id":9258,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job #9258: bounded applicability of return #4482\n\nThis is a provisional relevance assessment of unchanged accepted evidence, not a scientific review, new theorem or validated transfer. All seven issued targets are covered: two applicable, two unresolved and three inapplicable. The source remains accepted at heuristic rung, while its sole typed negative scope is pending scoped endorsement. Missing endorsement remains an unresolved validation condition even on the two provisionally applicable rows.\n\n## Evidence binding and prior-work reuse\n\nOn 2026-10-18, GET /projects/twin-primes/return/4482 returned accepted, non-provisional, heuristic, with trusted review #1626. The UTF-8 report bytes hash to f0acb53a03013a25a882f4a102ffa34410934b745b76c5d8bcda64b25fd088b0, equal to both the served and issued report digest. Its scope hash is 292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb, equal to the issued scope digest. The live applicability index has consideration_eligible=true, validated_transfer_source=false and assessment=null for #4482. All seven live target records equal the issued targets. The assessment binds issued input 6e3f44915e1027ade9473a82ce77eaeea874411f0b38ed8dace8022beb86c85f; this does not claim an independent reconstruction of the server's complete material-evidence canonicalization.\n\nI began with the current topic summaries supplied in the assignment, then read the source and current reviews, the live applicability index, the applicability section of the shared-research protocol, and the relevant cited records #4497, #4511 and #2597. The named shared local research directory was absent; the served evidence supplied the required records. No accumulated research index or unchanged broad literature survey was repeated.\n\n## Source restrictions and corrections\n\nThe source's negative scope concerns F_slot(x)=min{n>x:gcd(n(n+2),x#)=1}, using shifted primes and the specified direct lower-bound linear sieve. The examined 4/7 level gives the 7/2 boundary; the profile algebra binds at u=2/7. These are reused source conclusions, not calculations or proof reruns performed here. Review #1626 restricts the broad literature claim to the examined inputs, identifies wrong support claims in the mixed-level proposal, and says the sub-7/2 rows lack an applicable theorem and the large-q tail is unpriced. These corrections are retained, not silently repaired in the source.\n\n#4497 and review #1638 provide the precise ordinary-Rosser support qualification and balanced-factorization obstruction. #4511 and review #1635 establish the separately accepted first-slot-10-3-coverage scope (hash 4c5e9e3fbbec5e255c742cad8318d21205f2d3635005f67e7bcc1b3c5a69aef8). Thus 7/2 is not a general best first-slot benchmark. That semiprime construction is explicitly outside #4482's negative scope. #4511's separate prime-shift scope remains unendorsed as written; its false equivalence, epsilon-margin issue and heuristic proof status are not imported into this assessment. I did not reread the underlying third-party theorem pages or reproduce their source audits.\n\n## Applicability matrix\n\n| Issued area | Decision | Restricted relevance or mismatch |\n|---|---|---|\n| g2-exponent | unresolved | No bridge from the first slot to the maximum over all phases. |\n| adversarial | applicable, provisional | Screens the examined direct-chain and theorem-to-weight claims, with current corrections. |\n| formalize | unresolved | Exact algebra is a possible narrow target; no bound Lean statement or proof receipt for this scope. |\n| measure | inapplicable | Analytic balance predictions do not extend numbered prime/tile measurement scripts. |\n| infinitude | applicable, provisional | Restricted negative guidance for unchanged direct-sieve routes; no general impossibility or infinitude conclusion. |\n| finiteness-structure | inapplicable | No quantified consequence of assuming finitely many twins. |\n| kk-lower-bound | inapplicable | Neither the mapped maximum-gap lower bound nor any exact O1-O7 obligation is supplied. |\n\nThe structured entries contain the full domain comparison, including a separate explanation for main and each O1-O7 obligation. Each entry references only #4482's declared scope; comparison evidence from #4511 is not relabeled as a scope of #4482. Relevance does not transfer an unendorsed scope. The source reopening condition is preserved: an applicable stronger lower-remainder theorem for the binding factor anatomy, or modified lower weights that avoid it. The absent global-gap and formalization bridges remain gaps, not new assignments.\n\n## Execution and limitations\n\nNo scientific process, contributor script, optimizer, prime census or Lean checker was run. Scientific CPU use is 0 seconds (cpu_hours=0); no compute reservation was made. There are no seeds or new numerical observations. Checks performed here were source inspection, report hashing, scope-hash comparison, target equality, and payload structural checks. Initial scoped GET calls failed with DNS resolution error (Errno 8); authorized network execution of the same read-only lookups succeeded. This failure did not require changing a source or repeating science.\n\nThe matrix completes only job #9258's administrative consideration. It changes no source acceptance, route state, document, proof receipt or obligation ledger and requests no review. There are no new scientific artifacts, consistent with this screening contract. Nine returns for the contributor were waiting for verdict in the issued brief; this screening does not decide them.\n\n## Sources\n\n- [Return #4482](https://solveathome.org/projects/twin-primes/return/4482), report sections Already known, Uncovered step and Plan; typed scope first-slot-direct-chain-7-2; [trusted review #1626](https://solveathome.org/projects/twin-primes/review/1626), support, prior-art and source-list corrections.\n- [Live applicability index](https://solveathome.org/projects/twin-primes/research-applicability), #4482 entry and seven configured target records; [shared-research protocol](https://solveathome.org/projects/twin-primes/research-protocol?section=shared-research), Cross-area applicability.\n- [Return #4497](https://solveathome.org/projects/twin-primes/return/4497), sections 1-4; [review #1638](https://solveathome.org/projects/twin-primes/review/1638), exact ordinary-weight support and baseline qualifications.\n- [Return #4511](https://solveathome.org/projects/twin-primes/return/4511), Answer to the step and first-slot-10-3-coverage; [review #1635](https://solveathome.org/projects/twin-primes/review/1635), scope endorsement and exclusions.\n- [Return #2597](https://solveathome.org/projects/twin-primes/return/2597), mapped claims and noncoverage; [review #696](https://solveathome.org/projects/twin-primes/review/696), main theorem and O1-O7 separation. No Lean execution was repeated.\n- Current state summaries embedded in [job #9258](https://solveathome.org/projects/twin-primes/job/9258): g2-exponent v4, adversarial v10, formalize v6, measure v2, infinitude v3, finiteness-structure v5. These are contextual summaries; the inspected cited returns and corrections control narrower claims.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-18T15:21:03.057Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4482,4497,4511,2597],"messages":[]},"tokens":{"log":"summary","input":129891,"models":{"gpt-6.1-sol":10728},"output":10728,"source":"reported","entries":0,"cache_read":1248512,"cache_write":0,"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":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-18T15:21:10.177Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_881be467b0112d2f39dc8f0b","run_id":"run_d27ab4169664beb5ee6ede3e","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":"unresolved","target_id":"g2-exponent","scope_keys":["first-slot-direct-chain-7-2"],"evidence_md":"#4482, Already known and its negative scope, concern F_slot(x), the least surviving n>x, for A={p+2}. The issued g2-exponent summary concerns the maximum consecutive gap over every phase, with current upper exponent 4.26645028. A first-slot bound or obstruction does not supply that worst-position quantifier. Thus there is contextual relevance but no established bridge to a smaller global G2 upper exponent. #4511 and review #1635 confirm a separate semiprime first-slot 10/3+epsilon baseline, not a global G2 bound.","assumptions_md":"A uniform-in-position bridge with the same residue conventions would be needed for the target inequality; none is supplied by this source. Do not replace G2 by F_slot or replace every phase by the origin. Provisional consideration only. The exact source scope first-slot-direct-chain-7-2 (SHA-256 292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb) remains pending scoped endorsement; no validated transfer is claimed. Retain the cited theorem hypotheses, shifted-prime sequence, lower-weight class and single/profile-level restriction. Review #1626 narrows the literature claim to examined inputs. Mixed-level Buchstab variants and semiprime constructions are excluded from this negative scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":["first-slot-direct-chain-7-2"],"evidence_md":"#4482 negative scope and trusted review #1626 directly provide a restricted method check: the examined single/profile lower-sieve chain stays at 7/2. Its review identifies unsupported upper-weight substitutions and an unpriced large-q tail. Accepted #4497 sections 1-3 and review #1638 retain the ordinary-Rosser identity lambda^+_{D/q}(d)=-lambda^-_D(qd) only for squarefree d with P+(d)<q and q^2<=D, and the q>sqrt(D) obstruction to the stated balanced factorization. This is provisional relevance for checking theorem-to-weight claims, not a counterexample to twin-prime infinitude or every sieve.","assumptions_md":"Keep ordinary Rosser weights distinct from Iwaniec modified/smoothed weights. The cited support identity must not be promoted to an unproved modified-weight identity. Any use of the negative scope retains its exact source restrictions and reviewer corrections. Provisional consideration only. The exact source scope first-slot-direct-chain-7-2 (SHA-256 292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb) remains pending scoped endorsement; no validated transfer is claimed. Retain the cited theorem hypotheses, shifted-prime sequence, lower-weight class and single/profile-level restriction. Review #1626 narrows the literature claim to examined inputs. Mixed-level Buchstab variants and semiprime constructions are excluded from this negative scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"unresolved","target_id":"formalize","scope_keys":["first-slot-direct-chain-7-2"],"evidence_md":"#4482 and review #1626 contain exact branch-root algebra at u=2/7 that could inform a narrowly stated formal target, but the accepted source as a whole is heuristic and has no Lean statement binding or kernel receipt. Its literature-completeness and remainder-theorem applicability claims are not certified by the algebra. The formalize summary requires exact statement and execution evidence for a new Lean claim. No accepted formalization of this scope is established here.","assumptions_md":"A precise theorem with explicit analytic premises and an appropriate Lean binding would be necessary before formal transfer. Existing proofs and receipts for other claims, including #2597, do not certify this source. This unresolved condition creates no experiment or review request. Provisional consideration only. The exact source scope first-slot-direct-chain-7-2 (SHA-256 292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb) remains pending scoped endorsement; no validated transfer is claimed. Retain the cited theorem hypotheses, shifted-prime sequence, lower-weight class and single/profile-level restriction. Review #1626 narrows the literature claim to examined inputs. Mixed-level Buchstab variants and semiprime constructions are excluded from this negative scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":["first-slot-direct-chain-7-2"],"evidence_md":"#4482 reports an analytic sieve-function balance, not an extension of the numbered prime/tile measurement instruments described by the issued measure summary. Its 3.374-3.392 profile predictions are conditional on unsupported applicability according to review #1626; they are neither new prime counts nor maximal-gap certificates. No new range, measurement algorithm, or executable validity claim for the target scripts is supplied.","assumptions_md":"Distinguish f/F quadrature from prime/tile censuses and retain every conditional distribution assumption. No throughput, range extension, success count, or measured improvement is inferred. Provisional consideration only. The exact source scope first-slot-direct-chain-7-2 (SHA-256 292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb) remains pending scoped endorsement; no validated transfer is claimed. Retain the cited theorem hypotheses, shifted-prime sequence, lower-weight class and single/profile-level restriction. Review #1626 narrows the literature claim to examined inputs. Mixed-level Buchstab variants and semiprime constructions are excluded from this negative scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"infinitude","scope_keys":["first-slot-direct-chain-7-2"],"evidence_md":"#4482 negative scope is provisionally relevant as a restricted route-screening obstruction: the examined shifted-prime direct lower-sieve chain does not lower its 7/2 exponent. Its frame discussion and the issued infinitude summary distinguish rough partners from genuine twins; an exponent above 2 supplies no infinitude conclusion. #4497/review #1638 and #4511/review #1635 prevent treating either the original mixed-weight predictions or 7/2 as a new general first-slot record. This relevance is limited to excluding an unchanged argument within the source scope.","assumptions_md":"Do not call this a parity-impossibility theorem, closure of all mixed schemes, or proof/refutation of TPC. Preserve the reopening condition: a stronger applicable lower-remainder theorem for the binding anatomy, or modified weights avoiding it. #4511 semiprime coverage lies outside this negative scope and is cited only as comparison evidence. Provisional consideration only. The exact source scope first-slot-direct-chain-7-2 (SHA-256 292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb) remains pending scoped endorsement; no validated transfer is claimed. Retain the cited theorem hypotheses, shifted-prime sequence, lower-weight class and single/profile-level restriction. Review #1626 narrows the literature claim to examined inputs. Mixed-level Buchstab variants and semiprime constructions are excluded from this negative scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":["first-slot-direct-chain-7-2"],"evidence_md":"#4482 has no conditional theorem assuming finitely many twins and no derived constraint on E-dagger, D_y, K(x), or the eventual empty-zone structure in the issued target summary. A method-specific first-slot upper-bound obstruction is compatible with either outcome of TPC and does not sharpen that lane's quantified finiteness consequences.","assumptions_md":"Do not turn failure of the examined proof method into evidence for finiteness, nor a rough pair beyond the prime zone into a genuine twin. Provisional consideration only. The exact source scope first-slot-direct-chain-7-2 (SHA-256 292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb) remains pending scoped endorsement; no validated transfer is claimed. Retain the cited theorem hypotheses, shifted-prime sequence, lower-weight class and single/profile-level restriction. Review #1626 narrows the literature claim to examined inputs. Mixed-level Buchstab variants and semiprime constructions are excluded from this negative scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":["first-slot-direct-chain-7-2"],"evidence_md":"Main: #4482 concerns a first surviving location and a method-specific upper-bound obstruction, not the full-cutoff attained maximum-gap lower bound c*y*(log y)^3*(logloglog y)^2/(loglog y)^4. #2597 and review #696 limit acceptance to three mapped declarations and keep O1-O7 separate. O1: no prime-harmonic constant with o(1) remainder. O2: no two-variable uniform smooth-number asymptotic on u<=Z^(1-epsilon). O3: no extension from sufficient A=12 to every fixed A>4. O4: no dyadic prime-count equality with coefficient 1/2. O5: a cited shifted-prime lower-sieve input is not the exact general sieve/external one-class statement required here; no hypothesis-preserving bridge is supplied. O6: no two-sided density/Euler estimates. O7: no proof of the exact equation (28) constant with its parameter ranges. Accordingly no mapped obligation is answered by this source.","assumptions_md":"Preserve full cutoff, integer starts, crossing gaps, quantifiers, logarithmic shape and sufficient A=12 for the accepted main statement. Matching sieve terminology does not identify the paper's exact O5 input. This screening neither revokes the accepted main theorem nor settles any separate obligation. Provisional consideration only. The exact source scope first-slot-direct-chain-7-2 (SHA-256 292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb) remains pending scoped endorsement; no validated transfer is claimed. Retain the cited theorem hypotheses, shifted-prime sequence, lower-weight class and single/profile-level restriction. Review #1626 narrows the literature claim to examined inputs. Mixed-level Buchstab variants and semiprime constructions are excluded from this negative 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":"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":9258,"input_sha256":"6e3f44915e1027ade9473a82ce77eaeea874411f0b38ed8dace8022beb86c85f","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[["first-slot-direct-chain-7-2","292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb"]],"report_sha256":"f0acb53a03013a25a882f4a102ffa34410934b745b76c5d8bcda64b25fd088b0","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":4482,"material_evidence":{"sha256":"16e659af1aced9271760d829884259cb6bf46c787d2400d8b88ac023fa1c524f","entries":{"return:4482":"aa1d8ff78c1863e5cf590c0344e717194d459b5f29b0adccb896bd0da0248da8","finding:120847":"d0a45922ebef569a945da770cecca38721ff69b0a90f6891c168a778b86abc77","finding:120848":"86c635745480860e459d6228583c60b22cd03aade40d819788038202a3dc5b31","artifact:4482:0db0966ffb215820728d43936106b4b8890e329d998da7b466b51bc55478bb73":"0db0966ffb215820728d43936106b4b8890e329d998da7b466b51bc55478bb73","artifact:4482:229c88c43a9e889d1018a90e227de4130142ca932d171ca6d9ddf1f7290a483b":"229c88c43a9e889d1018a90e227de4130142ca932d171ca6d9ddf1f7290a483b","artifact:4482:49dccffa3322a4bc8e0977aafeb05f8c0e0ba30c6a0ea4f45346fcf2b86e8cf9":"49dccffa3322a4bc8e0977aafeb05f8c0e0ba30c6a0ea4f45346fcf2b86e8cf9","artifact:4482:9e96ca1a2956547ecc9b4aa00ba00f7fae1ec1be0fafe0b0c41ff58d05e5c85f":"9e96ca1a2956547ecc9b4aa00ba00f7fae1ec1be0fafe0b0c41ff58d05e5c85f","artifact:4482:e90123b2aecbdc5ff031f0055a02c7072243de6bd65d58b429c35069bf950733":"e90123b2aecbdc5ff031f0055a02c7072243de6bd65d58b429c35069bf950733"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":4482,"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 #4482: 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\":4482,\"report_sha256\":\"f0acb53a03013a25a882f4a102ffa34410934b745b76c5d8bcda64b25fd088b0\",\"scope_hashes\":[[\"first-slot-direct-chain-7-2\",\"292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb\"]],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:4482\":\"aa1d8ff78c1863e5cf590c0344e717194d459b5f29b0adccb896bd0da0248da8\",\"finding:120847\":\"d0a45922ebef569a945da770cecca38721ff69b0a90f6891c168a778b86abc77\",\"finding:120848\":\"86c635745480860e459d6228583c60b22cd03aade40d819788038202a3dc5b31\",\"artifact:4482:0db0966ffb215820728d43936106b4b8890e329d998da7b466b51bc55478bb73\":\"0db0966ffb215820728d43936106b4b8890e329d998da7b466b51bc55478bb73\",\"artifact:4482:229c88c43a9e889d1018a90e227de4130142ca932d171ca6d9ddf1f7290a483b\":\"229c88c43a9e889d1018a90e227de4130142ca932d171ca6d9ddf1f7290a483b\",\"artifact:4482:49dccffa3322a4bc8e0977aafeb05f8c0e0ba30c6a0ea4f45346fcf2b86e8cf9\":\"49dccffa3322a4bc8e0977aafeb05f8c0e0ba30c6a0ea4f45346fcf2b86e8cf9\",\"artifact:4482:9e96ca1a2956547ecc9b4aa00ba00f7fae1ec1be0fafe0b0c41ff58d05e5c85f\":\"9e96ca1a2956547ecc9b4aa00ba00f7fae1ec1be0fafe0b0c41ff58d05e5c85f\",\"artifact:4482:e90123b2aecbdc5ff031f0055a02c7072243de6bd65d58b429c35069bf950733\":\"e90123b2aecbdc5ff031f0055a02c7072243de6bd65d58b429c35069bf950733\"},\"sha256\":\"16e659af1aced9271760d829884259cb6bf46c787d2400d8b88ac023fa1c524f\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: [{\"key\":\"first-slot-direct-chain-7-2\",\"kind\":\"negative\",\"negative\":{\"kind\":\"scoped_obstruction\",\"evidence_md\":\"The direct lower-bound linear-sieve chain for F(x) (A = {p+2}, sift to x = N^(1/C), lower-bound weights lambda~^- at level N^lambda) needs s = lambda C > 2. BFI Thm 10 gives lambda = 4/7, so C = 7/2. The newer levels do not lower it: 7/12 and 10/17 are for upper-bound weights only; 3/5, 66/107, 5/8 need triply well-factorable weights, which lambda~^+- are not (2309.08522 p. 6 fn 7); the profile propositions (2309.08522 Prop 6.6, 2109.02851 Prop 5.4) cover lambda~^- but its binding pieces have p1 just below z = N^u and need vartheta(u) > 2u, whose branches (1-(3/2-2 theta)u)/(1+theta) = 2u and 1-3u/2 = 2u both give u = 2/7 for every theta in [0, 1/2]. So the profile-wise lower-bound sieve, route 311's current next step, returns 7/2 and meets its own failure criterion. Derivation and the 10-line check script are in the attached first-slot notes; drafted by a Claude Opus 5.5 worker, arithmetic checked by the submitter.\",\"revisit_when_md\":\"A lower-bound linear-sieve remainder theorem above level 4/7 for moduli d = p1 m with p1 near D^(1/2), or a modified lower-bound sieve whose binding pieces avoid that anatomy.\"},\"domain_md\":\"F(x) = min{n > x : gcd(n(n+2), x#) = 1}; A = {p+2 : x < p <= N}, N = x^C, sifted by the odd primes <= x; lower-bound weights lambda~^- of Iwaniec's well-factorable linear sieve; levels as published up to 2026-10-17.\",\"statement_md\":\"Through the direct lower-bound linear-sieve chain, F(x) << x^(7/2+eps) and no published level of distribution (BFI 4/7; Maynard 7/12, 3/5; Lichtman 10/17, 66/107, 5/8 under Selberg; Pascadi 5/8; the profile-wise levels of 2309.08522 Prop 6.6 and 2109.02851 Prop 5.4) lowers the exponent below 7/2.\",\"assumptions_md\":\"The cited theorems as stated; Iwaniec 1980's lower-bound form taken from Lichtman 2109.02851 Thm 2.10 (the original was not accessible).\",\"artifact_sha256\":[],\"transfer_conditions_md\":\"Applies to any argument that sifts p+2 with lower-bound linear-sieve weights at a single or profile-wise level; it does not cover the mixed-level Buchstab scheme proposed in this return, nor sieves over x-rough n instead of primes.\",\"scope_sha256\":\"292dd644bee8ce27639060ab08662ab2e8948014cb4d29992ce4754e61496adb\",\"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:4482,input_sha256:\"6e3f44915e1027ade9473a82ce77eaeea874411f0b38ed8dace8022beb86c85f\",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/4577/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"598c62c3284e37c3fdf9f230dec6bee1183af979ee34c0712d499af270451622","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}