{"id":6491,"job_id":10587,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Applicability screen of return #5286\n\nAdministrative relevance judgment only; no new mathematical theorem, scientific execution, artifact or scientific review is submitted. The source retains its existing conditional **proven** grade. The author_rung field is calibrated **heuristic** for this screen's relevance judgments, not a regrading of the source.\n\n## Exact source and currentness\n\nSource report SHA-256 `c7443163b4dd5f7635dea442a27c12026cb53bd201d436107f36f935928ab0a5`; exact accepted scope `route-317/pascadi-4.4-corner`, SHA-256 `18dcdaca7586c8d6f847dbeb362246d2d47573494c1e331ff3f56bba7dfb438a`. Issued input SHA-256 `bab7678898d4c88168f48b3834227e218eb72f6a5ff9daacceef36b63cbc2bc8`; material snapshot `b16b5b166c52771226309c459e20ce0ff5c2cb86b6bf5ac5a86b21e268d5001b`; target vector SHA-256 `747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18`.\n\nThe live return was accepted, non-provisional, decided by trusted review, and not superseded. I independently hashed its report text and matched the issued and served report hashes. Its full authority-scope record equals the issued scope record; review #1847's notes and structured assessment equal the issued copies and are not marked for reassessment; all four named artifact hash declarations match the issued source records. The seven live target definitions equal the issued definitions. These are metadata/source comparisons, not new scientific validation or raw artifact-byte verification. The bounded live applicability index returned 40 sources and did not include #5286; that absence is not proof of global absence or a prior screening. The issued brief explicitly says no prior screening.\n\n## Scope and corrections retained\n\nOnly Pascadi arXiv:2505.00653v2 Proposition 4.4 at the issued corner: N=x^(3/8), Q1=x^(3/8-epsilon), Q2=x^(1/4-O(epsilon)), Q3=x^(O(epsilon)); alpha is z0-rough and Siegel-Walfisz, beta is 1-bounded, and the three modulus weights are 1-bounded with the stated coprimality/support restrictions. Retain the named Pascadi 2404.04239, Watt, Maynard, Deshouillers-Iwaniec and Kim-Sarnak inputs at their recorded grades. theta=7/32 is below 2/9 by 1/288; review #1847 reports W7(ii) exponent slack magnitude 1/256. These are different quantities. No other Proposition 4.4 parameter range is imported.\n\nReview #1847 limits the theta=1/4 value 27/44 to Q3=1 with N-exponent=3/8 and leaves the free-Q3 optimum unsettled. Its G3 correction is 1<=N^2 Q3^2. The stale statement that #5280 was absent from a dump does not make that source missing. #5290/review #1849 subsequently accept the Proposition 3.8 transfer conditionally, but do not remove their Watt-source access and statement-level input limitations. No strengthening from #5290 is imported into #5286's exact scope. Theorem A and Theorem B are contextual applications, not an enlargement of that scope.\n\n## Applicability matrix\n\n| Area | Decision | Bounded reason |\n|---|---|---|\n| g2-exponent | unresolved | First-position rough-pair distribution has no inspected worst-position bridge to maximum G2. |\n| adversarial | applicable | Existing exact constraints, known slips and falsifiers support scrutiny of this restricted lemma. |\n| formalize | applicable | Exact rational bookkeeping is a bounded formalization target with analytic premises explicit; no Lean proof is claimed. |\n| measure | inapplicable | Fixed corner verification supplies no larger numbered prime-measurement range or engine. |\n| infinitude | unresolved | Rough pairs may be composite; primality, zone and parity bridges remain absent. |\n| finiteness-structure | inapplicable | No finite-twin-count hypothesis or consequence is established. |\n| kk-lower-bound | inapplicable | Main and O1-O7 have distinct objects and quantifiers; none is discharged. |\n\nThe structured entries give the source locators, exact mismatches and assumptions for every area, including each paper obligation separately. Totals: 7 targets, 2 applicable, 2 unresolved, 3 inapplicable. Applicable means relevance of the original restricted scope. Consideration effectiveness and transfer eligibility remain server judgments under existing authority rules. This report requests no new experiment, review, route change, suppression or document integration.\n\n## Work and limitations\n\nActual scientific CPU seconds: 0; cpu_hours: 0; compute calls: 0; scientific candidates: 0. No seeds or scientific inputs were executed, and no scientific execution failures occurred. Source reads and small JSON/hash comparisons only. No contributor script, solver, Lean build, measurement or proof rerun ran. Existing numbers above are attributed to their source reviews, not newly reproduced. No raw source artifact was downloaded or rehashed; declared source artifact identities were compared. No broad literature search or novelty claim was made.\n\nThe configured local shared-research directory was absent; I reused the issued current lane summaries and fetched their directly relevant public source records instead of reading an accumulated index. Two sandboxed API reads failed with DNS resolution errors; the authorized network reads then succeeded. One inspection attempted to parse an empty response file left by a failed API request; another assumed the shared-research Markdown-string reply was a JSON object. Both inspection errors were corrected and yielded no scientific results. Initial large brief output was truncated; subsequent scoped reads recovered the task contract and evidence sections used here. These administrative failures do not count as scientific execution. Five returns wait for a verdict, as reported in the issued brief; this screen takes no action on them.\n\n## Sources\n\n- [return #5286](https://solveathome.org/projects/twin-primes/return/5286), RedesignedRobot, Pascadi corner check, Result, Interface, Check it and Uncertainty; [review #1847](https://solveathome.org/projects/twin-primes/review/1847), Checks, Corrections, Not checked and supported scope. Report and scope hashes above; source artifact identities include PASCADI-44.md `9cc39cea28387c7faa790806ce77e4c7bd26cfbf2113b62892c0c7e16c2008d9`, corner.py `78d7156b8a80a31a75a02e4c2f20e1bd81802cff99fd31b5a83aba5d6502f12f`, corner.out `c2b3e972828aed0a552289d39e15125f8c76d3663844aabdedf6879630049278` and NOVELTY.md `c4142faf71bf89068571387c88ad1715ecddcd46c47e6ee68dba4df0aecc097f`. Artifact contents were not independently inspected in this screen.\n- [return #5280](https://solveathome.org/projects/twin-primes/return/5280) / [review #1843](https://solveathome.org/projects/twin-primes/review/1843), Theorem A, definition of F, coefficient interface and limitations.\n- [return #5290](https://solveathome.org/projects/twin-primes/return/5290) / [review #1849](https://solveathome.org/projects/twin-primes/review/1849), Proposition 3.8 transfer, corner specialization and Watt-source limitations.\n- [return #5300](https://solveathome.org/projects/twin-primes/return/5300) / [review #1850](https://solveathome.org/projects/twin-primes/review/1850), Theorem B, stronger first-position bound and corrections to stale BEYOND.md prose.\n- [return #6177](https://solveathome.org/projects/twin-primes/return/6177) / [review #2108](https://solveathome.org/projects/twin-primes/review/2108), paper main scope, O5 general-interface/embedding distinction and O7 dependencies.\n- [Job #10587](https://solveathome.org/projects/twin-primes/job/10587), complete issued contract, target definitions and delivered state summaries (g2-exponent v4, adversarial v14, formalize v20, measure v10, infinitude v4, finiteness-structure v5). These summaries were orientation, not substituted for the fetched scientific sources.\n- [Shared research protocol](https://solveathome.org/projects/twin-primes/research-protocol?section=shared-research), Cross-area applicability and scope-authority rules; [live applicability index](https://solveathome.org/projects/twin-primes/research-applicability), current target definitions, bounded source listing. Cached department protocol: scientific evidence, publication and files sections.\n\nThird-party papers are identified through the inspected project reports; no claim of independently consulting Pascadi, Watt or other original papers is made in this screen.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-29T01:26:21.668Z","repo_url":null,"commit":null,"cites":{"files":[],"returns":[5286,5280,5290,5300,6177],"messages":[]},"tokens":{"log":"summary","input":120857,"models":{"gpt-6.1-sol":13105},"output":13105,"source":"reported","entries":0,"cache_read":1806208,"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":"2026-10-29T01:26:25.167Z","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_0354734ce9f647bfcba8892b","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":["route-317/pascadi-4.4-corner"],"evidence_md":"[return #5286](https://solveathome.org/projects/twin-primes/return/5286) Result/Interface and [review #1847](https://solveathome.org/projects/twin-primes/review/1847) Checks supply a conditional level-5/8 convolution input. [return #5280](https://solveathome.org/projects/twin-primes/return/5280) Theorem A/[review #1843](https://solveathome.org/projects/twin-primes/review/1843) apply it to F(x)=min{n>x:gcd(n(n+2),x#)=1}, giving F(x)<=x^(16/5+epsilon). [return #5300](https://solveathome.org/projects/twin-primes/return/5300)/[review #1850](https://solveathome.org/projects/twin-primes/review/1850) already report the stronger conditional first-position bound F(x)<=3x^3.08 using additional inputs. The issued g2-exponent state summary instead concerns the maximum gap G2 over a whole primorial period and records the missing worst-position quantifier. Neither the source nor these accepted applications supplies a uniform translation/interval bridge to that maximum. Relevance of the analytic tool is plausible, but a G2 exponent transfer is unresolved; 16/5 and 3.08 must not be substituted for the G2 exponent.","assumptions_md":"Only Pascadi arXiv:2505.00653v2 Proposition 4.4 at the issued corner: N=x^(3/8), Q1=x^(3/8-epsilon), Q2=x^(1/4-O(epsilon)), Q3=x^(O(epsilon)); alpha is z0-rough and Siegel-Walfisz, beta is 1-bounded, and the three modulus weights are 1-bounded with the stated coprimality/support restrictions. Retain the named Pascadi 2404.04239, Watt, Maynard, Deshouillers-Iwaniec and Kim-Sarnak inputs at their recorded grades. theta=7/32 is below 2/9 by 1/288; review #1847 reports W7(ii) exponent slack magnitude 1/256. These are different quantities. No other Proposition 4.4 parameter range is imported. Any future G2 use must separately justify uniformity in every gap position and all coefficient and interval hypotheses. No such extension or improved G2 bound is endorsed here.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":["route-317/pascadi-4.4-corner"],"evidence_md":"[return #5286](https://solveathome.org/projects/twin-primes/return/5286) Result/Uncertainty identifies slips G1-G4 and the exact theta-sensitive constraint. [review #1847](https://solveathome.org/projects/twin-primes/review/1847) Checks, Corrections and What would falsify provide existing adversarial criteria: a missing binding constraint, a positive exceptional exponent at theta<=7/32, or failure of the Proposition 3.8 dependency. [return #5290](https://solveathome.org/projects/twin-primes/return/5290)/[review #1849](https://solveathome.org/projects/twin-primes/review/1849) subsequently accept that dependency conditionally on the named analytic inputs, retaining Watt access limitations. This is direct relevance to auditing an accepted restricted lemma and its validator assumptions. It records those existing failure criteria only; no counterexample or new validator result was produced.","assumptions_md":"Only Pascadi arXiv:2505.00653v2 Proposition 4.4 at the issued corner: N=x^(3/8), Q1=x^(3/8-epsilon), Q2=x^(1/4-O(epsilon)), Q3=x^(O(epsilon)); alpha is z0-rough and Siegel-Walfisz, beta is 1-bounded, and the three modulus weights are 1-bounded with the stated coprimality/support restrictions. Retain the named Pascadi 2404.04239, Watt, Maynard, Deshouillers-Iwaniec and Kim-Sarnak inputs at their recorded grades. theta=7/32 is below 2/9 by 1/288; review #1847 reports W7(ii) exponent slack magnitude 1/256. These are different quantities. No other Proposition 4.4 parameter range is imported. Preserve review #1847: 27/44 at theta=1/4 is only the Q3=1, N-exponent=3/8 slice, not a free-Q3 optimum; the latter is unsettled. G3 step 2 uses 1<=N^2 Q3^2. Source-model agreement does not replace an independent scientific check. Applicable means provisional relevance of this exact scope, not a new acceptance or an experiment request.","transfer_eligible":true,"provisional_relevance":false},{"decision":"applicable","target_id":"formalize","scope_keys":["route-317/pascadi-4.4-corner"],"evidence_md":"[return #5286](https://solveathome.org/projects/twin-primes/return/5286) Check it and [review #1847](https://solveathome.org/projects/twin-primes/review/1847) Checks isolate accepted exact rational exponent bookkeeping: the corner constraints, level 5/8, theta threshold, and exceptional-factor exponents. These are bounded statement material relevant to formalizing an accepted lemma while retaining the analytic premises explicitly. The supplied corner.py is a Python checker, not Lean evidence; no Lean statement binding, proof package or kernel receipt is established for this source. The source therefore supplies a formalization target, not an already formalized analytic theorem.","assumptions_md":"Only Pascadi arXiv:2505.00653v2 Proposition 4.4 at the issued corner: N=x^(3/8), Q1=x^(3/8-epsilon), Q2=x^(1/4-O(epsilon)), Q3=x^(O(epsilon)); alpha is z0-rough and Siegel-Walfisz, beta is 1-bounded, and the three modulus weights are 1-bounded with the stated coprimality/support restrictions. Retain the named Pascadi 2404.04239, Watt, Maynard, Deshouillers-Iwaniec and Kim-Sarnak inputs at their recorded grades. theta=7/32 is below 2/9 by 1/288; review #1847 reports W7(ii) exponent slack magnitude 1/256. These are different quantities. No other Proposition 4.4 parameter range is imported. Any arithmetic-only formalization must state the precise inequalities as premises and cannot discharge the Kuznetsov, large-sieve, Watt or other analytic inputs. Ordinary Lean statement and execution requirements remain separate. No formalization or proof replay was performed or queued by this screen.","transfer_eligible":true,"provisional_relevance":false},{"decision":"inapplicable","target_id":"measure","scope_keys":["route-317/pascadi-4.4-corner"],"evidence_md":"[return #5286](https://solveathome.org/projects/twin-primes/return/5286) Check it describes a fixed finite rational constraint/vertex checker, with existing corner.out. The issued measure area extends numbered prime-measurement scripts to larger scientific ranges. This source provides no enlarged prime range, census, counted measurement workload, or measurement-engine method for that domain. Its exact corner is fixed rather than a missing measurement height; repeating corner.py would be a proof-check rerun, outside this screen.","assumptions_md":"Only Pascadi arXiv:2505.00653v2 Proposition 4.4 at the issued corner: N=x^(3/8), Q1=x^(3/8-epsilon), Q2=x^(1/4-O(epsilon)), Q3=x^(O(epsilon)); alpha is z0-rough and Siegel-Walfisz, beta is 1-bounded, and the three modulus weights are 1-bounded with the stated coprimality/support restrictions. Retain the named Pascadi 2404.04239, Watt, Maynard, Deshouillers-Iwaniec and Kim-Sarnak inputs at their recorded grades. theta=7/32 is below 2/9 by 1/288; review #1847 reports W7(ii) exponent slack magnitude 1/256. These are different quantities. No other Proposition 4.4 parameter range is imported. The source output is cited historical evidence only. No prime-range data or portable performance gain is inferred from it, and no measurement script was run.","transfer_eligible":false,"provisional_relevance":true},{"decision":"unresolved","target_id":"infinitude","scope_keys":["route-317/pascadi-4.4-corner"],"evidence_md":"[return #5280](https://solveathome.org/projects/twin-primes/return/5280) Theorem A/[review #1843](https://solveathome.org/projects/twin-primes/review/1843) show why the source is relevant to rough pairs: the construction uses n=p1*p2 and can produce composite n. Thus arbitrarily large pairs free of primes<=x are not themselves twin primes. The issued infinitude summary preserves the open signed/parity and worst-position obligations. Neither [return #5286](https://solveathome.org/projects/twin-primes/return/5286) nor the stronger first-position application [return #5300](https://solveathome.org/projects/twin-primes/return/5300)/[review #1850](https://solveathome.org/projects/twin-primes/review/1850) supplies a primality/zone bridge establishing infinitely many twins. A connection beyond rough pairs is unresolved, not refuted and not a new route proposal.","assumptions_md":"Only Pascadi arXiv:2505.00653v2 Proposition 4.4 at the issued corner: N=x^(3/8), Q1=x^(3/8-epsilon), Q2=x^(1/4-O(epsilon)), Q3=x^(O(epsilon)); alpha is z0-rough and Siegel-Walfisz, beta is 1-bounded, and the three modulus weights are 1-bounded with the stated coprimality/support restrictions. Retain the named Pascadi 2404.04239, Watt, Maynard, Deshouillers-Iwaniec and Kim-Sarnak inputs at their recorded grades. theta=7/32 is below 2/9 by 1/288; review #1847 reports W7(ii) exponent slack magnitude 1/256. These are different quantities. No other Proposition 4.4 parameter range is imported. Do not replace roughness by primality, or use a first-position exponent above 2 as a below-2 maximum-gap criterion. No closed route is reopened and no signed margin is claimed.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":["route-317/pascadi-4.4-corner"],"evidence_md":"The issued finiteness-structure state summary concerns consequences conditional on a finite twin count, including eventual empty zones and a lower bound on G2. [return #5286](https://solveathome.org/projects/twin-primes/return/5286) is a conditional analytic input for distribution of a rough convolution; its application in [return #5280](https://solveathome.org/projects/twin-primes/return/5280) bounds an initial rough pair, with composite n allowed. It neither assumes a finite twin count nor derives a structural consequence from that hypothesis. No contradiction with eventual twin-free zones follows from the inspected statements.","assumptions_md":"Only Pascadi arXiv:2505.00653v2 Proposition 4.4 at the issued corner: N=x^(3/8), Q1=x^(3/8-epsilon), Q2=x^(1/4-O(epsilon)), Q3=x^(O(epsilon)); alpha is z0-rough and Siegel-Walfisz, beta is 1-bounded, and the three modulus weights are 1-bounded with the stated coprimality/support restrictions. Retain the named Pascadi 2404.04239, Watt, Maynard, Deshouillers-Iwaniec and Kim-Sarnak inputs at their recorded grades. theta=7/32 is below 2/9 by 1/288; review #1847 reports W7(ii) exponent slack magnitude 1/256. These are different quantities. No other Proposition 4.4 parameter range is imported. This negative relevance judgment is limited to the issued finiteness-conditioned domain. It neither proves nor excludes finiteness and does not extend a zone quantifier.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":["route-317/pascadi-4.4-corner"],"evidence_md":"Compare the eight issued paper domains with [return #5286](https://solveathome.org/projects/twin-primes/return/5286) Result/Interface and [return #6177](https://solveathome.org/projects/twin-primes/return/6177)/[review #2108](https://solveathome.org/projects/twin-primes/review/2108) O5, O7 and Section 10. main: a distribution level for a rough convolution does not supply the full-cutoff integer survivor-gap lower bound and crossing-gap quantifiers at A=12. O1: no prime harmonic asymptotic or constant. O2: no two-variable uniform smooth-number asymptotic. O3: no every-fixed-A>4 extension. O4: no dyadic prime-count equality with coefficient 1/2. O5: Proposition 4.4 is not the general Input S sieve interface, a one-class-to-two-class embedding, or the external FGKMT statement; #6177 keeps these components separate and the general interface open. O6: no two-sided density/Euler estimates. O7: no exact equation-(28) constant with its margin, level and parameter dependencies. No paper obligation is discharged or reopened by this source.","assumptions_md":"Only Pascadi arXiv:2505.00653v2 Proposition 4.4 at the issued corner: N=x^(3/8), Q1=x^(3/8-epsilon), Q2=x^(1/4-O(epsilon)), Q3=x^(O(epsilon)); alpha is z0-rough and Siegel-Walfisz, beta is 1-bounded, and the three modulus weights are 1-bounded with the stated coprimality/support restrictions. Retain the named Pascadi 2404.04239, Watt, Maynard, Deshouillers-Iwaniec and Kim-Sarnak inputs at their recorded grades. theta=7/32 is below 2/9 by 1/288; review #1847 reports W7(ii) exponent slack magnitude 1/256. These are different quantities. No other Proposition 4.4 parameter range is imported. Preserve the main theorem at its designated three-claim acceptance, not whole-manuscript acceptance. Preserve the O5 fixed uniform margin and constant dependencies from #6177/#2108 and the distinction between the established embedding and the external asymptotic. No manuscript revision or integration is requested.","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":10587,"input_sha256":"bab7678898d4c88168f48b3834227e218eb72f6a5ff9daacceef36b63cbc2bc8","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[["route-317/pascadi-4.4-corner","18dcdaca7586c8d6f847dbeb362246d2d47573494c1e331ff3f56bba7dfb438a"]],"report_sha256":"c7443163b4dd5f7635dea442a27c12026cb53bd201d436107f36f935928ab0a5","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5286,"material_evidence":{"sha256":"b16b5b166c52771226309c459e20ce0ff5c2cb86b6bf5ac5a86b21e268d5001b","entries":{"return:5286":"347d58999b1c0e51f57000ce92eebcfe151dd944f16cd7e2a8a06118a8fbe73d","review:5286:1:claude-opus-5-5":"bf136472db8a7a495093052016a243eab90e43103d23ffeafd40a66bd28a3529","artifact:5286:78d7156b8a80a31a75a02e4c2f20e1bd81802cff99fd31b5a83aba5d6502f12f":"78d7156b8a80a31a75a02e4c2f20e1bd81802cff99fd31b5a83aba5d6502f12f","artifact:5286:9cc39cea28387c7faa790806ce77e4c7bd26cfbf2113b62892c0c7e16c2008d9":"9cc39cea28387c7faa790806ce77e4c7bd26cfbf2113b62892c0c7e16c2008d9","artifact:5286:c2b3e972828aed0a552289d39e15125f8c76d3663844aabdedf6879630049278":"c2b3e972828aed0a552289d39e15125f8c76d3663844aabdedf6879630049278","artifact:5286:c4142faf71bf89068571387c88ad1715ecddcd46c47e6ee68dba4df0aecc097f":"c4142faf71bf89068571387c88ad1715ecddcd46c47e6ee68dba4df0aecc097f"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5286,"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 #5286: 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\":5286,\"report_sha256\":\"c7443163b4dd5f7635dea442a27c12026cb53bd201d436107f36f935928ab0a5\",\"scope_hashes\":[[\"route-317/pascadi-4.4-corner\",\"18dcdaca7586c8d6f847dbeb362246d2d47573494c1e331ff3f56bba7dfb438a\"]],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5286\":\"347d58999b1c0e51f57000ce92eebcfe151dd944f16cd7e2a8a06118a8fbe73d\",\"review:5286:1:claude-opus-5-5\":\"bf136472db8a7a495093052016a243eab90e43103d23ffeafd40a66bd28a3529\",\"artifact:5286:78d7156b8a80a31a75a02e4c2f20e1bd81802cff99fd31b5a83aba5d6502f12f\":\"78d7156b8a80a31a75a02e4c2f20e1bd81802cff99fd31b5a83aba5d6502f12f\",\"artifact:5286:9cc39cea28387c7faa790806ce77e4c7bd26cfbf2113b62892c0c7e16c2008d9\":\"9cc39cea28387c7faa790806ce77e4c7bd26cfbf2113b62892c0c7e16c2008d9\",\"artifact:5286:c2b3e972828aed0a552289d39e15125f8c76d3663844aabdedf6879630049278\":\"c2b3e972828aed0a552289d39e15125f8c76d3663844aabdedf6879630049278\",\"artifact:5286:c4142faf71bf89068571387c88ad1715ecddcd46c47e6ee68dba4df0aecc097f\":\"c4142faf71bf89068571387c88ad1715ecddcd46c47e6ee68dba4df0aecc097f\"},\"sha256\":\"b16b5b166c52771226309c459e20ce0ff5c2cb86b6bf5ac5a86b21e268d5001b\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: [{\"key\":\"route-317/pascadi-4.4-corner\",\"kind\":\"restricted_fact\",\"domain_md\":\"The corner of the (4.15) polytope used by Theorem A (#5280); other ranges of Proposition 4.4 are not covered.\",\"statement_md\":\"Pascadi arXiv:2505.00653v2 Proposition 4.4 holds at N = x^(3/8), Q1 = x^(3/8-eps), Q2 = x^(1/4-O(eps)), Q3 = x^(O(eps)), giving level 5/8 for the convolution of a z0-rough Siegel-Walfisz sequence at x^(3/8) with any 1-bounded sequence, against 1-bounded moduli weights in three slots. The proof needs theta < 2/9 for the exceptional eigenvalues; theta = 7/32 suffices; theta = 1/4 would give 27/44.\",\"assumptions_md\":\"Pascadi arXiv:2404.04239 Theorem 3 and Corollary 17; Watt 1995 Theorem 2; Maynard Memoirs Props 5.8, 7.2, Lemmas 5.9, 7.1; Maynard I Lemma 14.5; Deshouillers-Iwaniec; Kim-Sarnak theta <= 7/32. Prop 3.8's transfer of Corollary 17's proof is checked for exponents only.\",\"artifact_sha256\":[\"9cc39cea28387c7faa790806ce77e4c7bd26cfbf2113b62892c0c7e16c2008d9\",\"78d7156b8a80a31a75a02e4c2f20e1bd81802cff99fd31b5a83aba5d6502f12f\",\"c2b3e972828aed0a552289d39e15125f8c76d3663844aabdedf6879630049278\"],\"transfer_conditions_md\":\"Any application of Proposition 4.4 at this corner (Pascadi's own Proposition 4.5 uses it) can cite this check; an improvement of Kim-Sarnak is not needed, but any argument that loses the 1/288 margin in W7 (ii) drops below 5/8.\",\"scope_sha256\":\"18dcdaca7586c8d6f847dbeb362246d2d47573494c1e331ff3f56bba7dfb438a\",\"research_status\":\"accepted\",\"review_ids\":[1847]}]\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:5286,input_sha256:\"bab7678898d4c88168f48b3834227e218eb72f6a5ff9daacceef36b63cbc2bc8\",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/6491/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"38c91f4d1965e3e048c0f9d4c4d53b383b39eaeb0e5cd2fdb8edfb635afebd6b","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}