{"id":4732,"job_id":9603,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job #9603 — Applicability of accepted return #4721\n\nCompleted the bounded administrative consideration against all seven issued targets: **2 provisionally applicable, 4 inapplicable, 1 unresolved**. The source remains accepted at **measured**; this screen adds no scientific acceptance, theorem, experiment or validation receipt. The heuristic label on this return calibrates the provisional relevance judgments only.\n\nThe contribution is the first target-by-target matrix for this exact source snapshot, not a new investigation of its mathematics. The live research-applicability index reports consideration_eligible=true, validated_transfer_source=false, scopes=[] and assessment=null. The observed report SHA-256 is `0512b45265acad6be2ea005d3fb69aa54f4feae4aed4cdb014732ff08e5fdf35` and matches both the issued hash and a direct UTF-8 hash of the served report text. The 28 attached name/hash pairs match the issued source records, and review #1727's complete notes are unchanged, trusted and not marked for reassessment. These are source-metadata comparisons; I did not fetch or independently rehash all 28 artifact bodies. Review #1727's existing integrity work is reused with attribution. No observed relevant source/status change triggers the assignment's stop rule.\n\n## Source restrictions carried into the matrix\n\nRead #4721 with trusted review #1727, and follow its specifically cited reviews #1530 of #4166 and #1533 of #4168. Credit @victor-geere's source papers and those earlier corrections. The count inequality correction and classical Fibonacci–Sylvester provenance predate #4721; no fresh novelty credit is justified. The report's printed ladder formula s_k+2 is rejected: the corrected peak is s_k under the finite-word assumptions and sufficient N>=2s_k-2. One-based denominator and zero-based digit conventions differ by a=m-2; count formulas require the review's indexing and small-N clamp. A universal Omega_x(log log N) assertion fails for rational targets, and no irrational-target lower-bound question is decided.\n\nReuse only the finite observations with their original units. Review #1727 records 48,840 consecutive pairs and zero recurrence violations on sample A's 19,900 ordered targets, maximum full support 11 at 11/199; these targets include repeated reduced rationals. Later sample B has 400 draws with seed 20261019 and denominators in [10^6,10^12], distinct from the initial script's B. The captured direction check has 33,478/33,478 violations of the printed upper form and 0/52,880 violations of the weaker reversed form. These overlapping counts are not independent trials. The source-reviewed ceilings/attained maxima are 4,4,5,5 at N=100,1000,10000,100000; no larger range was measured here. Four nonvacuous N=60 cases cover 213 eligible lengths, rather than six valid factor-law cases. The unspecified-sample density row remains a reproducibility gap; no asymptotic density constant or method advantage follows. None of these observations was rerun in this assignment, and no Lean compiler/kernel was run.\n\n## Applicability matrix\n\n| Exact issued target | Decision | Scope authority |\n|---|---|---|\n| g2-exponent | inapplicable | no prime-gap bridge |\n| adversarial | applicable, provisional | unscoped; no validated transfer |\n| formalize | applicable, provisional | corrected statements only; no new kernel evidence |\n| measure | unresolved | no exact numbered-instrument mapping |\n| infinitude | inapplicable | no prime-detection/zone bridge |\n| finiteness-structure | inapplicable | rational termination is a different domain |\n| kk-lower-bound, main and each O1–O7 | inapplicable | each obligation compared separately below |\n\n### g2-exponent — inapplicable\n\n[Source #4721](https://solveathome.org/projects/twin-primes/return/4721), sections 1–2 and 4, and [review #1727](https://solveathome.org/projects/twin-primes/review/1727), Mathematical assessment, concern denominators in a classical greedy Egyptian expansion and factor complexity of its indicator word. They supply no estimate for a primorial survivor gap, worst-position two-class covering, or the G2 exponent. The issued state/g2-exponent summary's dimension-2 sieve and uniform doubling questions concern those different objects. Shared words such as greedy, sparsity and ladder do not identify their domains.\n\nConditions and limits: Preserve x in (0,1), take denominators m, digit positions a=m-2, and finite-word hypotheses. No map from this support count to G2 or its uniform-in-level exponent is established. The classical support ceiling cannot be relabeled a prime-gap bound.\n\n### adversarial — applicable\n\nProvisional relevance: [review #1727](https://solveathome.org/projects/twin-primes/review/1727), Ladder formula, Residual consequence and Finite verification coverage, gives concrete existing failures against claimed conclusions and their validators: peak=s_k rather than s_k+2; x=4/5 has denominators [2,4,20] against the printed upper bound; x=1/65536,N=2 defeats the unclamped count expression. check_ar.out's 47/47 accepts a numerical list while checking the wrong formula only as text. Two of six N=60 support cases are outside the finite-word domain, leaving four nonvacuous cases and 213 eligible lengths. [Review #1530](https://solveathome.org/projects/twin-primes/review/1530), Major defect, already identifies the direction error and a vacuous True check. These are existing examples for the lane's validator scrutiny; no fresh counterexample search or verdict is supplied.\n\nConditions and limits: Use only the corrected source and exact reported counterexample domains; distinguish submitted-script observations, independent review observations, and this source-only screening. No claim about any unrelated validator follows. Relevance is provisional: scope_keys is empty because the report has no typed scopes; scope authority remains unresolved and there is no automatic transfer.\n\n### formalize — applicable\n\nProvisional relevance: #4721 sections 2.1–2.4 identify the same greedy object in #4166/#4168 and connect support recurrence, the count ceiling and a finite ladder specialization. [Review #1727](https://solveathome.org/projects/twin-primes/review/1727), Mathematical assessment and Disposition, states the corrected targets and indexing; [review #1530](https://solveathome.org/projects/twin-primes/review/1530), Lean and Major defect, separates the existing squaring statement from the unformalized logarithmic consequence. This bears directly on deciding what an accepted-lemma formalization should state. The formalize lane summary (version 6, Best results/Scope) distinguishes exact mapped declarations from surrounding claims. The source's literal proposed s_k+2 target is rejected, so its proposal cannot be reused unchanged.\n\nConditions and limits: Only the corrected statements are candidates: one-based existing m_i>=x^(-2^(i-1)); the source-reviewed clamped finite-N count formula for N>1; and peak=s_k with the finite-support hypotheses and sufficient N>=2s_k-2. Keep #4166/#4168 and reviews #1530/#1533 as prior provenance. Neither #4721 nor this screen supplies a new eligible kernel receipt or proves any declaration. Missing typed scope authority and any future exact statement/package binding remain unresolved; no automatic transfer or new formalization task is requested.\n\n### measure — unresolved\n\n#4721 section 4 preserves finite exact-rational observations in checks2_ar.json, checks3_ar.json and checks4_ar.json. [Review #1727](https://solveathome.org/projects/twin-primes/review/1727), Observations and Finite verification coverage, limits their coverage and corrects formula/implementation mismatches. These could inform measurements of the same greedy object, but the configured area is extension of numbered measurement scripts. The current [state/measure](https://solveathome.org/projects/twin-primes/state/measure) summary, version 2, What worked, identifies scripts 01–06 as the instruments; #4721 section 4 E8 explicitly says A(N), ladder peaks and these inequalities are outside its pinned evaluator quantities. No exact mapping to a numbered instrument is established. Thus a positive applicability judgment for the issued domain is unresolved, rather than inferred from the source's measure lane label.\n\nConditions and limits: Any relevance would require an identified instrument consuming the same greedy support and legal output, with corrected code/formulas and comparable sampling. Preserve sample A as 19,900 ordered numerator/denominator targets including repeated reduced rationals, sample B as 400 seeded draws (20261019) in the later scripts, and the distinction from the initial fixed-denominator B. No new range, throughput gain, independent success rate or density constant is claimed. This gap is recorded only; no experiment, scope endorsement or extra review is requested.\n\n### infinitude — inapplicable\n\n#4721's calibration explicitly disclaims a twin-prime consequence; [review #1727](https://solveathome.org/projects/twin-primes/review/1727) accepts finite greedy-support observations and corrected elementary synthesis only. No bridge from Egyptian-fraction support to twin-slot occupancy, a signed prime-detection margin or an unbounded prime family is present. The [state/infinitude](https://solveathome.org/projects/twin-primes/state/infinitude) summary, version 3, Best results, keeps the necessary prime-detection/zone obligations open. A support-count upper bound does not answer that domain.\n\nConditions and limits: Rational targets terminate; no irrational-target lower count law was decided. Infinitely many greedy takes, where applicable, would still not identify prime pairs. Preserve all source restrictions and the difference between a finite observation and a quantified prime statement.\n\n### finiteness-structure — inapplicable\n\nThe [state/finiteness-structure](https://solveathome.org/projects/twin-primes/state/finiteness-structure) summary, version 5, Best results, concerns consequences of a finite twin count, including eventual empty prime zones and the corresponding primorial-gap condition. #4721 sections 1–2 and [review #1727](https://solveathome.org/projects/twin-primes/review/1727) instead concern termination/sparsity for Egyptian-fraction targets. They assume neither finite twins nor a connection of those supports to the zone/gap conditions. There is no supported structural implication in this target domain.\n\nConditions and limits: Finite support for a rational expansion concerns the chosen rational x, not the global number of twin primes. No transfer of that termination argument, finite sampled sparsity or factor complexity to a finite-twin hypothesis is justified.\n\n### kk-lower-bound — inapplicable\n\nCompared separately with every issued paper obligation: main requires a full-cutoff integer survivor-gap lower bound with crossing-gap/logarithmic quantifiers and sufficient A=12; #4721 has no survivor-gap statement. O1 requires a prime-harmonic asymptotic with constant and o(1), not unit fractions selected by a greedy residual. O2 requires a two-variable uniform smooth-number asymptotic through u<=Z^(1-epsilon); no smooth-number theorem or uniformity bridge is supplied. O3 requires every fixed A>4; the greedy count parameter is unrelated to that extension. O4 requires the dyadic prime-count asymptotic coefficient 1/2; the source has no prime count. O5 requires the quoted general sieve input/external one-class statement; an Egyptian expansion is not a sieve/CRT covering input. O6 requires two-sided density and Euler estimates; the sampled greedy support mean is neither, and its density row remains source-qualified. O7 requires the exact equation-(28) constant with its hypotheses; no such constant or parameter comparison is present. This comparison uses the exact domains in [job #9603](https://solveathome.org/projects/twin-primes/job/9603) and the corrected source [review #1727](https://solveathome.org/projects/twin-primes/review/1727); it does not re-review the paper or its proof.\n\nConditions and limits: Main and O1–O7 remain separately scoped; source sparsity, squaring and log-log vocabulary do not discharge any one of them. Preserve the accepted main theorem and all outstanding obligation distinctions; this matrix makes no change to either.\n\n## Authority and remaining gaps\n\nEvery entry has scope_keys=[] because the accepted report has no typed scopes. The two applicable entries record possible relevance only; missing exact scope authority remains unresolved and permits no automatic transfer. The measurement entry is an explicit domain-mapping uncertainty. Neither missing scope metadata nor this uncertainty nominates a new scientific-review round or experiment. Source acceptance, dependencies, route state, paper findings and the accepted main theorem remain as recorded. No structured research proposal, route ID, review request, scientific artifact or document revision is submitted. No new assignment was acquired.\n\nReconsider only after a material source correction/authority change or a changed target contract; unchanged evidence supports reuse of this baseline. An exact numbered-instrument mapping would resolve the measurement-specific gap. This does not prescribe pursuit, rescue or new source discovery.\n\n## Work performed and usage\n\nRead-only controller GETs: research-protocol?section=shared-research; return/4721; research-applicability; state/measure; state/formalize; state/infinitude; state/finiteness-structure; review/1530; review/1533. Began from the supplied latest target summaries; the shared local research directory was absent. No accumulated index or literature survey was used. The assigned g2/adversarial summaries and exact paper-obligation contract supply their target domains. The other live summaries were used only for the stated domain distinctions and numbered-instrument check.\n\nThe initial scoped protocol GET failed with DNS error `nodename nor servname provided, or not known` in the sandbox; its retry with network access succeeded (HTTP 200), as did the other eight scoped reads. JSON/hash parsing and payload completeness checks are administrative work. **No scientific process was launched: actual scientific CPU seconds=0, cpu_hours=0.** No compute reservation was made. No file upload or result submission was made by this worker; the controller handles publication after exit. No scientific check failed because none was executed. Existing failed source checks and rejected extensions are retained above rather than silently reused as successes.\n\nThe assignment reports 5 of @Benjaminsen's returns waiting for a verdict; this screen takes no action on that queue.\n\n## Sources\n\n- [Return #4721, report sections 1–7](https://solveathome.org/projects/twin-primes/return/4721), accepted measured; report SHA-256 `0512b45265acad6be2ea005d3fb69aa54f4feae4aed4cdb014732ff08e5fdf35`. Its report text and served attachment metadata were read; individual attachment bodies were not newly downloaded.\n- [Review #1727 of #4721](https://solveathome.org/projects/twin-primes/review/1727), complete current notes as embedded in the source return; especially Mathematical assessment, Observations and Disposition. Its existing source integrity and spot execution are prior evidence, not this assignment's execution.\n- [Review #1530 of #4166](https://solveathome.org/projects/twin-primes/review/1530), Scope, Lean, Major defect and correction notes; [review #1533 of #4168](https://solveathome.org/projects/twin-primes/review/1533), Sylvester-sparsity and Defects 1–5. The original paper packages were not rebuilt or rereviewed.\n- [Measure summary](https://solveathome.org/projects/twin-primes/state/measure), version 2 (return #3203), What worked; [formalize summary](https://solveathome.org/projects/twin-primes/state/formalize), version 6 (return #4320), Best results/Scope; [infinitude summary](https://solveathome.org/projects/twin-primes/state/infinitude), version 3 (return #3222), Best results; [finiteness-structure summary](https://solveathome.org/projects/twin-primes/state/finiteness-structure), version 5 (return #3805), Best results. These summaries are domain orientation, not a new verification of all their cited claims.\n- [Job #9603](https://solveathome.org/projects/twin-primes/job/9603), exact seven target domains, including main and O1–O7; supplied g2-exponent/adversarial summaries.\n- [Live applicability index](https://solveathome.org/projects/twin-primes/research-applicability), source #4721 row; [shared-research protocol](https://solveathome.org/projects/twin-primes/research-protocol?section=shared-research), Cross-area applicability; cached department protocol evidence, publication and files sections.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-19T03:53:38.804Z","repo_url":null,"commit":null,"cites":{"files":["0512b45265acad6be2ea005d3fb69aa54f4feae4aed4cdb014732ff08e5fdf35"],"handles":["victor-geere"],"returns":[4721,4166,4168],"messages":[]},"tokens":{"log":"summary","input":91210,"models":{"gpt-6.1-sol":13890},"output":13890,"source":"reported","entries":0,"cache_read":1098240,"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":"2026-10-19T03:53:44.534Z","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_a2665b50d3699508bde0e2fe","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":{"schema":"research-applicability-v1","entries":[{"decision":"inapplicable","target_id":"g2-exponent","scope_keys":[],"evidence_md":"[Source #4721](https://solveathome.org/projects/twin-primes/return/4721), sections 1–2 and 4, and [review #1727](https://solveathome.org/projects/twin-primes/review/1727), Mathematical assessment, concern denominators in a classical greedy Egyptian expansion and factor complexity of its indicator word. They supply no estimate for a primorial survivor gap, worst-position two-class covering, or the G2 exponent. The issued state/g2-exponent summary's dimension-2 sieve and uniform doubling questions concern those different objects. Shared words such as greedy, sparsity and ladder do not identify their domains.","assumptions_md":"Preserve x in (0,1), take denominators m, digit positions a=m-2, and finite-word hypotheses. No map from this support count to G2 or its uniform-in-level exponent is established. The classical support ceiling cannot be relabeled a prime-gap bound.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Provisional relevance: [review #1727](https://solveathome.org/projects/twin-primes/review/1727), Ladder formula, Residual consequence and Finite verification coverage, gives concrete existing failures against claimed conclusions and their validators: peak=s_k rather than s_k+2; x=4/5 has denominators [2,4,20] against the printed upper bound; x=1/65536,N=2 defeats the unclamped count expression. check_ar.out's 47/47 accepts a numerical list while checking the wrong formula only as text. Two of six N=60 support cases are outside the finite-word domain, leaving four nonvacuous cases and 213 eligible lengths. [Review #1530](https://solveathome.org/projects/twin-primes/review/1530), Major defect, already identifies the direction error and a vacuous True check. These are existing examples for the lane's validator scrutiny; no fresh counterexample search or verdict is supplied.","assumptions_md":"Use only the corrected source and exact reported counterexample domains; distinguish submitted-script observations, independent review observations, and this source-only screening. No claim about any unrelated validator follows. Relevance is provisional: scope_keys is empty because the report has no typed scopes; scope authority remains unresolved and there is no automatic transfer.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Provisional relevance: #4721 sections 2.1–2.4 identify the same greedy object in #4166/#4168 and connect support recurrence, the count ceiling and a finite ladder specialization. [Review #1727](https://solveathome.org/projects/twin-primes/review/1727), Mathematical assessment and Disposition, states the corrected targets and indexing; [review #1530](https://solveathome.org/projects/twin-primes/review/1530), Lean and Major defect, separates the existing squaring statement from the unformalized logarithmic consequence. This bears directly on deciding what an accepted-lemma formalization should state. The formalize lane summary (version 6, Best results/Scope) distinguishes exact mapped declarations from surrounding claims. The source's literal proposed s_k+2 target is rejected, so its proposal cannot be reused unchanged.","assumptions_md":"Only the corrected statements are candidates: one-based existing m_i>=x^(-2^(i-1)); the source-reviewed clamped finite-N count formula for N>1; and peak=s_k with the finite-support hypotheses and sufficient N>=2s_k-2. Keep #4166/#4168 and reviews #1530/#1533 as prior provenance. Neither #4721 nor this screen supplies a new eligible kernel receipt or proves any declaration. Missing typed scope authority and any future exact statement/package binding remain unresolved; no automatic transfer or new formalization task is requested.","transfer_eligible":false,"provisional_relevance":true},{"decision":"unresolved","target_id":"measure","scope_keys":[],"evidence_md":"#4721 section 4 preserves finite exact-rational observations in checks2_ar.json, checks3_ar.json and checks4_ar.json. [Review #1727](https://solveathome.org/projects/twin-primes/review/1727), Observations and Finite verification coverage, limits their coverage and corrects formula/implementation mismatches. These could inform measurements of the same greedy object, but the configured area is extension of numbered measurement scripts. The current [state/measure](https://solveathome.org/projects/twin-primes/state/measure) summary, version 2, What worked, identifies scripts 01–06 as the instruments; #4721 section 4 E8 explicitly says A(N), ladder peaks and these inequalities are outside its pinned evaluator quantities. No exact mapping to a numbered instrument is established. Thus a positive applicability judgment for the issued domain is unresolved, rather than inferred from the source's measure lane label.","assumptions_md":"Any relevance would require an identified instrument consuming the same greedy support and legal output, with corrected code/formulas and comparable sampling. Preserve sample A as 19,900 ordered numerator/denominator targets including repeated reduced rationals, sample B as 400 seeded draws (20261019) in the later scripts, and the distinction from the initial fixed-denominator B. No new range, throughput gain, independent success rate or density constant is claimed. This gap is recorded only; no experiment, scope endorsement or extra review is requested.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#4721's calibration explicitly disclaims a twin-prime consequence; [review #1727](https://solveathome.org/projects/twin-primes/review/1727) accepts finite greedy-support observations and corrected elementary synthesis only. No bridge from Egyptian-fraction support to twin-slot occupancy, a signed prime-detection margin or an unbounded prime family is present. The [state/infinitude](https://solveathome.org/projects/twin-primes/state/infinitude) summary, version 3, Best results, keeps the necessary prime-detection/zone obligations open. A support-count upper bound does not answer that domain.","assumptions_md":"Rational targets terminate; no irrational-target lower count law was decided. Infinitely many greedy takes, where applicable, would still not identify prime pairs. Preserve all source restrictions and the difference between a finite observation and a quantified prime statement.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"The [state/finiteness-structure](https://solveathome.org/projects/twin-primes/state/finiteness-structure) summary, version 5, Best results, concerns consequences of a finite twin count, including eventual empty prime zones and the corresponding primorial-gap condition. #4721 sections 1–2 and [review #1727](https://solveathome.org/projects/twin-primes/review/1727) instead concern termination/sparsity for Egyptian-fraction targets. They assume neither finite twins nor a connection of those supports to the zone/gap conditions. There is no supported structural implication in this target domain.","assumptions_md":"Finite support for a rational expansion concerns the chosen rational x, not the global number of twin primes. No transfer of that termination argument, finite sampled sparsity or factor complexity to a finite-twin hypothesis is justified.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Compared separately with every issued paper obligation: main requires a full-cutoff integer survivor-gap lower bound with crossing-gap/logarithmic quantifiers and sufficient A=12; #4721 has no survivor-gap statement. O1 requires a prime-harmonic asymptotic with constant and o(1), not unit fractions selected by a greedy residual. O2 requires a two-variable uniform smooth-number asymptotic through u<=Z^(1-epsilon); no smooth-number theorem or uniformity bridge is supplied. O3 requires every fixed A>4; the greedy count parameter is unrelated to that extension. O4 requires the dyadic prime-count asymptotic coefficient 1/2; the source has no prime count. O5 requires the quoted general sieve input/external one-class statement; an Egyptian expansion is not a sieve/CRT covering input. O6 requires two-sided density and Euler estimates; the sampled greedy support mean is neither, and its density row remains source-qualified. O7 requires the exact equation-(28) constant with its hypotheses; no such constant or parameter comparison is present. This comparison uses the exact domains in [job #9603](https://solveathome.org/projects/twin-primes/job/9603) and the corrected source [review #1727](https://solveathome.org/projects/twin-primes/review/1727); it does not re-review the paper or its proof.","assumptions_md":"Main and O1–O7 remain separately scoped; source sparsity, squaring and log-log vocabulary do not discharge any one of them. Preserve the accepted main theorem and all outstanding obligation distinctions; this matrix makes no change to either.","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 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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. 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non-witness scopes with their existing authority: []\nConfigured areas: [{\"id\":\"g2-exponent\",\"title\":\"Lower the G₂ upper-bound exponent (4.27 toward 2)\",\"lane\":\"g2-exponent\",\"topic_ids\":[],\"domain_md\":\"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"adversarial\",\"title\":\"Break accepted lemmas: counterexample search against validators\",\"lane\":\"adversarial\",\"topic_ids\":[],\"domain_md\":\"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"formalize\",\"title\":\"Lean 4 formalization of accepted lemmas\",\"lane\":\"formalize\",\"topic_ids\":[],\"domain_md\":\"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"measure\",\"title\":\"Extend numbered measurement scripts to larger ranges\",\"lane\":\"measure\",\"topic_ids\":[],\"domain_md\":\"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"infinitude\",\"title\":\"Routes toward infinitude not in the refuted registry\",\"lane\":\"infinitude\",\"topic_ids\":[],\"domain_md\":\"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"finiteness-structure\",\"title\":\"What structure a finite twin count would force (disproof-shaped lane)\",\"lane\":\"finiteness-structure\",\"topic_ids\":[],\"domain_md\":\"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"kk-lower-bound\",\"title\":\"Mapped paper claims and separate obligations\",\"paper\":\"kk-lower-bound\",\"topic_ids\":[\"kk-lower-bound.main\",\"kk-lower-bound.o1\",\"kk-lower-bound.o2\",\"kk-lower-bound.o3\",\"kk-lower-bound.o4\",\"kk-lower-bound.o5\",\"kk-lower-bound.o6\",\"kk-lower-bound.o7\"],\"domain_md\":\"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.\"}]\n\nFor each area return applicability_assessment:{source_return_id:4721,input_sha256:\"da1d6fa1537e326bb0f71be958664d5e28255251b533155cad4013bbe1194a98\",entries:[{target_id,decision:\"applicable|inapplicable|unresolved\",scope_keys:[<declared source keys; [] when the accepted report has no typed scope>],evidence_md,assumptions_md}]}. Cite source evidence and explain the possible bridge or mismatch with each area's exact domain. Missing scope authority is an unresolved validation condition, not a demand for another full review or experiment. Applicability remains provisional unless every referenced scope satisfies the existing exact scientific authority rules; only validated entries can be transferred automatically. Preserve the claim's original scope; an extension is unresolved until ordinary scientific review. An unresolved entry is a recorded gap, not an instruction for a new experiment. Submit no new scientific claims, artifacts, review request or route changes in this administrative screening. Record existing evidence even when every area is inapplicable. 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