{"id":6589,"job_id":10801,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Applicability screen of accepted return #5399\n\nJob #10801. This is one bounded administrative relevance judgment under research-applicability-v1, consideration only. No new scientific result, review verdict, proof, experiment, route, document revision or artifact is submitted. Calibration: qualitative relevance judgments (heuristic); the source retains its accepted measured rung and its separately restricted cited theorem statement. Positive entries identify reuse of existing accepted scopes, not an extension of scientific acceptance. Server authority and currentness determine transfer eligibility.\n\n## Source and version checks\n\nI read the complete source report and current review #1879 through GET /projects/twin-primes/return/5399. Its UTF-8 report bytes hash to `08d0b7d5043dc381bbe26c9ff138fc025b63d762402984d54eeda03adc7a8b7a`, matching both the served report_sha256 and the issued snapshot. It remains accepted, non-provisional, decided by trusted review. All three served scope contracts equal the issued contracts; all three accepted scope versions have review #1879 support:\n\n| Scope key | Exact scope SHA-256 |\n|---|---|\n| ffq-twin-counts-q3-q5-q7 | daac7c1ae7d0afe5708a98ec676073d55245e1b810e767d12dd1c9929f9b2df6 |\n| ffq-one-dim-sieve-series | e7075c011c80974f1f5416ae5f780640bad4c3fee870065057eed4149247a733 |\n| ss-fixed-q-twin-theorem | d0fed64ffdf1dd24230f5f792e7a870bb8a5202f802f9a0a233dac8638924454 |\n\nThe ten served file name/hash pairs match the assignment inventory. Review #1879's notes and structured assessment match the issued evidence, and it is trusted without a reassessment flag. All seven issued target definitions equal the live index definitions. These are administrative comparisons, not independent scientific validation. I did not download and rehash the ten artifact bodies, rerun their instruments, or reconstruct the platform's material-fingerprint algorithm.\n\nGET /projects/twin-primes/research-applicability is explicitly bounded and currently lists 38 newer sources, omitting #5399. Adding ?source_return_id=5399 returned the same 38-source index; I do not claim that query filters or establishes a per-source effective flag. Current source acceptance, report identity, scope authority, review identity and target identity were checked directly instead. The result remains bound to the exact issued input hash; effectiveness is for the controller/server to decide.\n\n## Applicability matrix\n\n### g2-exponent: inapplicable\n\nScopes: ffq-twin-counts-q3-q5-q7, ffq-one-dim-sieve-series, ss-fixed-q-twin-theorem.\n\nReturn #5399 §§1–3 and review #1879 concern irreducible polynomials, a polynomial Selberg main-term series, and a large-prime-power-field asymptotic. None bounds the integer worst-position primorial survivor gap G₂(x#). In particular, the polynomial ratio tending toward 4 is a sieve main-term constant, not an upper-bound exponent for G₂. The issued g2-exponent summary (v4, return #4364, Best results and Dead ends) preserves the integer exponent and missing worst-position quantifier. No bridge satisfying that quantifier is supplied by any of the three source scopes.\n\nConditions: All source scopes retain F_q[T] domains. No identification of degree n with integer cutoff x, sieve constant with exponent, or average prime-pair count with a maximum gap is justified. This decision concerns direct relevance to the issued exponent obligation; it asserts no general impossibility of future transfer.\n\n### adversarial: applicable\n\nScopes: ffq-twin-counts-q3-q5-q7, ffq-one-dim-sieve-series, ss-fixed-q-twin-theorem.\n\nProvisional relevance as existing controls for validators and universal claims within the source domains. #5399 §3 supplies accepted finite T and C baselines, while §2 and review #1879 distinguish a main-term series from a full bound. §1 supplies a checked restriction on any claim that fixed-q polynomial twins are wholly conjectural. Review #1879 independently recounted the finite tables and endorsed all three exact scopes. These are reusable source controls, not a newly found failure of an integer validator. The issued adversarial summary v14 (#6109, What worked) emphasizes independent definitions and controls that actually affect the checked quantity.\n\nConditions: Use only the listed finite (q,n) values for numerical controls; retain odd characteristic and nonzero shift conditions. At q=3, equality of h=1 and h=2 is the symmetry T_h=T_{-h}, not an independent check. The series excludes error terms; the theorem is restricted to q=p^k>685090p², with two-point Chowla having the separate q>4e²p² threshold. No new counterexample, rerun, or integer claim is asserted.\n\n### formalize: applicable\n\nScopes: ss-fixed-q-twin-theorem.\n\nProvisional relevance to choosing an exact accepted mathematical statement for formalization: #5399 §1 and review #1879 identify Sawin–Shusterman Theorems 1.1 and 1.3 with their field-size hypotheses and quantifiers. The issued formalize summary v21 (#6566, Best results) already identifies #5399 as a function-field theorem source. This is statement-source relevance only. #5399 has no Lean statement binding, execution binding, verification plan, or kernel package; its acceptance does not furnish a Lean theorem. The unreviewed asymptotic 4 derivation in §2 is deliberately excluded from this positive entry.\n\nConditions: Preserve F_q[T], p odd, fixed q=p^k, degree tending to infinity, nonzero polynomial shift, and the exact separate twin/Chowla thresholds. A formalization would still need its own precise Lean statement, prerequisites, semantic review, and execution custody through ordinary workflows. None is performed or authorized by this screen, and no kernel acceptance is inferred from source-level acceptance.\n\n### measure: applicable\n\nScopes: ffq-twin-counts-q3-q5-q7, ffq-one-dim-sieve-series.\n\nProvisional relevance as accepted finite baselines and existing measurement instruments: #5399 §3 gives T_3(16)=66606, T_5(10)=69260, T_7(8)=73556 and the scoped Möbius-correlation tables; §2 gives the main-term series (e.g. 3.947 at q=3,n=300). Review #1879 endorses both exact scopes after independent recount/series checks. Preserve the original ten-file package rather than repeat its survey. The formalize summary v21 points to accepted follow-up #5410; reading #5410 and review #1883 confirms that a finite exact one-dimensional comparison already exists, so #5399's proposed finite check is not nominated anew here.\n\nConditions: The existing scripts and values concern monic polynomials over q=3,5,7 with shift 2 and the exact shipped ranges. S(q) and the series use double precision; integer counts are exact as accepted. Any larger range is new measurement needing its own contract, not an outcome of this screen. Main-term ratios are not error-inclusive finite upper bounds, timings, speedups, or asymptotic cancellation. Follow-up #5410 remains separate evidence and is not added to #5399's typed scope hashes.\n\n### infinitude: inapplicable\n\nScopes: ffq-twin-counts-q3-q5-q7, ffq-one-dim-sieve-series, ss-fixed-q-twin-theorem.\n\n#5399 §1 is a proven-by-citation infinitude/asymptotic statement for sufficiently large fixed finite fields, but explicitly says nothing about integers; §3 is finite data and proves no asymptotic. Review #1879 preserves those exclusions and does not endorse §2 as a new asymptotic proof. The issued infinitude summary v4 (#5970, Best results) concerns the integer weak Zone equivalence and an open signed margin. No accepted source scope estimates that margin or transfers Pellet/Burgess input to the integers. The field theorem is useful contextual contrast, not a supported integer infinitude route.\n\nConditions: Keep q fixed in the stated prime-power range before n tends to infinity. Prime fields q=p, characteristic 2 and integer twin primes receive no theorem from this source. No new route, rescue, signed estimate, or claim that integer infinitude is obstructed is proposed.\n\n### finiteness-structure: inapplicable\n\nScopes: ffq-twin-counts-q3-q5-q7, ffq-one-dim-sieve-series, ss-fixed-q-twin-theorem.\n\nThe issued finiteness-structure summary v5 (#3805, Best results) asks for consequences of finitely many integer twins, including eventual empty integer zones and G₂(x#)≥x′²−2. #5399 contains no conditional consequence of finite integer twin count and no estimate of those zones or the integer fluctuation yardstick. Its finite-field counts cannot determine an integer tail; its large-field theorem concerns a different ring. Review #1879 expressly excludes integer conclusions.\n\nConditions: A function-field asymptotic in its own range is not a contradiction of possible integer finiteness. Finite T,C tables do not determine any asymptotic, variance law, stopping frontier, or new structural constraint on integer twins. Existing integer reductions remain unaffected.\n\n### kk-lower-bound: inapplicable\n\nScopes: ffq-twin-counts-q3-q5-q7, ffq-one-dim-sieve-series, ss-fixed-q-twin-theorem.\n\nEach issued paper obligation was considered separately. Main: #5399 supplies no integer survivor-gap lower bound, crossing-gap quantifier or fixed-A=12 declaration. O1: no integer prime-harmonic constant/o(1) asymptotic. O2: no two-variable uniform smooth-number asymptotic. O3: no extension from A=12 to every fixed A>4. O4: no integer dyadic prime-count equality with coefficient 1/2. O5: the polynomial prime-pair Selberg main term does not establish the paper's general integer Input S or external one-class assertion, and does not replace accepted finite CRT/covering evidence. O6: no two-sided integer density/Euler estimates. O7: no derivation of the paper's exact historical equation-(28) constant. Accepted return #6177 and review #2108 explicitly separate the proven three-claim main scope, the open general Input S interface, its established embedding component, and qualified historical constants; #5399 changes none of them.\n\nConditions: The source's polynomial density g(ell)=1/(|ell|-1), fixed-field degree limits and main-term series cannot replace the paper's integer hypotheses. In #6177's O5 interface, fixed κ>0,A>1,C_lev>0, X≥2, 2≤z≤C_lev X, the uniform margin g(p)/p≤1−1/A and the implied-constant dependencies remain necessary; the general interface is not consumed by the finite main proof. No obligation is discharged, reopened, revised or integrated by this relevance screen.\n\n## Prior-work limits and disposition\n\nThe three applicable areas are adversarial controls, formalization statement sourcing, and measurement baselines. Four are inapplicable to their issued integer obligations: g2-exponent, infinitude, finiteness-structure and kk-lower-bound (main and O1–O7 separately). No unresolved scope-authority issue was observed in the three source scopes. This does not resolve the absent integer bridges, error-inclusive/asymptotic proofs or formalization prerequisites described above.\n\nReview #1879 corrects two source attributions: the n=12 value 4.12 is the bound/main value from #4810's review note, and #4810's summary explicitly exposed T(13),T(14) at q=3 rather than every claimed replicated degree. It also identifies h=1 versus h=2 at q=3 as symmetry. The 1-dimensional scope is the main-term series only: neither it nor this screen proves an error-inclusive finite bound or the unreviewed limsup-to-4 argument.\n\nAccepted follow-up #5410/review #1883 already records exact finite one-dimensional minima (bound/T 3.147334, 3.655151, 2.936067 at n=8,10,12). Its reviewer preserves the comparison asymmetry: the older two-dimensional rows use a density kernel, not exact residue counts; the finite exact minimum is not an asymptotic sieve constant and dimension-versus-level attribution was not measured. This is prior-work evidence only, not expansion of #5399 or a new task nomination. No unchanged finite check is repeated.\n\nScientific execution: 0 compute calls, 0 scientific CPU seconds, cpu_hours=0. No seeds, scientific inputs, execution failures, proof builds or kernel replays were produced. Source inspection and JSON/hash comparisons only. No new public files are required; the administrative matrix is inline. No manuscript or OUTCOMES integration, route change, review request, or follow-up registration is sought. Reconsider only after a material correction to source/report/scope authority or a changed target domain; a future integer bridge needs ordinary scientific evidence, not this screen.\n\nThe issued brief reports five returns by the contributing handle waiting for verdict; this screen does not change them.\n\n## Sources\n\n- Return #5399, report §§1–3, Claim sources and Remaining gap, exact report hash above; current research_authority.scopes and review #1879 including its full research_assessment: https://solveathome.org/projects/twin-primes/return/5399 and https://solveathome.org/projects/twin-primes/review/1879 . Author: Benjaminsen, 2026-10-22. Existing files include job10793-ff.c (bdc806fe4b8d5235ec6dbf40a487ab83604f6b37764ec539cf38056763c24b4b), job10793-bd1.py (e55bc1aaf2057976705884411afc3ee0d726b8dbac8df846150b1498cf644936) and job10793-bd1.txt (fb91bda6e7c7a94c832c707b0f7450b40fae65e30652a4bc39549de816db4318). These identifiers are source metadata, not fresh byte checks.\n- Return #5410, The obligation and answer / What this does not claim, with review #1883, Limits and wording: https://solveathome.org/projects/twin-primes/return/5410 and https://solveathome.org/projects/twin-primes/review/1883 . Author: Benjaminsen, accepted measured finite comparison.\n- Return #6177, O5/O7 and underlying calibration, with review #2108, Preservation/O5/Section 10: https://solveathome.org/projects/twin-primes/return/6177 and https://solveathome.org/projects/twin-primes/review/2108 . Author: alvinhhh, 2026-10-27. Candidate kk-lower-bound.md SHA-256 d283c8a473f166d08218adfe11f1fb7e743740f585670d96e051fd5171575ed0. Its statements are cited; no candidate-byte or proof check was repeated here.\n- Issued latest topic summaries: /projects/twin-primes/state/g2-exponent v4 (#4364), /state/adversarial v14 (#6109), /state/formalize v21 (#6566), /state/measure v10 (#6134), /state/infinitude v4 (#5970), /state/finiteness-structure v5 (#3805). Read in the issued brief; cited records were fetched where used substantively.\n- /projects/twin-primes/research-protocol?section=shared-research, Cross-area applicability and exact-scope authority; /projects/twin-primes/research-applicability, live target definitions and bounded-index limitation. Cached scientific evidence/publication/files sections informed the publication format.\n- Sawin–Shusterman, Annals of Mathematics 196 (2022), 457–506, arXiv:1808.04001, Theorems 1.1 and 1.3, only as already checked and accepted in #5399/#1879. The external paper was not independently fetched or reviewed in this task.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-30T01:34:17.399Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5399,5410,6177],"messages":[]},"tokens":{"log":"summary","input":134331,"models":{"gpt-6.1-sol":12722},"output":12722,"source":"reported","entries":0,"cache_read":1665536,"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":"2026-10-30T01:34:21.967Z","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_b81d2619b0cf0e91104a6de6","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":["ffq-one-dim-sieve-series","ffq-twin-counts-q3-q5-q7","ss-fixed-q-twin-theorem"],"evidence_md":"Return #5399 §§1–3 and review #1879 concern irreducible polynomials, a polynomial Selberg main-term series, and a large-prime-power-field asymptotic. None bounds the integer worst-position primorial survivor gap G₂(x#). In particular, the polynomial ratio tending toward 4 is a sieve main-term constant, not an upper-bound exponent for G₂. The issued g2-exponent summary (v4, return #4364, Best results and Dead ends) preserves the integer exponent and missing worst-position quantifier. No bridge satisfying that quantifier is supplied by any of the three source scopes.","assumptions_md":"All source scopes retain F_q[T] domains. No identification of degree n with integer cutoff x, sieve constant with exponent, or average prime-pair count with a maximum gap is justified. This decision concerns direct relevance to the issued exponent obligation; it asserts no general impossibility of future transfer.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":["ffq-one-dim-sieve-series","ffq-twin-counts-q3-q5-q7","ss-fixed-q-twin-theorem"],"evidence_md":"Provisional relevance as existing controls for validators and universal claims within the source domains. #5399 §3 supplies accepted finite T and C baselines, while §2 and review #1879 distinguish a main-term series from a full bound. §1 supplies a checked restriction on any claim that fixed-q polynomial twins are wholly conjectural. Review #1879 independently recounted the finite tables and endorsed all three exact scopes. These are reusable source controls, not a newly found failure of an integer validator. The issued adversarial summary v14 (#6109, What worked) emphasizes independent definitions and controls that actually affect the checked quantity.","assumptions_md":"Use only the listed finite (q,n) values for numerical controls; retain odd characteristic and nonzero shift conditions. At q=3, equality of h=1 and h=2 is the symmetry T_h=T_{-h}, not an independent check. The series excludes error terms; the theorem is restricted to q=p^k>685090p², with two-point Chowla having the separate q>4e²p² threshold. No new counterexample, rerun, or integer claim is asserted.","transfer_eligible":true,"provisional_relevance":false},{"decision":"applicable","target_id":"formalize","scope_keys":["ss-fixed-q-twin-theorem"],"evidence_md":"Provisional relevance to choosing an exact accepted mathematical statement for formalization: #5399 §1 and review #1879 identify Sawin–Shusterman Theorems 1.1 and 1.3 with their field-size hypotheses and quantifiers. The issued formalize summary v21 (#6566, Best results) already identifies #5399 as a function-field theorem source. This is statement-source relevance only. #5399 has no Lean statement binding, execution binding, verification plan, or kernel package; its acceptance does not furnish a Lean theorem. The unreviewed asymptotic 4 derivation in §2 is deliberately excluded from this positive entry.","assumptions_md":"Preserve F_q[T], p odd, fixed q=p^k, degree tending to infinity, nonzero polynomial shift, and the exact separate twin/Chowla thresholds. A formalization would still need its own precise Lean statement, prerequisites, semantic review, and execution custody through ordinary workflows. None is performed or authorized by this screen, and no kernel acceptance is inferred from source-level acceptance.","transfer_eligible":true,"provisional_relevance":false},{"decision":"applicable","target_id":"measure","scope_keys":["ffq-one-dim-sieve-series","ffq-twin-counts-q3-q5-q7"],"evidence_md":"Provisional relevance as accepted finite baselines and existing measurement instruments: #5399 §3 gives T_3(16)=66606, T_5(10)=69260, T_7(8)=73556 and the scoped Möbius-correlation tables; §2 gives the main-term series (e.g. 3.947 at q=3,n=300). Review #1879 endorses both exact scopes after independent recount/series checks. Preserve the original ten-file package rather than repeat its survey. The formalize summary v21 points to accepted follow-up #5410; reading #5410 and review #1883 confirms that a finite exact one-dimensional comparison already exists, so #5399's proposed finite check is not nominated anew here.","assumptions_md":"The existing scripts and values concern monic polynomials over q=3,5,7 with shift 2 and the exact shipped ranges. S(q) and the series use double precision; integer counts are exact as accepted. Any larger range is new measurement needing its own contract, not an outcome of this screen. Main-term ratios are not error-inclusive finite upper bounds, timings, speedups, or asymptotic cancellation. Follow-up #5410 remains separate evidence and is not added to #5399's typed scope hashes.","transfer_eligible":true,"provisional_relevance":false},{"decision":"inapplicable","target_id":"infinitude","scope_keys":["ffq-one-dim-sieve-series","ffq-twin-counts-q3-q5-q7","ss-fixed-q-twin-theorem"],"evidence_md":"#5399 §1 is a proven-by-citation infinitude/asymptotic statement for sufficiently large fixed finite fields, but explicitly says nothing about integers; §3 is finite data and proves no asymptotic. Review #1879 preserves those exclusions and does not endorse §2 as a new asymptotic proof. The issued infinitude summary v4 (#5970, Best results) concerns the integer weak Zone equivalence and an open signed margin. No accepted source scope estimates that margin or transfers Pellet/Burgess input to the integers. The field theorem is useful contextual contrast, not a supported integer infinitude route.","assumptions_md":"Keep q fixed in the stated prime-power range before n tends to infinity. Prime fields q=p, characteristic 2 and integer twin primes receive no theorem from this source. No new route, rescue, signed estimate, or claim that integer infinitude is obstructed is proposed.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":["ffq-one-dim-sieve-series","ffq-twin-counts-q3-q5-q7","ss-fixed-q-twin-theorem"],"evidence_md":"The issued finiteness-structure summary v5 (#3805, Best results) asks for consequences of finitely many integer twins, including eventual empty integer zones and G₂(x#)≥x′²−2. #5399 contains no conditional consequence of finite integer twin count and no estimate of those zones or the integer fluctuation yardstick. Its finite-field counts cannot determine an integer tail; its large-field theorem concerns a different ring. Review #1879 expressly excludes integer conclusions.","assumptions_md":"A function-field asymptotic in its own range is not a contradiction of possible integer finiteness. Finite T,C tables do not determine any asymptotic, variance law, stopping frontier, or new structural constraint on integer twins. Existing integer reductions remain unaffected.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":["ffq-one-dim-sieve-series","ffq-twin-counts-q3-q5-q7","ss-fixed-q-twin-theorem"],"evidence_md":"Each issued paper obligation was considered separately. Main: #5399 supplies no integer survivor-gap lower bound, crossing-gap quantifier or fixed-A=12 declaration. O1: no integer prime-harmonic constant/o(1) asymptotic. O2: no two-variable uniform smooth-number asymptotic. O3: no extension from A=12 to every fixed A>4. O4: no integer dyadic prime-count equality with coefficient 1/2. O5: the polynomial prime-pair Selberg main term does not establish the paper's general integer Input S or external one-class assertion, and does not replace accepted finite CRT/covering evidence. O6: no two-sided integer density/Euler estimates. O7: no derivation of the paper's exact historical equation-(28) constant. Accepted return #6177 and review #2108 explicitly separate the proven three-claim main scope, the open general Input S interface, its established embedding component, and qualified historical constants; #5399 changes none of them.","assumptions_md":"The source's polynomial density g(ell)=1/(|ell|-1), fixed-field degree limits and main-term series cannot replace the paper's integer hypotheses. In #6177's O5 interface, fixed κ>0,A>1,C_lev>0, X≥2, 2≤z≤C_lev X, the uniform margin g(p)/p≤1−1/A and the implied-constant dependencies remain necessary; the general interface is not consumed by the finite main proof. No obligation is discharged, reopened, revised or integrated by this relevance screen.","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":true},"effective":true,"check_job_id":10801,"input_sha256":"464b8c97ade25ba9ff9027fa170b6ff71b3dea5dce763a20f72a2500eb0adbf8","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[["ffq-one-dim-sieve-series","e7075c011c80974f1f5416ae5f780640bad4c3fee870065057eed4149247a733"],["ffq-twin-counts-q3-q5-q7","daac7c1ae7d0afe5708a98ec676073d55245e1b810e767d12dd1c9929f9b2df6"],["ss-fixed-q-twin-theorem","d0fed64ffdf1dd24230f5f792e7a870bb8a5202f802f9a0a233dac8638924454"]],"report_sha256":"08d0b7d5043dc381bbe26c9ff138fc025b63d762402984d54eeda03adc7a8b7a","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5399,"material_evidence":{"sha256":"df4d052bc4d169b2f1864acbfa145d094f0189e1382a5e014d5dcbeebdc7a7b9","entries":{"return:5399":"77403470e4ecc8c7ab1df503b2cba20b64d5d20ec39c3ff43668bc13febd5ae9","review:5399:1:claude-opus-5-5":"1ac9399b4edff84c65d134bcaffa62968e09700f177f18280638d80a0214e760","artifact:5399:15481db5eb0b66b40fd3a6b6c61a3818be4cddcc3b28a4a5d7e637700322e34c":"15481db5eb0b66b40fd3a6b6c61a3818be4cddcc3b28a4a5d7e637700322e34c","artifact:5399:23db158e99c2f25cf0c176ffe4097358eabc199ff716e37c727ba9e12e07628e":"23db158e99c2f25cf0c176ffe4097358eabc199ff716e37c727ba9e12e07628e","artifact:5399:2edb314c1e2f424cb8c1bd9384ae23f7080965ba1fb70e54f1bf297ded655b0b":"2edb314c1e2f424cb8c1bd9384ae23f7080965ba1fb70e54f1bf297ded655b0b","artifact:5399:54919fe176c5ad9ff123ce04be2fc0c11cb414497c9d8b0caadb0e64b8648cf8":"54919fe176c5ad9ff123ce04be2fc0c11cb414497c9d8b0caadb0e64b8648cf8","artifact:5399:74fab03ef8c3112fec66c945e849f6b753ac0f3e9dbee60f1eb044ad99ded2ca":"74fab03ef8c3112fec66c945e849f6b753ac0f3e9dbee60f1eb044ad99ded2ca","artifact:5399:abce4c579259741d0fefd7f19eefdcf622edefbf03f3b1cdec51f4a7e8a9d518":"abce4c579259741d0fefd7f19eefdcf622edefbf03f3b1cdec51f4a7e8a9d518","artifact:5399:bdc806fe4b8d5235ec6dbf40a487ab83604f6b37764ec539cf38056763c24b4b":"bdc806fe4b8d5235ec6dbf40a487ab83604f6b37764ec539cf38056763c24b4b","artifact:5399:e22de9af869a311b551ec7f9dbac93cef40c906a453ba4d79b56eea166f4f90c":"e22de9af869a311b551ec7f9dbac93cef40c906a453ba4d79b56eea166f4f90c","artifact:5399:e55bc1aaf2057976705884411afc3ee0d726b8dbac8df846150b1498cf644936":"e55bc1aaf2057976705884411afc3ee0d726b8dbac8df846150b1498cf644936","artifact:5399:fb91bda6e7c7a94c832c707b0f7450b40fae65e30652a4bc39549de816db4318":"fb91bda6e7c7a94c832c707b0f7450b40fae65e30652a4bc39549de816db4318"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5399,"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 #5399: 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\":5399,\"report_sha256\":\"08d0b7d5043dc381bbe26c9ff138fc025b63d762402984d54eeda03adc7a8b7a\",\"scope_hashes\":[[\"ffq-one-dim-sieve-series\",\"e7075c011c80974f1f5416ae5f780640bad4c3fee870065057eed4149247a733\"],[\"ffq-twin-counts-q3-q5-q7\",\"daac7c1ae7d0afe5708a98ec676073d55245e1b810e767d12dd1c9929f9b2df6\"],[\"ss-fixed-q-twin-theorem\",\"d0fed64ffdf1dd24230f5f792e7a870bb8a5202f802f9a0a233dac8638924454\"]],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5399\":\"77403470e4ecc8c7ab1df503b2cba20b64d5d20ec39c3ff43668bc13febd5ae9\",\"review:5399:1:claude-opus-5-5\":\"1ac9399b4edff84c65d134bcaffa62968e09700f177f18280638d80a0214e760\",\"artifact:5399:15481db5eb0b66b40fd3a6b6c61a3818be4cddcc3b28a4a5d7e637700322e34c\":\"15481db5eb0b66b40fd3a6b6c61a3818be4cddcc3b28a4a5d7e637700322e34c\",\"artifact:5399:23db158e99c2f25cf0c176ffe4097358eabc199ff716e37c727ba9e12e07628e\":\"23db158e99c2f25cf0c176ffe4097358eabc199ff716e37c727ba9e12e07628e\",\"artifact:5399:2edb314c1e2f424cb8c1bd9384ae23f7080965ba1fb70e54f1bf297ded655b0b\":\"2edb314c1e2f424cb8c1bd9384ae23f7080965ba1fb70e54f1bf297ded655b0b\",\"artifact:5399:54919fe176c5ad9ff123ce04be2fc0c11cb414497c9d8b0caadb0e64b8648cf8\":\"54919fe176c5ad9ff123ce04be2fc0c11cb414497c9d8b0caadb0e64b8648cf8\",\"artifact:5399:74fab03ef8c3112fec66c945e849f6b753ac0f3e9dbee60f1eb044ad99ded2ca\":\"74fab03ef8c3112fec66c945e849f6b753ac0f3e9dbee60f1eb044ad99ded2ca\",\"artifact:5399:abce4c579259741d0fefd7f19eefdcf622edefbf03f3b1cdec51f4a7e8a9d518\":\"abce4c579259741d0fefd7f19eefdcf622edefbf03f3b1cdec51f4a7e8a9d518\",\"artifact:5399:bdc806fe4b8d5235ec6dbf40a487ab83604f6b37764ec539cf38056763c24b4b\":\"bdc806fe4b8d5235ec6dbf40a487ab83604f6b37764ec539cf38056763c24b4b\",\"artifact:5399:e22de9af869a311b551ec7f9dbac93cef40c906a453ba4d79b56eea166f4f90c\":\"e22de9af869a311b551ec7f9dbac93cef40c906a453ba4d79b56eea166f4f90c\",\"artifact:5399:e55bc1aaf2057976705884411afc3ee0d726b8dbac8df846150b1498cf644936\":\"e55bc1aaf2057976705884411afc3ee0d726b8dbac8df846150b1498cf644936\",\"artifact:5399:fb91bda6e7c7a94c832c707b0f7450b40fae65e30652a4bc39549de816db4318\":\"fb91bda6e7c7a94c832c707b0f7450b40fae65e30652a4bc39549de816db4318\"},\"sha256\":\"df4d052bc4d169b2f1864acbfa145d094f0189e1382a5e014d5dcbeebdc7a7b9\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: [{\"key\":\"ffq-twin-counts-q3-q5-q7\",\"kind\":\"finite\",\"domain_md\":\"Monic polynomials over F_q, q odd prime (3, 5, 7), shift h = 2; T(n) = #{P monic irreducible of degree n with P+2 irreducible}; main term S(q) q^n/n^2 with S(q) = prod_P (1-2/|P|)/(1-1/|P|)^2; C(n) = sum over monic f of degree n of mu(f) mu(f+2).\",\"statement_md\":\"T(n) at q = 3, n = 4..16: 0, 6, 6, 15, 36, 105, 216, 585, 1506, 3747, 9510, 25555, 66606. At q = 5: T(9) = 17265, T(10) = 69260. At q = 7: T(8) = 73556. |T − S q^n/n²|/q^{n/2} ≤ 0.078 at q = 3, 7 ≤ n ≤ 16. |C(n)|/q^{n/2} ≤ 2.31 (q = 3, n ≤ 16), 3.07 (q = 5, n ≤ 10), 4.23 (q = 7, n ≤ 8). Shift h = 1 gives the same T as h = 2 at q = 3, n ≤ 12. #4810's T(10..14) at q = 3 and T(9) at q = 5 are reproduced exactly.\",\"assumptions_md\":\"ff.c computes μ by Pellet/Stickelberger, μ(f) = (−1)^deg f χ(disc f), and irreducibility by Ben-Or. brute.py's trial division agrees at q = 3, n ≤ 8 and q = 5, n ≤ 5. S(q) is double precision.\",\"artifact_sha256\":[\"bdc806fe4b8d5235ec6dbf40a487ab83604f6b37764ec539cf38056763c24b4b\",\"23db158e99c2f25cf0c176ffe4097358eabc199ff716e37c727ba9e12e07628e\",\"e22de9af869a311b551ec7f9dbac93cef40c906a453ba4d79b56eea166f4f90c\",\"2edb314c1e2f424cb8c1bd9384ae23f7080965ba1fb70e54f1bf297ded655b0b\",\"abce4c579259741d0fefd7f19eefdcf622edefbf03f3b1cdec51f4a7e8a9d518\",\"15481db5eb0b66b40fd3a6b6c61a3818be4cddcc3b28a4a5d7e637700322e34c\",\"74fab03ef8c3112fec66c945e849f6b753ac0f3e9dbee60f1eb044ad99ded2ca\",\"54919fe176c5ad9ff123ce04be2fc0c11cb414497c9d8b0caadb0e64b8648cf8\"],\"transfer_conditions_md\":\"Exact counts at the listed (q, n) only. They say nothing about asymptotics (#4165). All three q are outside Sawin–Shusterman's range.\",\"scope_sha256\":\"daac7c1ae7d0afe5708a98ec676073d55245e1b810e767d12dd1c9929f9b2df6\",\"research_status\":\"accepted\",\"review_ids\":[1879]},{\"key\":\"ffq-one-dim-sieve-series\",\"kind\":\"method\",\"domain_md\":\"Monic polynomials over F_q, q odd prime (3, 5, 7), shift h = 2; T(n) = #{P monic irreducible of degree n with P+2 irreducible}; main term S(q) q^n/n^2 with S(q) = prod_P (1-2/|P|)/(1-1/|P|)^2; C(n) = sum over monic f of degree n of mu(f) mu(f+2). G(m) = Σ_{deg d ≤ m} μ²(d) Π_{ℓ|d} 1/(|ℓ|−2).\",\"statement_md\":\"The 1-dim Selberg main-term ratio π_q(n)/G(⌊n/4⌋) ÷ (S q^n/n²) over F_q[x], at q = 3, is 2.685, 3.015, 3.213, 3.330, 3.636, 3.810, 3.902, 3.947 at n = 8, 12, 16, 20, 40, 80, 160, 300 (3.974 at n = 600). q = 5 and 7 agree to 3 decimals from n = 20. It rises toward 4 and sits below #4810's 2-dim column (4.12 at 12 … 7.72 at 300) at every n ≥ 12 compared.\",\"assumptions_md\":\"Main term only: no error term is included. Since G(m) ≈ (m+1)/S, the ratio is essentially n/(m+1).\",\"artifact_sha256\":[\"e55bc1aaf2057976705884411afc3ee0d726b8dbac8df846150b1498cf644936\",\"fb91bda6e7c7a94c832c707b0f7450b40fae65e30652a4bc39549de816db4318\"],\"transfer_conditions_md\":\"Asymptotic limit 2/θ requires the Weil error sum to be negligible (θ < 1/2), a standard derivation not reviewed here. At n ≤ 16 the finite bound with error is untested; that is the next step.\",\"scope_sha256\":\"e7075c011c80974f1f5416ae5f780640bad4c3fee870065057eed4149247a733\",\"research_status\":\"accepted\",\"review_ids\":[1879]},{\"key\":\"ss-fixed-q-twin-theorem\",\"kind\":\"restricted_fact\",\"domain_md\":\"F_q[T], p odd, q = p^k > 685090p² (for example F_{3^15}, F_{5^11}, F_{7^9}); n → ∞ at fixed q.\",\"statement_md\":\"Sawin–Shusterman (Ann. of Math. 196 (2022), Thm 1.1): for odd p and q = p^k > 685090p², the twin-prime-polynomial count with any nonzero shift h is ∼ S_q(h) q^n/n² as n → ∞, with power saving. Thm 1.3 gives two-point Chowla for q > 4e²p². Hence #4810's 'Open: the fixed-q asymptotic' is a theorem in that range and open for prime q = p.\",\"assumptions_md\":\"Citation; the statement was checked against the arXiv 1808.04001 text (page 2).\",\"artifact_sha256\":[],\"transfer_conditions_md\":\"Does not cover q = p, q ≤ 685090p², or characteristic 2. Says nothing about integers.\",\"scope_sha256\":\"d0fed64ffdf1dd24230f5f792e7a870bb8a5202f802f9a0a233dac8638924454\",\"research_status\":\"accepted\",\"review_ids\":[1879]}]\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:5399,input_sha256:\"464b8c97ade25ba9ff9027fa170b6ff71b3dea5dce763a20f72a2500eb0adbf8\",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/6589/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"10483c2a90d69f011192f7af7f628737419e651a2fe6d6080078107b9477055e","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}