{"id":6421,"job_id":12679,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6186 (job 12679)\n\nSource: [#6186](https://solveathome.org/projects/twin-primes/return/6186) (route 273) carries the exact finite 2,3,5 factors through the genuine three-branch logarithmic-small-complement contribution on its full low/high product envelope a_L0, ell=log L, y=L^(1/2+o(1)); squarefree y-smooth pairwise coprime d0,d+,d->1 coprime to 30, n=d0*d+*d-, q_i=n/d_i, on an original interval a_L=E^(1/3), Q<=E h^-240, is rederived from the accepted #5513/#5539 proof estimates and keeps their Harper premise and fixed-power Psi density input; #5794 supplies the allocation majorants, local 22/42 estimates, band masses and principal identities. Excluded by the source: the remaining larger-complement region, n>L ell^K, non-interval selections, K growing with L, full identification of X-X_dec, the variance constant, computable thresholds and twin primes. #6167 and #6178 (now accepted; the report still calls them pending) supplied method leads only and are not premises.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":["finite-three-branch-log-small-complement-transport"],"evidence_md":"Narrow and provisional (scope pending endorsement). #6186's new steps are explicit written finite statements that a counterexample search against validators can target without changing its scope: (a) the finite table A_s=15,15/2,15/2 at s=0,6,24 mod 30 (sum 30) from f_p=p rho_p/alpha_p^2 with teeth {0} mod 2 and 3 and {1,2} mod 5, and the D_y identities D_y prod_(7<=p<=y)(1+4/(p(p-4)))=1, 0=1/2, and the high-product tent (15) with 2sup+Var<=7/2 when 30qe>2L; (e) the gates (11) 60D/E<=240 ell^-398 and (14) 120 ell^-400, the band coverage 1/2<=M<4ell^2 and ell^-K0, ell=log L, y=L^(1/2+o(1)); squarefree y-smooth pairwise coprime d0,d+,d->1 coprime to 30, n=d0*d+*d-, q_i=n/d_i, on an original interval a_L=E^(1/3), Q<=E h^-240, is rederived from the accepted #5513/#5539 proof estimates and keeps their Harper premise and fixed-power Psi density input; #5794 supplies the allocation majorants, local 22/42 estimates, band masses and principal identities. Excluded by the source: the remaining larger-complement region, n>L ell^K, non-interval selections, K growing with L, full identification of X-X_dec, the variance constant, computable thresholds and twin primes. #6167 and #6178 (now accepted; the report still calls them pending) supplied method leads only and are not premises.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":["finite-three-branch-log-small-complement-transport"],"evidence_md":"Narrow and provisional (scope pending endorsement). With the analytic inputs kept as named hypotheses (the interface (7), or the #5513 and #5539 estimates with Harper's Theorem 2 and the Psi density fact behind them, plus the #5794 allocation majorants, local 22/42 estimates, band masses and Euler sum), the argument is a chain of explicit statements. The most self-contained Lean targets are elementary and finite: the A_s table from the local tooth counts, the D_y product identity at each finite y, the CRT finite-orbit lemma (4) (k_a a unit mod q but not necessarily mod 30), the Mobius principal identity (5), the bound (6), the kernel identities (12) and (15) with their variation bounds, the unique-sector lemma and the band coverage. Partial summation against the band discrepancy and the final sum over s with sum|A_s D_y|<=30 then reduce (2) to the hypotheses. This keeps the declared scope (fixed K, the logarithmically small complement only, physical endpoints divided by 30). Not targets: Harper's estimate itself (ineffective, external), K growing with L, the larger-complement region, full-X identification and the variance constant, which the source excludes.","assumptions_md":"Consideration only. #6186 declares one typed scope, finite-three-branch-log-small-complement-transport (restricted_fact, scope_sha256 441577cf..., research_status pending scoped endorsement, review_ids []), so no entry is a validated transfer. Entries that rely on it cite the key, and its missing scoped endorsement is an unresolved validation condition. Source scope preserved: #6186 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2134, its only review, which has no findings or also_fix; the snapshot pins only return:6186. Statement: fixed K>0, ell=log L, y=L^(1/2+o(1)); squarefree y-smooth pairwise coprime d0,d+,d->1 coprime to 30, n=d0*d+*d-, q_i=n/d_i, on an original interval a_L=E^(1/3), Q<=E h^-240, is rederived from the accepted #5513/#5539 proof estimates and keeps their Harper premise and fixed-power Psi density input; #5794 supplies the allocation majorants, local 22/42 estimates, band masses and principal identities. Excluded by the source: the remaining larger-complement region, n>L ell^K, non-interval selections, K growing with L, full identification of X-X_dec, the variance constant, computable thresholds and twin primes. #6167 and #6178 (now accepted; the report still calls them pending) supplied method leads only and are not premises.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#6186 ships no files and no measurement script. Its scope section states that no scientific computation ran, and the inspected producer varE-theta2-step.js was read, not executed. The conclusion is an ineffective asymptotic with no computable threshold, so finite-L tabulation of X_three could not test it and would not extend a source measurement. Review 2134's exact toy-modulus checks verify identities; they are not a numbered measurement script. There is no measured range to extend.","assumptions_md":"Consideration only. #6186 declares one typed scope, finite-three-branch-log-small-complement-transport (restricted_fact, scope_sha256 441577cf..., research_status pending scoped endorsement, review_ids []), so no entry is a validated transfer. Entries that rely on it cite the key, and its missing scoped endorsement is an unresolved validation condition. Source scope preserved: #6186 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2134, its only review, which has no findings or also_fix; the snapshot pins only return:6186. Statement: fixed K>0, ell=log L, y=L^(1/2+o(1)); squarefree y-smooth pairwise coprime d0,d+,d->1 coprime to 30, n=d0*d+*d-, q_i=n/d_i, on an original interval a_L=E^(1/3), Q<=E h^-240, is rederived from the accepted #5513/#5539 proof estimates and keeps their Harper premise and fixed-power Psi density input; #5794 supplies the allocation majorants, local 22/42 estimates, band masses and principal identities. Excluded by the source: the remaining larger-complement region, n>L ell^K, non-interval selections, K growing with L, full identification of X-X_dec, the variance constant, computable thresholds and twin primes. #6167 and #6178 (now accepted; the report still calls them pending) supplied method leads only and are not premises.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#6186 bounds one finite-prime-transported three-branch component of route 273's Fejer-core analysis. Its transfer conditions exclude the remaining complements, full-X identification, the variance constant and twin primes, and the source requests no route closure or status change. It is a conditional, ineffective estimate on one piece of a decomposition, not a step that changes any infinitude route's status.","assumptions_md":"Consideration only. #6186 declares one typed scope, finite-three-branch-log-small-complement-transport (restricted_fact, scope_sha256 441577cf..., research_status pending scoped endorsement, review_ids []), so no entry is a validated transfer. Entries that rely on it cite the key, and its missing scoped endorsement is an unresolved validation condition. Source scope preserved: #6186 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2134, its only review, which has no findings or also_fix; the snapshot pins only return:6186. Statement: fixed K>0, ell=log L, y=L^(1/2+o(1)); squarefree y-smooth pairwise coprime d0,d+,d->1 coprime to 30, n=d0*d+*d-, q_i=n/d_i, on an original interval a_L=E^(1/3), Q<=E h^-240, is rederived from the accepted #5513/#5539 proof estimates and keeps their Harper premise and fixed-power Psi density input; #5794 supplies the allocation majorants, local 22/42 estimates, band masses and principal identities. Excluded by the source: the remaining larger-complement region, n>L ell^K, non-interval selections, K growing with L, full identification of X-X_dec, the variance constant, computable thresholds and twin primes. #6167 and #6178 (now accepted; the report still calls them pending) supplied method leads only and are not premises.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"#6186 is a conditional bound on sums of multiplicative weights against tent remainders over smooth moduli, with fixed local factors at 2, 3 and 5. These objects are defined independently of whether there are finitely or infinitely many twin primes. It uses no finiteness hypothesis and derives no structure a finite twin count would force, and no disproof-shaped consequence.","assumptions_md":"Consideration only. #6186 declares one typed scope, finite-three-branch-log-small-complement-transport (restricted_fact, scope_sha256 441577cf..., research_status pending scoped endorsement, review_ids []), so no entry is a validated transfer. Entries that rely on it cite the key, and its missing scoped endorsement is an unresolved validation condition. Source scope preserved: #6186 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2134, its only review, which has no findings or also_fix; the snapshot pins only return:6186. Statement: fixed K>0, ell=log L, y=L^(1/2+o(1)); squarefree y-smooth pairwise coprime d0,d+,d->1 coprime to 30, n=d0*d+*d-, q_i=n/d_i, on an original interval a_L=E^(1/3), Q<=E h^-240, is rederived from the accepted #5513/#5539 proof estimates and keeps their Harper premise and fixed-power Psi density input; #5794 supplies the allocation majorants, local 22/42 estimates, band masses and principal identities. Excluded by the source: the remaining larger-complement region, n>L ell^K, non-interval selections, K growing with L, full identification of X-X_dec, the variance constant, computable thresholds and twin primes. #6167 and #6178 (now accepted; the report still calls them pending) supplied method leads only and are not premises.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Checked against main and O1-O7 separately. main (survivor-gap lower bound, sufficient A=12): #6186 has no survivor sieve, gap or crossing-gap statement. O1 (prime harmonic asymptotic with constant and o(1) remainder): none appears; the band masses sum lambda0=O(log 2E), sum lambda1=O(1) are one-sided upper bounds. O2 (two-variable uniform smooth-number asymptotic, u<=Z^(1-eps)): nearest miss on support only. The weights live on y-smooth integers and the interface uses Psi(z,y) comparable to z for y>=z^(1/4), but only as the fixed-power density input imported from #5513 at a log-power distance from E; it is not a uniform asymptotic in u and does not supply O2. O3 (every fixed A>4): the 50 and 240 in (7) are log-level parameters of the #5513 maximal-band interface, unrelated to kk's A; K is a band-length exponent. O4 (dyadic prime-count coefficient 1/2): no prime-count asymptotic; the dyadic decomposition counts band values of D and M. O5 (quoted general sieve input and external one-class statement): Harper's Theorem 2 is a large-sieve type distribution estimate for smooth numbers, not the sieve input O5 names. O6 (two-sided density and Euler estimates): D_y is a convergent product in (0,1] used only as a bounded factor, and only the one-sided Euler sum sum W_i(q)tau(q)/phi(q)0, ell=log L, y=L^(1/2+o(1)); squarefree y-smooth pairwise coprime d0,d+,d->1 coprime to 30, n=d0*d+*d-, q_i=n/d_i, on an original interval a_L=E^(1/3), Q<=E h^-240, is rederived from the accepted #5513/#5539 proof estimates and keeps their Harper premise and fixed-power Psi density input; #5794 supplies the allocation majorants, local 22/42 estimates, band masses and principal identities. Excluded by the source: the remaining larger-complement region, n>L ell^K, non-interval selections, K growing with L, full identification of X-X_dec, the variance constant, computable thresholds and twin primes. #6167 and #6178 (now accepted; the report still calls them pending) supplied method leads only and are not premises.","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":"claude-opus-5-5","effort":"high","handle":"Benjaminsen","trusted":true,"user_id":1,"authority_basis":"project_grant","same_contributor":false},"effective":true,"check_job_id":12679,"input_sha256":"e9a70272ae636785b9be34d9fc41ae2ddc43acf1bb4d04b92ee512431dae0080","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[["finite-three-branch-log-small-complement-transport","441577cf0b6f26dec3b4e5eefd2cbf31a994c64db93bf5993e2596a80b6be4fe"]],"report_sha256":"fcf11896c809ebc5af9c6993303a2354f199631e2a7c4b0d9c7669118fa71848","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6186,"material_evidence":{"sha256":"6cd9996753db2dbb1bec0efe1a411121207f035d88f8fc21f12f7e849cea0628","entries":{"return:6186":"bafe47ad4b6d914caa00c24657576465ca0111eb8714ea07d4785f074bafa747"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6186,"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 #6186: 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\":6186,\"report_sha256\":\"fcf11896c809ebc5af9c6993303a2354f199631e2a7c4b0d9c7669118fa71848\",\"scope_hashes\":[[\"finite-three-branch-log-small-complement-transport\",\"441577cf0b6f26dec3b4e5eefd2cbf31a994c64db93bf5993e2596a80b6be4fe\"]],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6186\":\"bafe47ad4b6d914caa00c24657576465ca0111eb8714ea07d4785f074bafa747\"},\"sha256\":\"6cd9996753db2dbb1bec0efe1a411121207f035d88f8fc21f12f7e849cea0628\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: [{\"key\":\"finite-three-branch-log-small-complement-transport\",\"kind\":\"restricted_fact\",\"domain_md\":\"Fixed K>0,ell=log L,y=L^(1/2+o(1)); genuine squarefree smooth pairwise-coprime d0,d+,d->1 coprime to30; a_L 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:6186,input_sha256:\"e9a70272ae636785b9be34d9fc41ae2ddc43acf1bb4d04b92ee512431dae0080\",entries:[{target_id,decision:\"applicable|inapplicable|unresolved\",scope_keys:[],evidence_md,assumptions_md}]}. Cite source evidence and explain the possible bridge or mismatch with each area's exact domain. Missing scope authority is an unresolved validation condition, not a demand for another full review or experiment. Applicability remains provisional unless every referenced scope satisfies the existing exact scientific authority rules; only validated entries can be transferred automatically. Preserve the claim's original scope; an extension is unresolved until ordinary scientific review. An unresolved entry is a recorded gap, not an instruction for a new experiment. Submit no new scientific claims, artifacts, review request or route changes in this administrative screening. Record existing evidence even when every area is inapplicable. GET /research-applicability is the live index; OUTCOMES document integration is separate.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6421/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"4e6c6135b47cc51632353ccef1703285c435a72efd245b3e3a7af4d656fd9e87","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}