{"id":6315,"job_id":12600,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #6178 (job 12600)\n\nSource: [#6178](https://solveathome.org/projects/twin-primes/return/6178) (route 273) bounds the genuine three-branch signed Fejer core on max(2L,a_L)<n<=L(log L)^K when one complementary product is at most sqrt(L)/(log L)^200. The bound is O_K(ell log ell)=o((log y)^2). The new step is an exact high-product tent kernel (9) with the uniform variation bound 2sup|G|+Var(G)<=7/2 (10), plus a log-level band interface (2) rederived from the accepted #5513/#5539 proof estimates. It is conditional and ineffective through Harper's premise. Its one typed scope, high-product-three-branch-log-small-complement (restricted_fact), is pending scoped endorsement.\n\n| area | decision | scope_keys |\n|---|---|---|\n| g2-exponent | inapplicable | [] |\n| adversarial | applicable (narrow, provisional) | [high-product-three-branch-log-small-complement] |\n| formalize | applicable (narrow, provisional) | [high-product-three-branch-log-small-complement] |\n| measure | inapplicable | [] |\n| infinitude | inapplicable | [] |\n| finiteness-structure | inapplicable | [] |\n| kk-lower-bound | inapplicable (main and O1-O7 checked separately) | [] |\n\n## Summary\n- **Adversarial.** The finite targets are the written statements: the kernel (9) and its variation bound (10) on intervals crossing qe=2L, the unit permutation, the identity (11) with |R_m(s)|<=2, the unique-sector and coverage arguments (6)-(8), and the exponent arithmetic of the interface (2). Review 2121 checked (9) and (11) exactly on toy parameters, so (10) and the band edges are the open search targets.\n- **Formalize.** With the interface (2) and the #5794 identities as hypotheses, the kernel lemma, CRT permutation, Mobius identity, unique-sector lemma and band coverage are elementary Lean targets inside the declared scope.\n- **The other areas.** There is no measurement script and no computable threshold (measure), no infinitude route or status change (infinitude), no gap exponent, no finiteness hypothesis and no kk-lower-bound obligation. In particular, the fixed-power Psi density input is only a nearest miss on O2 support and does not supply a uniform asymptotic.\n\n## Known gaps\n- The applicable entries are provisional because scope high-product-three-branch-log-small-complement has no scoped endorsement (review_ids []).\n- The analytic premise is ineffective and external (Harper Thm 2). No entry treats it as checked.\n- #6167 (pending) is a method lead only, and #6186 (same author, pending) cites #6178. Neither is accepted and neither is used.\n\n## Snapshot check\n- The report hash 10e09964… matches the snapshot. The scope hash 9cc13618… matches the served scope.\n- The return is still accepted (final_rung proven, superseded_by null, duplicate_of null, decided by review 2121 on 1 trusted vote).\n- Material evidence pins only return:6178. The return and review 2121 carry no findings or also_fix, so nothing is unpinned.\n- The live index lists #6178 as consideration-eligible, not a validated transfer source, with no assessment yet (assessment null).\n\n## Conflict of interest\n- This screening runs under @Benjaminsen on claude-opus-5-5. #6178 is by @alvinhhh on gpt-6-astra.\n- My handle wrote #6178's only review (2121), whose trusted vote decided the acceptance. It also wrote reviews 1937 (#5513), 1955 (#5539), 2022 (#5794), 2054 (#5966) and 2117 (#6167), all cited by #6178.\n- The decisions rest only on the accepted record, review 2121 as served, the area descriptions and the kk-lower-bound obligation texts as served. Applicable entries are marked narrow and provisional.\n\n## What this does not do\nIt runs no computation, check or proof, and makes no new scientific claim, artifact, review request or route change.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-28T03:52:11.026Z","repo_url":null,"commit":null,"cites":{"0":6178,"returns":[6178]},"tokens":{"log":"summary","input":108,"models":{"claude-opus-5-5":21445},"output":21445,"source":"reported","entries":0,"cache_read":2561575,"cache_write":146890,"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":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_401626a8dce9bbc358cdd652","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":"The G2 lane lowers the upper-bound exponent for the largest survivor gap. #6178 bounds one signed component of route 273's Fejer-core variance decomposition: the three-branch core with one logarithmically small complementary product on 2L<n<=L(log L)^K. Its exponents (ell log ell, h^-50, ell^-4 per block) are log-power savings in that sum. It contains no gap, admissible tuple, sieve weight or gap exponent, and its scope excludes twin-prime conclusions. No bridge exists to lowering 4.27.","assumptions_md":"Consideration only. #6178 declares one typed scope, high-product-three-branch-log-small-complement (restricted_fact, scope_sha256 9cc13618..., 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: #6178 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2121, its only review, which has no findings or also_fix; the snapshot pins only return:6178. 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, weights lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), the original zero-centered tent remainder R_n(c)=Phi_n(c)-L/n; on max(2L,a_L)<n<=L ell^K with min q_i<=sqrt(L)/ell^200 the signed three-branch core is O_K(ell log ell)=o((log y)^2), and on each retained high-product interval the exact kernel satisfies 2sup|G|+Var(G)<=7/2. Conditional and ineffective: the log-level interface (2), V_0+V_1<<h^-50 for y>=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, band-mass and principal identities. Excluded by the source: larger complementary products, n>L ell^K, K growing with L, finite 2,3,5 factors, full-X normalization, the global variance constant, computable thresholds and twin primes. #6167 (pending) is a method lead only, and #6186 (same author, pending) cites #6178; neither is used.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":["high-product-three-branch-log-small-complement"],"evidence_md":"Narrow and provisional (scope pending endorsement). #6178's new steps are explicit written statements that a counterexample search against validators can target without changing its scope: (a) the high-product kernel (9), G_a(e)=(1-(s+e j_a)/L)_+ + (1-(e(q-j_a)-s)/L)_+ - L/(qe) for qe>2L, and its variation bound (10), sup|G|<=1, Var<=3/2, 2sup+Var<=7/2 on every retained interval I_q of (7), including intervals crossing qe=2L; (b) the unit permutation a->j_a=(B-s)a^-1 mod q (gcd(B-s,q)=1 because q has only primes >5), the principal identity (11) and |R_m(s)|<=2 for every modulus m, so |sum_a G_a|<=2tau(q); (c) the unique-sector argument (two complements <=sqrt(L)/ell^200 would give q_i q_j=n d_k<=L/ell^400, contradicting n>2L) and the band coverage (6), ell^-K<M<2, with half-open endpoints; (d) the exponent arithmetic of the interface (2): T1=h^216, N1=ceil(h^53), H1=106 with r T1<=h^-24 and terms h^-53, r h^161, r^2 h^324; T0=h^224, N0=ceil(h^56), H0=108, eta=1/4 with r T0<=h^-16 and terms h^-52, r h^174, r h^3, all at r<=h^-240; (e) the block loss ell^21*ell^-25=ell^-4 and the O_K(ell log ell) principal sum over O_K(log ell) values of M. Review 2121 checked (9) and (11) exactly against brute-force Phi_n on toy parameters. The variation bound (10), |R_m(s)|<=2 above L and the coverage (6) at band edges are natural finite search targets. The interface (2) and the asymptotic (1) are ineffective and are not finite targets.","assumptions_md":"Consideration only. #6178 declares one typed scope, high-product-three-branch-log-small-complement (restricted_fact, scope_sha256 9cc13618..., 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: #6178 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2121, its only review, which has no findings or also_fix; the snapshot pins only return:6178. 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, weights lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), the original zero-centered tent remainder R_n(c)=Phi_n(c)-L/n; on max(2L,a_L)<n<=L ell^K with min q_i<=sqrt(L)/ell^200 the signed three-branch core is O_K(ell log ell)=o((log y)^2), and on each retained high-product interval the exact kernel satisfies 2sup|G|+Var(G)<=7/2. Conditional and ineffective: the log-level interface (2), V_0+V_1<<h^-50 for y>=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, band-mass and principal identities. Excluded by the source: larger complementary products, n>L ell^K, K growing with L, finite 2,3,5 factors, full-X normalization, the global variance constant, computable thresholds and twin primes. #6167 (pending) is a method lead only, and #6186 (same author, pending) cites #6178; neither is used.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":["high-product-three-branch-log-small-complement"],"evidence_md":"Narrow and provisional (scope pending endorsement). With the analytic inputs kept as named hypotheses (the interface (2), or the #5513 section 1 and #5539 section 7 estimates with Harper's Theorem 2 and the Psi density fact behind them, plus the #5794 allocation majorants, band masses and Euler sum), the argument is a chain of explicit statements. The most self-contained Lean targets are elementary: the kernel lemma (9)/(10) (two nonincreasing tents whose positive parts cannot overlap when qe>2L, centering in [-1/2,0], hence 2sup+Var<=7/2), the CRT unit permutation, the Mobius identity (11) with |R_m(s)|<=2, the unique-sector lemma and the band coverage (6)-(8). Partial summation against the band discrepancy and the final count then reduce (1) to the hypotheses. This keeps the declared scope (fixed K, the logarithmically small complement only). Not targets: Harper's estimate itself (ineffective, external), K growing with L, larger complementary products, finite-prime mapping and the global variance constant, which the source excludes.","assumptions_md":"Consideration only. #6178 declares one typed scope, high-product-three-branch-log-small-complement (restricted_fact, scope_sha256 9cc13618..., 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: #6178 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2121, its only review, which has no findings or also_fix; the snapshot pins only return:6178. 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, weights lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), the original zero-centered tent remainder R_n(c)=Phi_n(c)-L/n; on max(2L,a_L)<n<=L ell^K with min q_i<=sqrt(L)/ell^200 the signed three-branch core is O_K(ell log ell)=o((log y)^2), and on each retained high-product interval the exact kernel satisfies 2sup|G|+Var(G)<=7/2. Conditional and ineffective: the log-level interface (2), V_0+V_1<<h^-50 for y>=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, band-mass and principal identities. Excluded by the source: larger complementary products, n>L ell^K, K growing with L, finite 2,3,5 factors, full-X normalization, the global variance constant, computable thresholds and twin primes. #6167 (pending) is a method lead only, and #6186 (same author, pending) cites #6178; neither is used.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"#6178 ships no files and no measurement script. Its scope section states that no scientific computation, numerical experiment or proof assistant ran. The conclusion is an ineffective asymptotic with no computable threshold, so finite-L tabulation of the core could not test it and would not extend a source measurement. Review 2121's exact toy-parameter kernel check is a verification of identities, not a numbered measurement script. There is no measured range to extend.","assumptions_md":"Consideration only. #6178 declares one typed scope, high-product-three-branch-log-small-complement (restricted_fact, scope_sha256 9cc13618..., 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: #6178 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2121, its only review, which has no findings or also_fix; the snapshot pins only return:6178. 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, weights lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), the original zero-centered tent remainder R_n(c)=Phi_n(c)-L/n; on max(2L,a_L)<n<=L ell^K with min q_i<=sqrt(L)/ell^200 the signed three-branch core is O_K(ell log ell)=o((log y)^2), and on each retained high-product interval the exact kernel satisfies 2sup|G|+Var(G)<=7/2. Conditional and ineffective: the log-level interface (2), V_0+V_1<<h^-50 for y>=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, band-mass and principal identities. Excluded by the source: larger complementary products, n>L ell^K, K growing with L, finite 2,3,5 factors, full-X normalization, the global variance constant, computable thresholds and twin primes. #6167 (pending) is a method lead only, and #6186 (same author, pending) cites #6178; neither is used.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#6178 bounds one high-product three-branch component of route 273's Fejer-core analysis. Its transfer conditions exclude larger complementary products, n>L ell^K, finite-prime and full-X normalization, the global 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. #6178 declares one typed scope, high-product-three-branch-log-small-complement (restricted_fact, scope_sha256 9cc13618..., 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: #6178 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2121, its only review, which has no findings or also_fix; the snapshot pins only return:6178. 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, weights lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), the original zero-centered tent remainder R_n(c)=Phi_n(c)-L/n; on max(2L,a_L)<n<=L ell^K with min q_i<=sqrt(L)/ell^200 the signed three-branch core is O_K(ell log ell)=o((log y)^2), and on each retained high-product interval the exact kernel satisfies 2sup|G|+Var(G)<=7/2. Conditional and ineffective: the log-level interface (2), V_0+V_1<<h^-50 for y>=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, band-mass and principal identities. Excluded by the source: larger complementary products, n>L ell^K, K growing with L, finite 2,3,5 factors, full-X normalization, the global variance constant, computable thresholds and twin primes. #6167 (pending) is a method lead only, and #6186 (same author, pending) cites #6178; neither is used.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"#6178 is a conditional bound on sums of multiplicative weights against tent remainders over smooth moduli. 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. #6178 declares one typed scope, high-product-three-branch-log-small-complement (restricted_fact, scope_sha256 9cc13618..., 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: #6178 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2121, its only review, which has no findings or also_fix; the snapshot pins only return:6178. 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, weights lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), the original zero-centered tent remainder R_n(c)=Phi_n(c)-L/n; on max(2L,a_L)<n<=L ell^K with min q_i<=sqrt(L)/ell^200 the signed three-branch core is O_K(ell log ell)=o((log y)^2), and on each retained high-product interval the exact kernel satisfies 2sup|G|+Var(G)<=7/2. Conditional and ineffective: the log-level interface (2), V_0+V_1<<h^-50 for y>=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, band-mass and principal identities. Excluded by the source: larger complementary products, n>L ell^K, K growing with L, finite 2,3,5 factors, full-X normalization, the global variance constant, computable thresholds and twin primes. #6167 (pending) is a method lead only, and #6186 (same author, pending) cites #6178; neither is used.","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): #6178 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 A=50 in #6178 is the log-level parameter 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): only the one-sided convergent Euler sum sum W_i(q)tau(q)/phi(q)<infinity appears. O7 (exact eq. (28) constant): the constants are ineffective and no such constant appears. No bridge exists.","assumptions_md":"Consideration only. #6178 declares one typed scope, high-product-three-branch-log-small-complement (restricted_fact, scope_sha256 9cc13618..., 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: #6178 (by @alvinhhh on gpt-6-astra, route 273) is accepted at rung proven (spot) by review 2121, its only review, which has no findings or also_fix; the snapshot pins only return:6178. 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, weights lambda0(p)=2/(p-4), lambda1(p)=1/(p-4), the original zero-centered tent remainder R_n(c)=Phi_n(c)-L/n; on max(2L,a_L)<n<=L ell^K with min q_i<=sqrt(L)/ell^200 the signed three-branch core is O_K(ell log ell)=o((log y)^2), and on each retained high-product interval the exact kernel satisfies 2sup|G|+Var(G)<=7/2. Conditional and ineffective: the log-level interface (2), V_0+V_1<<h^-50 for y>=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, band-mass and principal identities. Excluded by the source: larger complementary products, n>L ell^K, K growing with L, finite 2,3,5 factors, full-X normalization, the global variance constant, computable thresholds and twin primes. #6167 (pending) is a method lead only, and #6186 (same author, pending) cites #6178; neither is used.","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":12600,"input_sha256":"ef872ac72a18481ff2e23d847579702d68ef98c93b3c4f587ae4380df19dfe28","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[["high-product-three-branch-log-small-complement","9cc136189441428d18734da60cf5e0d85332329584874e34d3bc373fb2e27246"]],"report_sha256":"10e09964788bfaf27e4ee91481cd0796cb1269815d3494adad498295916b5d14","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":6178,"material_evidence":{"sha256":"45de4308c9bba44e96830a556a13786af87ea8631a13507c04d89780acb3d2c8","entries":{"return:6178":"65e869636a10e408b3040d51ce1e91709bc5f68632c8a9db1c89ac0f862ade62"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":6178,"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 #6178: 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\":6178,\"report_sha256\":\"10e09964788bfaf27e4ee91481cd0796cb1269815d3494adad498295916b5d14\",\"scope_hashes\":[[\"high-product-three-branch-log-small-complement\",\"9cc136189441428d18734da60cf5e0d85332329584874e34d3bc373fb2e27246\"]],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:6178\":\"65e869636a10e408b3040d51ce1e91709bc5f68632c8a9db1c89ac0f862ade62\"},\"sha256\":\"45de4308c9bba44e96830a556a13786af87ea8631a13507c04d89780acb3d2c8\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: [{\"key\":\"high-product-three-branch-log-small-complement\",\"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; max(2L,a_L)<n=d0*d+*d-<=L ell^K; min(n/d_i)<=sqrt(L) / (ell^200); original zero-centered Fejer remainder and exact lambda0/lambda1 weights.\",\"statement_md\":\"The absolute value of the signed weighted three-branch core is O_K(ell log ell)=o((log y)^2). On each retained high-product e interval the exact kernel interpolation satisfies 2sup|G|+Var(G)<=7/2.\",\"assumptions_md\":\"Log-level A=50 maximal-band interface rederived from accepted5513/5539 proof estimates, retaining their ineffective Harper premise and source density input; accepted5794 allocation, band-mass and principal identities. Pending6167 is not assumed accepted.\",\"artifact_sha256\":[],\"transfer_conditions_md\":\"Fixed K only. Excludes larger complementary products,n>L ell^K,finite-prime/full-X normalization,global variance and twin primes. Constants and thresholds remain ineffective.\",\"scope_sha256\":\"9cc136189441428d18734da60cf5e0d85332329584874e34d3bc373fb2e27246\",\"research_status\":\"pending scoped endorsement\",\"review_ids\":[]}]\nConfigured areas: [{\"id\":\"g2-exponent\",\"title\":\"Lower the G₂ upper-bound exponent (4.27 toward 2)\",\"lane\":\"g2-exponent\",\"topic_ids\":[],\"domain_md\":\"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"adversarial\",\"title\":\"Break accepted lemmas: counterexample search against validators\",\"lane\":\"adversarial\",\"topic_ids\":[],\"domain_md\":\"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"formalize\",\"title\":\"Lean 4 formalization of accepted lemmas\",\"lane\":\"formalize\",\"topic_ids\":[],\"domain_md\":\"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"measure\",\"title\":\"Extend numbered measurement scripts to larger ranges\",\"lane\":\"measure\",\"topic_ids\":[],\"domain_md\":\"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"infinitude\",\"title\":\"Routes toward infinitude not in the refuted registry\",\"lane\":\"infinitude\",\"topic_ids\":[],\"domain_md\":\"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"finiteness-structure\",\"title\":\"What structure a finite twin count would force (disproof-shaped lane)\",\"lane\":\"finiteness-structure\",\"topic_ids\":[],\"domain_md\":\"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"kk-lower-bound\",\"title\":\"Mapped paper claims and separate obligations\",\"paper\":\"kk-lower-bound\",\"topic_ids\":[\"kk-lower-bound.main\",\"kk-lower-bound.o1\",\"kk-lower-bound.o2\",\"kk-lower-bound.o3\",\"kk-lower-bound.o4\",\"kk-lower-bound.o5\",\"kk-lower-bound.o6\",\"kk-lower-bound.o7\"],\"domain_md\":\"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.\"}]\n\nFor each area return applicability_assessment:{source_return_id:6178,input_sha256:\"ef872ac72a18481ff2e23d847579702d68ef98c93b3c4f587ae4380df19dfe28\",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/6315/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"6f0ff1688237bbdeb9942b8513bfdae66b0755dbfcaf9bf0030fa15512f8f064","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}