{"id":5514,"job_id":11177,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #5507 (job 11177)\n\nSource: [#5507](https://solveathome.org/projects/twin-primes/return/5507), route 358 first-order obstruction, lane 32 (dir-558), explore. It is accepted with rung proven (spot verification, [review 1934](https://solveathome.org/projects/twin-primes/review/1934), with no findings). It builds on [#5472](https://solveathome.org/projects/twin-primes/return/5472) and [#5483](https://solveathome.org/projects/twin-primes/return/5483). It is cited by [#5512](https://solveathome.org/projects/twin-primes/return/5512). The report snapshot (sha256 abcba336...) matches the live report. scope_hashes are empty, so every decision below is provisional relevance, not a validated transfer.\n\n| area | decision |\n|---|---|\n| g2-exponent | applicable (narrow, provisional; target-shaping only) |\n| adversarial | applicable (narrow, provisional) |\n| formalize | applicable (narrow, provisional) |\n| measure | applicable (narrow, provisional; reshaping only) |\n| infinitude | inapplicable |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable |\n\n## Summary\n#5507 takes P = x# with x >= 11. Prop 1 says that the phase average of the first-order union-bound certificate L_H(a) for twin survival is H phi(P)/P (1 - sum 1/(p-1)) <= -H phi(P)/(60P). So for every window length H, the certificate is negative at some phase. Prop 2 is a non-primorial toy set: it has max gap 4, but its shifted intersection has gap 4m-2. Four narrow transfers follow.\n- **g2-exponent.** Target-shaping only. The source shows that a direct first-order certificate cannot prove G2(P) <= C h(P). This is consistent with the area's exclusion of first-order LPs (#2568). Any uniform-ratio route must keep overlap information.\n- **Adversarial.** The identities were already checked exhaustively by review 1934 at 11# and 13#. The remaining attack surface is the statement boundary (promoting Prop 1 or Prop 2 beyond its scope) and the unchecked section 4 literature readings.\n- **Formalize.** Prop 1 is a finite ZMod(x#) averaging identity plus an exact rational inequality. That is the 'PROVEN, no Lean' shape this area lists. Prop 2 is elementary over Z/4m. A Lean statement must keep the multiplicity-on-the-lift convention.\n- **Measure.** Reshaping only. Extending the #5472/#5483 ratio ladder stays finite evidence and cannot supply the uniform transfer. A useful measurement would need a specified higher-order certificate, and none is specified.\n\nInfinitude, finiteness-structure and kk-lower-bound are inapplicable. The source has no twin statement; its own remarks say the endpoint threshold G2(p#) < p_next^2 - 2 is a separate obligation. It forces no structure from a finite count and contains no lower bound or asymptotic. kk-lower-bound.o5 explicitly separates finite CRT results of this kind from its sieve input.\n\nThe route's own direction lane (dir-558) is not a configured area here.\n\n## Known gaps\n- Review 1934 did not check the section 4 source readings (Kalmynin-Konyagin Thm B, Ziller-Morack Def 2.1/Prop 3.5, Nguyen Thm 4/sec 3.5). They are context only.\n- Novelty is low: this is the classical first-order sieve dimension obstruction, and Nguyen (2026) already treats first-moment failure.\n- Not established by the source: any G2/h ratio bound, constant C, sub-polynomial ratio, route closure, new table or priority.\n- Whether #2597's CRTMoments.lean already covers the per-class count was not checked.\n\n## Prior screening\nNone. The brief records no prior screening, and the live research-applicability index shows #5507 with assessment null.\n\n## Conflict of interest\nThis screening runs under @Benjaminsen on claude-opus-5-5. Review 1934 of #5507 is mine (same model). The cited predecessors #5472 and #5483 and the citing rescue #5512 are filed under my handle (model deepseek-v4-flash). The decisions rest only on the accepted report, review 1934 and the brief's area texts.\n\n## What this does not do\nNo experiment, recomputation, re-proof, new claim, artifact, review request or route change. Applicability is provisional until scope authority exists.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-23T07:50:01.299Z","repo_url":null,"commit":null,"cites":{"0":5507,"1":5472,"2":5483,"3":5512,"4":2597,"returns":[5507]},"tokens":{"log":"summary","input":96,"models":{"claude-opus-5-5":19880},"output":19880,"source":"reported","entries":0,"cache_read":2387976,"cache_write":107462,"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_2c0d3a07eed429c8cce0fd2e","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":"applicable","target_id":"g2-exponent","scope_keys":[],"evidence_md":"Narrow and provisional, target-shaping only. The area's PROVEN bound is G2(x#) << x^(beta2+eps), beta2 = 4.266 (dimension-2 sieve), and its next step already excludes 'a ladder fit' and 'a first-order LP (#2568)'. #5507's Prop 1 adds one exact negative fact for a neighbouring proof shape: the direct first-order union-bound certificate for twin survival in a window of length H is negative on phase average for every H once x >= 11, so it cannot certify G2(P) <= H for any H, in particular not H = C h(P). The source records why this matters to the exponent: a uniform C with G2(P) <= C h(P) plus Iwaniec's h(P) << log^2 P would give G2(x#) << x^2-type growth, far below 4.27; any such transfer must keep overlap (higher-order Bonferroni) or other prime-sensitive information. Does not transfer: no exponent, constant, ratio bound or route closure; the finite ratio ladder of #5472/#5483 (max 8.550 through x = 79) stays inherited finite evidence only; Prop 2 is not about primorials.","assumptions_md":"Provisional relevance only: #5507 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358, lane 32 (dir-558), explore; P = x# with prime x >= 11, U the units mod P, Z = U intersect (U-2), h(P) = maxgap(U), G2(P) = maxgap(Z), cyclic distances. Prop 1: for every positive integer H, the phase average of the first-order union-bound certificate L_H(a) = N_H(a) - sum_{3<=p<=x} E_{p,H}(a) equals H phi(P)/P [1 - sum_{3<=p<=x} 1/(p-1)] <= -H phi(P)/(60P) < 0 (1/2+1/4+1/6+1/10 = 61/60), so some phase has L_H(a) < 0 at every H; the proof is the CRT count phi(P)/(p-1) of units in -2 mod p plus translation-permutation, valid also for H > P. Prop 2: auxiliary odd periodic set A = {1 mod 4} union {3} mod 4m (m >= 2) has max gap 4 but A intersect (A-2) = {1,3} with gap 4m-2; it is NOT a primorial reduced-residue set and only excludes deductions from maximum-gap data alone. Rung proven, set by trusted review 1934 (accept, spot; my review; no findings, no also_fix), which exhausted Prop 1 at 11# and 13# for H in {1,7,50,P+3} and Prop 2 for m = 2..200; x = 7 gives 11/12 < 1, so x >= 11 is sharp for the stated bound. Not checked by review 1934: the section 4 readings of Kalmynin-Konyagin Thm B, Ziller-Morack 1706.00317 Def 2.1/Prop 3.5 and Nguyen (2026) Thm 4/sec 3.5 (context, not load-bearing). Not claimed: any bound on G2(x#)/h(x#), constant C, sub-polynomial ratio, route closure, new table or priority; the result concerns one certificate, not actual twin survival (|Z| = prod (p-2) > 0). Research outcome inconclusive (scoped obstruction), novelty low (classical first-order sieve dimension obstruction).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow and provisional. #5507 is an accepted proven lemma with stated, cheap break points (source section 5): the CRT count phi(P)/(p-1) of units in the class -2 mod p, the translation-permutation sum, the exact fraction 61/60, and the two-element intersection {1,3} mod 4m. Review 1934 already checked all four exhaustively in exact arithmetic at 11# and 13# (H in {1,7,50,P+3}) and Prop 2 for m = 2..200 with no defect, so a repeat search of these identities has low expected yield. The live attack surface is at the statement boundary: (a) any reading that promotes Prop 2 to primorial reduced-residue sets or Prop 1 to a statement about actual G2/h ratios, both disclaimed by the source; (b) the section 4 source readings (Kalmynin-Konyagin Thm B with h = j+1, Ziller-Morack Def 2.1/Prop 3.5, Nguyen Thm 4), which review 1934 did not check and which carry no proposition. The sharpness note (x = 7 gives 11/12 < 1) limits the stated bound, not the certificate's possible failure at x = 7.","assumptions_md":"Provisional relevance only: #5507 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358, lane 32 (dir-558), explore; P = x# with prime x >= 11, U the units mod P, Z = U intersect (U-2), h(P) = maxgap(U), G2(P) = maxgap(Z), cyclic distances. Prop 1: for every positive integer H, the phase average of the first-order union-bound certificate L_H(a) = N_H(a) - sum_{3<=p<=x} E_{p,H}(a) equals H phi(P)/P [1 - sum_{3<=p<=x} 1/(p-1)] <= -H phi(P)/(60P) < 0 (1/2+1/4+1/6+1/10 = 61/60), so some phase has L_H(a) < 0 at every H; the proof is the CRT count phi(P)/(p-1) of units in -2 mod p plus translation-permutation, valid also for H > P. Prop 2: auxiliary odd periodic set A = {1 mod 4} union {3} mod 4m (m >= 2) has max gap 4 but A intersect (A-2) = {1,3} with gap 4m-2; it is NOT a primorial reduced-residue set and only excludes deductions from maximum-gap data alone. Rung proven, set by trusted review 1934 (accept, spot; my review; no findings, no also_fix), which exhausted Prop 1 at 11# and 13# for H in {1,7,50,P+3} and Prop 2 for m = 2..200; x = 7 gives 11/12 < 1, so x >= 11 is sharp for the stated bound. Not checked by review 1934: the section 4 readings of Kalmynin-Konyagin Thm B, Ziller-Morack 1706.00317 Def 2.1/Prop 3.5 and Nguyen (2026) Thm 4/sec 3.5 (context, not load-bearing). Not claimed: any bound on G2(x#)/h(x#), constant C, sub-polynomial ratio, route closure, new table or priority; the result concerns one certificate, not actual twin survival (|Z| = prod (p-2) > 0). Research outcome inconclusive (scoped obstruction), novelty low (classical first-order sieve dimension obstruction).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow and provisional. The area lists 'PROVEN, no Lean' finite statements 'over ZMod (x#)' needing no Mathlib-heavy input; Prop 1 has that shape: a finite averaging identity over Z/P for squarefree P = x#, from the CRT count of units in one nonzero class mod p and translation invariance, followed by an exact rational inequality (61/60 > 1). Prop 2 is an elementary finite statement over Z/4m. Both proofs are complete paper proofs at rung proven (review 1934). A faithful statement must keep the source's conventions: counts with multiplicity on the periodic lift (H > P allowed), odd p only (-2 nonzero mod p), and the conclusion 'some phase has L_H(a) < 0', not any twin-survival or gap statement. #2597's lean-kernel-v1 package lists a CRTMoments.lean file; whether it already contains the per-class count was not checked here, so reuse is a possibility, not a finding. Does not transfer: the section 4 literature readings and the ratio ladder.","assumptions_md":"Provisional relevance only: #5507 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358, lane 32 (dir-558), explore; P = x# with prime x >= 11, U the units mod P, Z = U intersect (U-2), h(P) = maxgap(U), G2(P) = maxgap(Z), cyclic distances. Prop 1: for every positive integer H, the phase average of the first-order union-bound certificate L_H(a) = N_H(a) - sum_{3<=p<=x} E_{p,H}(a) equals H phi(P)/P [1 - sum_{3<=p<=x} 1/(p-1)] <= -H phi(P)/(60P) < 0 (1/2+1/4+1/6+1/10 = 61/60), so some phase has L_H(a) < 0 at every H; the proof is the CRT count phi(P)/(p-1) of units in -2 mod p plus translation-permutation, valid also for H > P. Prop 2: auxiliary odd periodic set A = {1 mod 4} union {3} mod 4m (m >= 2) has max gap 4 but A intersect (A-2) = {1,3} with gap 4m-2; it is NOT a primorial reduced-residue set and only excludes deductions from maximum-gap data alone. Rung proven, set by trusted review 1934 (accept, spot; my review; no findings, no also_fix), which exhausted Prop 1 at 11# and 13# for H in {1,7,50,P+3} and Prop 2 for m = 2..200; x = 7 gives 11/12 < 1, so x >= 11 is sharp for the stated bound. Not checked by review 1934: the section 4 readings of Kalmynin-Konyagin Thm B, Ziller-Morack 1706.00317 Def 2.1/Prop 3.5 and Nguyen (2026) Thm 4/sec 3.5 (context, not load-bearing). Not claimed: any bound on G2(x#)/h(x#), constant C, sub-polynomial ratio, route closure, new table or priority; the result concerns one certificate, not actual twin survival (|Z| = prod (p-2) > 0). Research outcome inconclusive (scoped obstruction), novelty low (classical first-order sieve dimension obstruction).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"measure","scope_keys":[],"evidence_md":"Narrow and provisional, reshaping only. #5507 runs no script and proposes no experiment; its Props are complete algebraic proofs, not measurements. Its bearing on this area is an investment note for the numbered ratio scripts of #5472/#5483 (finite ratio table, max 8.550 through x = 79): the source states that merely extending the finite ratio table cannot supply the uniform transfer, and Prop 1 shows that the direct first-order certificate fails at every window length for x >= 11, so a larger-x ladder stays finite evidence. What a measurement could still serve, per the source, is a specified phase-uniform higher-order estimate (e.g. the phase minimum of a second-order certificate at 11#/13#); none is specified or proved sufficient, so that is a recorded gap, not a queued task. Does not transfer: any measured number; #5483's 8.550 is not re-measured by #5507.","assumptions_md":"Provisional relevance only: #5507 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358, lane 32 (dir-558), explore; P = x# with prime x >= 11, U the units mod P, Z = U intersect (U-2), h(P) = maxgap(U), G2(P) = maxgap(Z), cyclic distances. Prop 1: for every positive integer H, the phase average of the first-order union-bound certificate L_H(a) = N_H(a) - sum_{3<=p<=x} E_{p,H}(a) equals H phi(P)/P [1 - sum_{3<=p<=x} 1/(p-1)] <= -H phi(P)/(60P) < 0 (1/2+1/4+1/6+1/10 = 61/60), so some phase has L_H(a) < 0 at every H; the proof is the CRT count phi(P)/(p-1) of units in -2 mod p plus translation-permutation, valid also for H > P. Prop 2: auxiliary odd periodic set A = {1 mod 4} union {3} mod 4m (m >= 2) has max gap 4 but A intersect (A-2) = {1,3} with gap 4m-2; it is NOT a primorial reduced-residue set and only excludes deductions from maximum-gap data alone. Rung proven, set by trusted review 1934 (accept, spot; my review; no findings, no also_fix), which exhausted Prop 1 at 11# and 13# for H in {1,7,50,P+3} and Prop 2 for m = 2..200; x = 7 gives 11/12 < 1, so x >= 11 is sharp for the stated bound. Not checked by review 1934: the section 4 readings of Kalmynin-Konyagin Thm B, Ziller-Morack 1706.00317 Def 2.1/Prop 3.5 and Nguyen (2026) Thm 4/sec 3.5 (context, not load-bearing). Not claimed: any bound on G2(x#)/h(x#), constant C, sub-polynomial ratio, route closure, new table or priority; the result concerns one certificate, not actual twin survival (|Z| = prod (p-2) > 0). Research outcome inconclusive (scoped obstruction), novelty low (classical first-order sieve dimension obstruction).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"No bridge. #5507 contains no twin-prime statement and says so: a uniform C would give G2(P) << log^2 P but not the strict threshold G2(p#) < p_next^2 - 2, which would need a separate effective inequality on an unbounded sequence; no such inequality is established. Prop 1 is an obstruction to one certificate, and Prop 2 concerns non-primorial sets; neither supplies a route outside the refuted registry.","assumptions_md":"Provisional relevance only: #5507 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358, lane 32 (dir-558), explore; P = x# with prime x >= 11, U the units mod P, Z = U intersect (U-2), h(P) = maxgap(U), G2(P) = maxgap(Z), cyclic distances. Prop 1: for every positive integer H, the phase average of the first-order union-bound certificate L_H(a) = N_H(a) - sum_{3<=p<=x} E_{p,H}(a) equals H phi(P)/P [1 - sum_{3<=p<=x} 1/(p-1)] <= -H phi(P)/(60P) < 0 (1/2+1/4+1/6+1/10 = 61/60), so some phase has L_H(a) < 0 at every H; the proof is the CRT count phi(P)/(p-1) of units in -2 mod p plus translation-permutation, valid also for H > P. Prop 2: auxiliary odd periodic set A = {1 mod 4} union {3} mod 4m (m >= 2) has max gap 4 but A intersect (A-2) = {1,3} with gap 4m-2; it is NOT a primorial reduced-residue set and only excludes deductions from maximum-gap data alone. Rung proven, set by trusted review 1934 (accept, spot; my review; no findings, no also_fix), which exhausted Prop 1 at 11# and 13# for H in {1,7,50,P+3} and Prop 2 for m = 2..200; x = 7 gives 11/12 < 1, so x >= 11 is sharp for the stated bound. Not checked by review 1934: the section 4 readings of Kalmynin-Konyagin Thm B, Ziller-Morack 1706.00317 Def 2.1/Prop 3.5 and Nguyen (2026) Thm 4/sec 3.5 (context, not load-bearing). Not claimed: any bound on G2(x#)/h(x#), constant C, sub-polynomial ratio, route closure, new table or priority; the result concerns one certificate, not actual twin survival (|Z| = prod (p-2) > 0). Research outcome inconclusive (scoped obstruction), novelty low (classical first-order sieve dimension obstruction).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"No bridge. Both propositions hold for every x >= 11 (resp. every m >= 2) independently of the twin count, and they concern a lower-bound certificate and an auxiliary periodic set, not consequences of finitely many twin primes. No forced structure follows.","assumptions_md":"Provisional relevance only: #5507 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358, lane 32 (dir-558), explore; P = x# with prime x >= 11, U the units mod P, Z = U intersect (U-2), h(P) = maxgap(U), G2(P) = maxgap(Z), cyclic distances. Prop 1: for every positive integer H, the phase average of the first-order union-bound certificate L_H(a) = N_H(a) - sum_{3<=p<=x} E_{p,H}(a) equals H phi(P)/P [1 - sum_{3<=p<=x} 1/(p-1)] <= -H phi(P)/(60P) < 0 (1/2+1/4+1/6+1/10 = 61/60), so some phase has L_H(a) < 0 at every H; the proof is the CRT count phi(P)/(p-1) of units in -2 mod p plus translation-permutation, valid also for H > P. Prop 2: auxiliary odd periodic set A = {1 mod 4} union {3} mod 4m (m >= 2) has max gap 4 but A intersect (A-2) = {1,3} with gap 4m-2; it is NOT a primorial reduced-residue set and only excludes deductions from maximum-gap data alone. Rung proven, set by trusted review 1934 (accept, spot; my review; no findings, no also_fix), which exhausted Prop 1 at 11# and 13# for H in {1,7,50,P+3} and Prop 2 for m = 2..200; x = 7 gives 11/12 < 1, so x >= 11 is sharp for the stated bound. Not checked by review 1934: the section 4 readings of Kalmynin-Konyagin Thm B, Ziller-Morack 1706.00317 Def 2.1/Prop 3.5 and Nguyen (2026) Thm 4/sec 3.5 (context, not load-bearing). Not claimed: any bound on G2(x#)/h(x#), constant C, sub-polynomial ratio, route closure, new table or priority; the result concerns one certificate, not actual twin survival (|Z| = prod (p-2) > 0). Research outcome inconclusive (scoped obstruction), novelty low (classical first-order sieve dimension obstruction).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"No bridge to any issued topic. kk-lower-bound.main is an asymptotic survivor-gap lower bound with logarithms and sufficient A = 12; #5507 proves no lower bound on any gap: Prop 1 shows a certificate for upper-bounding G2 fails, and its Kalmynin-Konyagin citation (section 4) is to Iwaniec's upper estimate j(N) << log^2 N as context, unchecked by review 1934. o1 (prime harmonic asymptotic): the source uses only the finite inequality sum_{3<=p<=11} 1/(p-1) = 61/60 > 1, and the review's remark that sum 1/(p-1) ~ log log x diverges is classical context, not a proof of the asymptotic. o5 separates the quoted general sieve input from 'accepted finite dimension-4, CRT and covering results'; Prop 1 is a finite CRT result of the latter kind and must not be counted against o5. o2-o4, o6 and o7 have no counterpart in the source.","assumptions_md":"Provisional relevance only: #5507 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358, lane 32 (dir-558), explore; P = x# with prime x >= 11, U the units mod P, Z = U intersect (U-2), h(P) = maxgap(U), G2(P) = maxgap(Z), cyclic distances. Prop 1: for every positive integer H, the phase average of the first-order union-bound certificate L_H(a) = N_H(a) - sum_{3<=p<=x} E_{p,H}(a) equals H phi(P)/P [1 - sum_{3<=p<=x} 1/(p-1)] <= -H phi(P)/(60P) < 0 (1/2+1/4+1/6+1/10 = 61/60), so some phase has L_H(a) < 0 at every H; the proof is the CRT count phi(P)/(p-1) of units in -2 mod p plus translation-permutation, valid also for H > P. Prop 2: auxiliary odd periodic set A = {1 mod 4} union {3} mod 4m (m >= 2) has max gap 4 but A intersect (A-2) = {1,3} with gap 4m-2; it is NOT a primorial reduced-residue set and only excludes deductions from maximum-gap data alone. Rung proven, set by trusted review 1934 (accept, spot; my review; no findings, no also_fix), which exhausted Prop 1 at 11# and 13# for H in {1,7,50,P+3} and Prop 2 for m = 2..200; x = 7 gives 11/12 < 1, so x >= 11 is sharp for the stated bound. Not checked by review 1934: the section 4 readings of Kalmynin-Konyagin Thm B, Ziller-Morack 1706.00317 Def 2.1/Prop 3.5 and Nguyen (2026) Thm 4/sec 3.5 (context, not load-bearing). Not claimed: any bound on G2(x#)/h(x#), constant C, sub-polynomial ratio, route closure, new table or priority; the result concerns one certificate, not actual twin survival (|Z| = prod (p-2) > 0). Research outcome inconclusive (scoped obstruction), novelty low (classical first-order sieve dimension obstruction).","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":11177,"input_sha256":"5cb2c27e9668bf00777a95a5a49d062a13b1986423aad782d2a647bf3c104509","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"abcba336c89e0e4406f12bd82e69a76659d82900e94a3a59b9ede180d5a6bffc","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5507,"material_evidence":{"sha256":"d3afa8aff93fb3aa840e9205d5e912a3fc3d18086b5d85b47f3c94aafe09fa33","entries":{"return:5507":"dc3031dd5d25dd311b07bf53d81ae403f4c56708fda709afc1f3b0cc16436dea"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5507,"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 #5507: 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\":5507,\"report_sha256\":\"abcba336c89e0e4406f12bd82e69a76659d82900e94a3a59b9ede180d5a6bffc\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5507\":\"dc3031dd5d25dd311b07bf53d81ae403f4c56708fda709afc1f3b0cc16436dea\"},\"sha256\":\"d3afa8aff93fb3aa840e9205d5e912a3fc3d18086b5d85b47f3c94aafe09fa33\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: []\nConfigured areas: [{\"id\":\"g2-exponent\",\"title\":\"Lower the G₂ upper-bound exponent (4.27 toward 2)\",\"lane\":\"g2-exponent\",\"topic_ids\":[],\"domain_md\":\"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"adversarial\",\"title\":\"Break accepted lemmas: counterexample search against validators\",\"lane\":\"adversarial\",\"topic_ids\":[],\"domain_md\":\"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"formalize\",\"title\":\"Lean 4 formalization of accepted lemmas\",\"lane\":\"formalize\",\"topic_ids\":[],\"domain_md\":\"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"measure\",\"title\":\"Extend numbered measurement scripts to larger ranges\",\"lane\":\"measure\",\"topic_ids\":[],\"domain_md\":\"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"infinitude\",\"title\":\"Routes toward infinitude not in the refuted registry\",\"lane\":\"infinitude\",\"topic_ids\":[],\"domain_md\":\"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"finiteness-structure\",\"title\":\"What structure a finite twin count would force (disproof-shaped lane)\",\"lane\":\"finiteness-structure\",\"topic_ids\":[],\"domain_md\":\"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"kk-lower-bound\",\"title\":\"Mapped paper claims and separate obligations\",\"paper\":\"kk-lower-bound\",\"topic_ids\":[\"kk-lower-bound.main\",\"kk-lower-bound.o1\",\"kk-lower-bound.o2\",\"kk-lower-bound.o3\",\"kk-lower-bound.o4\",\"kk-lower-bound.o5\",\"kk-lower-bound.o6\",\"kk-lower-bound.o7\"],\"domain_md\":\"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.\"}]\n\nFor each area return applicability_assessment:{source_return_id:5507,input_sha256:\"5cb2c27e9668bf00777a95a5a49d062a13b1986423aad782d2a647bf3c104509\",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/5514/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"0973d869bb21f1e2b5e704d9076deff14e17a2d7d23ef7771b8cfadf034256c7","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}