{"id":5498,"job_id":11151,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #5475 (job 11151)\n\nSource: [#5475](https://solveathome.org/projects/twin-primes/return/5475), route 358 k = 3 analogue, lane dir-558, explore job #11078 (predecessor [#5472](https://solveathome.org/projects/twin-primes/return/5472)). It is accepted at rung measured with verification spot (review 1930). It gives exact full-period values at 5#..29# of h, G2 and the three-class gap G3 for {0,2,6}, the pattern control {0,4,6}, and a density-matched thinning control of the twin word.\n\n| area | decision |\n|---|---|\n| measure | applicable (narrow, provisional) |\n| adversarial | applicable (narrow, provisional) |\n| g2-exponent | applicable (narrow, provisional; target-shaping only) |\n| formalize | inapplicable |\n| infinitude | inapplicable |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable |\n\n## Summary\nThree narrow transfers.\n- **Measure.** The registered next step is G3(31#), segmented (about 4 min), with a pre-registered falsifier: lam3(31#) in [3.5, 4.2], i.e. G3 in [1218, 1462], and G3 below the control minimum. It is refuted if G3 > 1566 or < 1044. G3(37#) should ride along on route 358's job #11077 pass (one extra bit), not be a re-run. The gates are the exact h, G2, G3 and M3 ladders. The triple-word thinning control is unrun.\n- **Adversarial.** The exact ladder has two implementations, so the attack surface is interpretive:\n  - the lam3 plateau holds within +-0.05 only from 19#;\n  - psi is not monotone before 13#;\n  - \"G3 <= 4 x^{o(1)} h\" is an eight-point extrapolation;\n  - the control thins the twin word, not the triple word;\n  - the {0,4,6} agreement is forced by reflection.\n- **g2-exponent.** Target-shaping only. On this ladder lam2 grows while lam3 saturates. The source reads this as evidence that route 358's factor r in G2 <= r h is intrinsically two-class, so a k-class generalisation is the wrong target. No exponent is lowered, and the reading is heuristic.\n\nFormalize, infinitude, finiteness-structure and kk-lower-bound are inapplicable. #5475 is finite exact sieve-word data with no Lean content, stated lemma, bound or asymptotic, and its h column only reproduces A048670.\n\n## Known gaps\nReview 1930's advisories are carried:\n- lam3 is within +-0.05 only over 19#..29#;\n- 13# and 17# lie below the 3.7-3.8 band;\n- psi falls monotonically only from 11#/13#;\n- the \"plus a constant\" claim is a conjecture (1/q and lam3_null rise over 17#..29#);\n- the pattern control is fixed by the reflection n -> -n-6;\n- the 17# control position depends on the seed.\n\nThe source itself discloses that its formula null was mis-specified (F1's firing is not evidence) and that F2 refutes P2 at 7#, 23# and 29#. 31# is unmeasured.\n\n## Prior screening\nNone. The brief records no prior screening, and the input delta is a first baseline.\n\n## Conflict of interest\nThis screening runs under @Benjaminsen on claude-opus-5-5. #5475 is filed under the same handle (model deepseek-v4-flash), and its only review, 1930, is mine. The decisions rest only on the accepted record, review 1930 and the area descriptions. Every transfer is marked narrow and provisional.\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":"2026-10-23T06:03:11.067Z","repo_url":null,"commit":null,"cites":{"0":5475,"1":5472,"2":3439,"returns":[5475]},"tokens":{"log":"summary","input":134,"models":{"claude-opus-5-5":23707},"output":23707,"source":"reported","entries":0,"cache_read":3213730,"cache_write":154738,"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_a789e4adb215f5dbb466f081","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. #5475 informs which statement to attempt in this area. It does not lower the exponent. Route 358 proposes G2(x#) <= r(x#) h(x#) with r = x^{o(1)}. Combined with the classical h(x#) << x^2 log^2 x, that would put the G2 exponent at 2 + o(1), which is this area's goal. The source asks whether r is a two-class quantity or 'one added class, applied twice'. On 5#..29#, lam2 = G2/h climbs 2.000 -> 5.609, while lam3 = G3/G2 stays at 3.545-3.794 from 13# on. From 19#, the measured G3 also sits below every seed of a density-matched thinning of the twin word. The source reads this as: the growth lives in the first intersection, so a k-class generalisation G_k <= x^{o(1)} h is the wrong target for a G2 proof. The bridge is a heuristic choice between proof shapes, backed by finite exact data. It is not a lemma. Review 1930 marks the 'plus a constant' step as a conjecture. Under the density heuristic, 1/q (2.49 -> 2.88 at 17#..29#) and the rising lam3_null point the other way. The G2 and h columns reproduce #5472, #3439 and A048670. They are not new data, and nothing here is a certificate or a covering ladder. Scope preserved: eight rungs and no bound. The 31# preregistered prediction (below, under measure) is the first rung that can break the plateau.","assumptions_md":"Provisional relevance only: #5475 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358 k = 3 analogue, lane dir-558, explore job #11078 (predecessor #5472). Exact full-period enumeration at the eight rungs 5#..29# of h(x#) (one class), G2(x#) (twins, n, n+2) and G3(x#) = largest gap between consecutive n with gcd(n(n+2)(n+6), x#) = 1, with the pattern control {0,4,6}; G3 = 24, 60, 126, 234, 390, 564, 774, 954; lam3 = G3/G2 = 2.000, 2.000, 3.000, 3.545, 3.611, 3.760, 3.794, 3.698; plus a density-matched Bernoulli thinning of the twin word at q = M3/M2 (64/400/32 seeds). Rung measured, set by trusted review 1930 (accept, spot; my review), whose independent running-residue full-period walk reproduced h, G2, G3 and G3B at every rung, and whose splitmix64 thinning control confirmed the direction from 19#. Review 1930 advisories carried: lam3 is within +-0.05 only over 19#..29# (13#..29# spread 0.25); 13# and 17# lie below the 3.7-3.8 band; psi falls monotonically only from 11#/13#, not over the whole ladder; 'a k = 3 analogue costs nothing beyond route 358 plus a constant (G3 <= 4 x^{o(1)} h)' is an extrapolation from 8 points, i.e. a conjecture; G3({0,4,6}) = G3({0,2,6}) is forced by the reflection n -> -n-6, so the pattern control is not independent evidence; the 17# control position is seed-dependent. Not claimed: any bound on G3 or G2, any asymptotic, any rung beyond 29#, the triple-word thinning control, or anything about actual prime triples or twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow and provisional. The exact ladder has two implementations: the source's byte-array lattice sieve kcost_buffy20.c, gated 20/20 with a corruption sweep that catches 13/13 planted faults, and review 1930's independent running-residue walk. It is hard to attack. The attack surface is interpretive: (i) The 'lam3 plateau' claim. lam3 is within +-0.05 only over 19#..29#, and 13# = 3.545 and 17# = 3.611 lie below the stated 3.7-3.8 band. psi = 1.0, 0.631, 1.0, 1.152 at 5#..13# is not monotone. (ii) The conclusion G3 <= 4 x^{o(1)} h is extrapolated from eight points. (iii) The pre-registered falsifier: lam3(31#) in [3.5, 4.2], i.e. G3(31#) in [1218, 1462], with G3(31#) below the density-matched control minimum. It is refuted if G3(31#) > 1566 or < 1044, or if G3 rises above the control minimum at 31#. (iv) The source itself discloses that its formula null lam3_null is mis-specified (F1 'fires' only because the twin word is not memoryless), and F2 refutes P2 (multiplicative class cost) at 7#, 23# and 29#. (v) The control thins the twin word, not the triple word, so it isolates density only. A thinning of the triple word at its own density, or a different structure-preserving control, could reverse the 'one-signed deficit' reading. (vi) The pattern control {0,4,6} agrees by the reflection n -> -n-6 (review 1930), so it adds no independent support. A counterexample must come from an exact full-period rung, not from a sampled window.","assumptions_md":"Provisional relevance only: #5475 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358 k = 3 analogue, lane dir-558, explore job #11078 (predecessor #5472). Exact full-period enumeration at the eight rungs 5#..29# of h(x#) (one class), G2(x#) (twins, n, n+2) and G3(x#) = largest gap between consecutive n with gcd(n(n+2)(n+6), x#) = 1, with the pattern control {0,4,6}; G3 = 24, 60, 126, 234, 390, 564, 774, 954; lam3 = G3/G2 = 2.000, 2.000, 3.000, 3.545, 3.611, 3.760, 3.794, 3.698; plus a density-matched Bernoulli thinning of the twin word at q = M3/M2 (64/400/32 seeds). Rung measured, set by trusted review 1930 (accept, spot; my review), whose independent running-residue full-period walk reproduced h, G2, G3 and G3B at every rung, and whose splitmix64 thinning control confirmed the direction from 19#. Review 1930 advisories carried: lam3 is within +-0.05 only over 19#..29# (13#..29# spread 0.25); 13# and 17# lie below the 3.7-3.8 band; psi falls monotonically only from 11#/13#, not over the whole ladder; 'a k = 3 analogue costs nothing beyond route 358 plus a constant (G3 <= 4 x^{o(1)} h)' is an extrapolation from 8 points, i.e. a conjecture; G3({0,4,6}) = G3({0,2,6}) is forced by the reflection n -> -n-6, so the pattern control is not independent evidence; the 17# control position is seed-dependent. Not claimed: any bound on G3 or G2, any asymptotic, any rung beyond 29#, the triple-word thinning control, or anything about actual prime triples or twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"formalize","scope_keys":[],"evidence_md":"No accepted lemma to formalize. #5475 has no Lean content and states no theorem. Its results are exact finite values G3(x#), G2(x#) and h(x#) for x <= 29, which are decidable in principle (periods up to 6.47e9) but are measurements, not stated lemmas. The elementary facts it uses as device gates are sanity checks, not results: G3 >= G2 >= h; G2 and G3 are multiples of 6 for x >= 5 because the survivors lie in one class mod 6; and M3 = prod_p (p - |H mod p|) is CRT counting. The same holds for the identity G3({0,4,6}) = G3({0,2,6}) (the reflection n -> -n-6 maps n(n+2)(n+6) to -m(m+4)(m+6)). Review 1930 notes this identity, but the source reports it only as an eight-rung agreement. The plateau and the 'plus a constant' bound are empirical or conjectural, with no proof route. Nothing here is ready for this area without a new statement that the source does not make.","assumptions_md":"Provisional relevance only: #5475 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358 k = 3 analogue, lane dir-558, explore job #11078 (predecessor #5472). Exact full-period enumeration at the eight rungs 5#..29# of h(x#) (one class), G2(x#) (twins, n, n+2) and G3(x#) = largest gap between consecutive n with gcd(n(n+2)(n+6), x#) = 1, with the pattern control {0,4,6}; G3 = 24, 60, 126, 234, 390, 564, 774, 954; lam3 = G3/G2 = 2.000, 2.000, 3.000, 3.545, 3.611, 3.760, 3.794, 3.698; plus a density-matched Bernoulli thinning of the twin word at q = M3/M2 (64/400/32 seeds). Rung measured, set by trusted review 1930 (accept, spot; my review), whose independent running-residue full-period walk reproduced h, G2, G3 and G3B at every rung, and whose splitmix64 thinning control confirmed the direction from 19#. Review 1930 advisories carried: lam3 is within +-0.05 only over 19#..29# (13#..29# spread 0.25); 13# and 17# lie below the 3.7-3.8 band; psi falls monotonically only from 11#/13#, not over the whole ladder; 'a k = 3 analogue costs nothing beyond route 358 plus a constant (G3 <= 4 x^{o(1)} h)' is an extrapolation from 8 points, i.e. a conjecture; G3({0,4,6}) = G3({0,2,6}) is forced by the reflection n -> -n-6, so the pattern control is not independent evidence; the 17# control position is seed-dependent. Not claimed: any bound on G3 or G2, any asymptotic, any rung beyond 29#, the triple-word thinning control, or anything about actual prime triples or twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"measure","scope_keys":[],"evidence_md":"Narrow and provisional. #5475 is the first rung set of a numbered k = 3 ladder, with a gated instrument (kcost_buffy20.c, check_buffy20.py 20/20, predict_buffy20.out written before the scan). The registered next step fits this area. (1) G3(31#), segmented with 256 MB chunks (buffy19's ladder31 stride, about 4 min), because one byte per slot needs 33 GB. Pre-registered: lam3(31#) = G3(31#)/348 in [3.5, 4.2], i.e. G3 in [1218, 1462] with centre 1322, and G3(31#) below the density-matched control minimum. Refuted if G3 > 1566 or < 1044, or if G3 rises above the control minimum. Predicted psi(31#) about 0.75. (2) G3(37#) as a ride-along on route 358's own 37# pass (job #11077, P/6 = 1.24e12 slots, 2.1 CPU-h). It needs one extra bit for 6i+11 and should not be a re-run. Gates a larger range must reproduce: h = A048670 a(3..10) = 6, 10, 14, 22, 26, 34, 40, 46; G2 = 12, 30, 42, 66, 108, 150, 204, 258 (#5472, #3439); G3 = 24, 60, 126, 234, 390, 564, 774, 954; M3 = 2, 8, 64, 640, 8960, 143360, 2867200, 74547200; G3_mult = 1 at every rung; and the q = 1 control returning G2(23#) = 204 exactly. Controls: report quantiles and seed counts; the 17# position depends on the seed (review 1930: 57/64 seeds above G3, against the source's below-q05 reading). The triple-word thinning at its own density is an unrun control. Scope preserved: exact single rungs and no asymptotic.","assumptions_md":"Provisional relevance only: #5475 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358 k = 3 analogue, lane dir-558, explore job #11078 (predecessor #5472). Exact full-period enumeration at the eight rungs 5#..29# of h(x#) (one class), G2(x#) (twins, n, n+2) and G3(x#) = largest gap between consecutive n with gcd(n(n+2)(n+6), x#) = 1, with the pattern control {0,4,6}; G3 = 24, 60, 126, 234, 390, 564, 774, 954; lam3 = G3/G2 = 2.000, 2.000, 3.000, 3.545, 3.611, 3.760, 3.794, 3.698; plus a density-matched Bernoulli thinning of the twin word at q = M3/M2 (64/400/32 seeds). Rung measured, set by trusted review 1930 (accept, spot; my review), whose independent running-residue full-period walk reproduced h, G2, G3 and G3B at every rung, and whose splitmix64 thinning control confirmed the direction from 19#. Review 1930 advisories carried: lam3 is within +-0.05 only over 19#..29# (13#..29# spread 0.25); 13# and 17# lie below the 3.7-3.8 band; psi falls monotonically only from 11#/13#, not over the whole ladder; 'a k = 3 analogue costs nothing beyond route 358 plus a constant (G3 <= 4 x^{o(1)} h)' is an extrapolation from 8 points, i.e. a conjecture; G3({0,4,6}) = G3({0,2,6}) is forced by the reflection n -> -n-6, so the pattern control is not independent evidence; the 17# control position is seed-dependent. Not claimed: any bound on G3 or G2, any asymptotic, any rung beyond 29#, the triple-word thinning control, or anything about actual prime triples or twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"Not a route toward infinitude in this source's scope. G3(x#), G2(x#) and h(x#) are maximal gaps in periodic sieve words modulo x#. Upper bounds on them would control where admissible residues must recur in [1, x#], not whether prime triples or twin primes occur, because sieving only by primes <= x leaves large primes unaccounted for. The source states that nothing here bounds G3 in absolute terms (G3/h grows 4.0 -> 20.7). It is finite exact data with no asymptotic and gives no lower bound on twin counts.","assumptions_md":"Provisional relevance only: #5475 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358 k = 3 analogue, lane dir-558, explore job #11078 (predecessor #5472). Exact full-period enumeration at the eight rungs 5#..29# of h(x#) (one class), G2(x#) (twins, n, n+2) and G3(x#) = largest gap between consecutive n with gcd(n(n+2)(n+6), x#) = 1, with the pattern control {0,4,6}; G3 = 24, 60, 126, 234, 390, 564, 774, 954; lam3 = G3/G2 = 2.000, 2.000, 3.000, 3.545, 3.611, 3.760, 3.794, 3.698; plus a density-matched Bernoulli thinning of the twin word at q = M3/M2 (64/400/32 seeds). Rung measured, set by trusted review 1930 (accept, spot; my review), whose independent running-residue full-period walk reproduced h, G2, G3 and G3B at every rung, and whose splitmix64 thinning control confirmed the direction from 19#. Review 1930 advisories carried: lam3 is within +-0.05 only over 19#..29# (13#..29# spread 0.25); 13# and 17# lie below the 3.7-3.8 band; psi falls monotonically only from 11#/13#, not over the whole ladder; 'a k = 3 analogue costs nothing beyond route 358 plus a constant (G3 <= 4 x^{o(1)} h)' is an extrapolation from 8 points, i.e. a conjecture; G3({0,4,6}) = G3({0,2,6}) is forced by the reflection n -> -n-6, so the pattern control is not independent evidence; the 17# control position is seed-dependent. Not claimed: any bound on G3 or G2, any asymptotic, any rung beyond 29#, the triple-word thinning control, or anything about actual prime triples or twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"No finiteness consequence. #5475's objects are maximal gaps of the sieve words modulo x# for x <= 29. They exist and have the same values whether or not there are infinitely many twin primes, and the source says nothing about what a finite twin count would force.","assumptions_md":"Provisional relevance only: #5475 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358 k = 3 analogue, lane dir-558, explore job #11078 (predecessor #5472). Exact full-period enumeration at the eight rungs 5#..29# of h(x#) (one class), G2(x#) (twins, n, n+2) and G3(x#) = largest gap between consecutive n with gcd(n(n+2)(n+6), x#) = 1, with the pattern control {0,4,6}; G3 = 24, 60, 126, 234, 390, 564, 774, 954; lam3 = G3/G2 = 2.000, 2.000, 3.000, 3.545, 3.611, 3.760, 3.794, 3.698; plus a density-matched Bernoulli thinning of the twin word at q = M3/M2 (64/400/32 seeds). Rung measured, set by trusted review 1930 (accept, spot; my review), whose independent running-residue full-period walk reproduced h, G2, G3 and G3B at every rung, and whose splitmix64 thinning control confirmed the direction from 19#. Review 1930 advisories carried: lam3 is within +-0.05 only over 19#..29# (13#..29# spread 0.25); 13# and 17# lie below the 3.7-3.8 band; psi falls monotonically only from 11#/13#, not over the whole ladder; 'a k = 3 analogue costs nothing beyond route 358 plus a constant (G3 <= 4 x^{o(1)} h)' is an extrapolation from 8 points, i.e. a conjecture; G3({0,4,6}) = G3({0,2,6}) is forced by the reflection n -> -n-6, so the pattern control is not independent evidence; the 17# control position is seed-dependent. Not claimed: any bound on G3 or G2, any asymptotic, any rung beyond 29#, the triple-word thinning control, or anything about actual prime triples or twin primes.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Not one of the mapped claims. kk-lower-bound.main is the survivor-gap lower bound with full cutoff and sufficient A = 12, and O1-O7 are its asymptotic obligations. The nearest is O5, the external one-class statement. #5475 measures h(x#) only as a reproduced by-product (A048670 a(3..10)) and the multi-class gaps G2 and G3 at eight finite rungs. These are finite exact values. They are not a lower bound with quantifiers and logarithms, and per O5/O6 a finite statement does not close an asymptotic obligation. The trivial ordering G3 >= G2 >= h is finite and elementary and is not an O-input. Nothing transfers to main or to O1-O7.","assumptions_md":"Provisional relevance only: #5475 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 358 k = 3 analogue, lane dir-558, explore job #11078 (predecessor #5472). Exact full-period enumeration at the eight rungs 5#..29# of h(x#) (one class), G2(x#) (twins, n, n+2) and G3(x#) = largest gap between consecutive n with gcd(n(n+2)(n+6), x#) = 1, with the pattern control {0,4,6}; G3 = 24, 60, 126, 234, 390, 564, 774, 954; lam3 = G3/G2 = 2.000, 2.000, 3.000, 3.545, 3.611, 3.760, 3.794, 3.698; plus a density-matched Bernoulli thinning of the twin word at q = M3/M2 (64/400/32 seeds). Rung measured, set by trusted review 1930 (accept, spot; my review), whose independent running-residue full-period walk reproduced h, G2, G3 and G3B at every rung, and whose splitmix64 thinning control confirmed the direction from 19#. Review 1930 advisories carried: lam3 is within +-0.05 only over 19#..29# (13#..29# spread 0.25); 13# and 17# lie below the 3.7-3.8 band; psi falls monotonically only from 11#/13#, not over the whole ladder; 'a k = 3 analogue costs nothing beyond route 358 plus a constant (G3 <= 4 x^{o(1)} h)' is an extrapolation from 8 points, i.e. a conjecture; G3({0,4,6}) = G3({0,2,6}) is forced by the reflection n -> -n-6, so the pattern control is not independent evidence; the 17# control position is seed-dependent. Not claimed: any bound on G3 or G2, any asymptotic, any rung beyond 29#, the triple-word thinning control, or anything about actual prime triples or twin primes.","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":true},"effective":true,"check_job_id":11151,"input_sha256":"ccb964fb793660e8549d5002992a58347c757c55ec8cfd3a6f35ef426fd24da3","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"bfb5bdc0ef805bded510ae72dd10f9afffbe60700180e5a3ea3a4b76bad7b725","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5475,"material_evidence":{"sha256":"cb4c845e3ad050de5130dac2d150a3a6e7758378fe55f9b888be0ab88dab587c","entries":{"return:5475":"e454e8b2ce368e9c80537e25fbdb701dad661d0fed086e11e6f9688badf16a82","artifact:5475:05979b4176df9d5b42d12aada2b6ae6b6ffd228c43a11071499ff60782af2d3f":"05979b4176df9d5b42d12aada2b6ae6b6ffd228c43a11071499ff60782af2d3f","artifact:5475:1ee442e8171364898fb79452d4ff3294d21f5626911cdcc56361d6c5e0ff2a78":"1ee442e8171364898fb79452d4ff3294d21f5626911cdcc56361d6c5e0ff2a78","artifact:5475:1fe42434f3d2173461ccef7e83ed49523972768f3196aac75b3567ab8042c1f1":"1fe42434f3d2173461ccef7e83ed49523972768f3196aac75b3567ab8042c1f1","artifact:5475:4896cb56cb1aaec18726488f07d3dfbd695d04f05066d7c90a02c542b97887d7":"4896cb56cb1aaec18726488f07d3dfbd695d04f05066d7c90a02c542b97887d7","artifact:5475:48e44f456bdc718acc7737c1eda587bc83fdb73c2d106fcb1d51468de1b33b60":"48e44f456bdc718acc7737c1eda587bc83fdb73c2d106fcb1d51468de1b33b60","artifact:5475:51bacd2e9588d702b4212b913a4068ff7c68cac209997858c81022ea0743fbda":"51bacd2e9588d702b4212b913a4068ff7c68cac209997858c81022ea0743fbda","artifact:5475:6580aba7b998bb4f7ecdd7b454af367ac6f622301eaaf1e21f151b59a91e2720":"6580aba7b998bb4f7ecdd7b454af367ac6f622301eaaf1e21f151b59a91e2720","artifact:5475:65a471442f104ec82a898ff40c505f9d3705c4517a643b2a2b86990655121a3b":"65a471442f104ec82a898ff40c505f9d3705c4517a643b2a2b86990655121a3b","artifact:5475:76c68cc1552f86181b4bd9255d8c1bb4f5870cbaee185e7598c97b8d68a0c37c":"76c68cc1552f86181b4bd9255d8c1bb4f5870cbaee185e7598c97b8d68a0c37c","artifact:5475:7fe9561cd3ffdd7ace1f6075844d4737b549008b45d7f21cd0a11d1c367fb8af":"7fe9561cd3ffdd7ace1f6075844d4737b549008b45d7f21cd0a11d1c367fb8af","artifact:5475:a5f3054f65d38cc404d74f9f679aaedebaad7597dc109123c5cdc10763e09ad7":"a5f3054f65d38cc404d74f9f679aaedebaad7597dc109123c5cdc10763e09ad7","artifact:5475:ada18baa58555e2b725e820fac3bb9063f1e5ad1be5b74b382935a58329e70c9":"ada18baa58555e2b725e820fac3bb9063f1e5ad1be5b74b382935a58329e70c9","artifact:5475:bb23e84747fd3526d63fd28bab96ef6b591aab41f4cd821bba66ee82cba962a4":"bb23e84747fd3526d63fd28bab96ef6b591aab41f4cd821bba66ee82cba962a4","artifact:5475:bfb5bdc0ef805bded510ae72dd10f9afffbe60700180e5a3ea3a4b76bad7b725":"bfb5bdc0ef805bded510ae72dd10f9afffbe60700180e5a3ea3a4b76bad7b725","artifact:5475:c3a4b5df1f976890dfcc207c18bc4eaa1e614225fe0bca450a4d9f107a8fa6fc":"c3a4b5df1f976890dfcc207c18bc4eaa1e614225fe0bca450a4d9f107a8fa6fc","artifact:5475:c412aa2db7888b1749b5e215bf709d625f4b60b48f6a10ce82ce5fa8c7c60693":"c412aa2db7888b1749b5e215bf709d625f4b60b48f6a10ce82ce5fa8c7c60693","artifact:5475:d2dbf5ba5ed35846d38b05e9114de3ba65de31fd6ba312822228a0789903e25e":"d2dbf5ba5ed35846d38b05e9114de3ba65de31fd6ba312822228a0789903e25e","artifact:5475:d99222d04dd62296a0b85e5463e71f7f7869e76cbafc0d72093c754b9ce8f1e1":"d99222d04dd62296a0b85e5463e71f7f7869e76cbafc0d72093c754b9ce8f1e1","artifact:5475:f8a507aec3fe6a10617c45b661e481733e29a46831388e437035db7c548a1529":"f8a507aec3fe6a10617c45b661e481733e29a46831388e437035db7c548a1529"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5475,"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 #5475: 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\":5475,\"report_sha256\":\"bfb5bdc0ef805bded510ae72dd10f9afffbe60700180e5a3ea3a4b76bad7b725\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5475\":\"e454e8b2ce368e9c80537e25fbdb701dad661d0fed086e11e6f9688badf16a82\",\"artifact:5475:05979b4176df9d5b42d12aada2b6ae6b6ffd228c43a11071499ff60782af2d3f\":\"05979b4176df9d5b42d12aada2b6ae6b6ffd228c43a11071499ff60782af2d3f\",\"artifact:5475:1ee442e8171364898fb79452d4ff3294d21f5626911cdcc56361d6c5e0ff2a78\":\"1ee442e8171364898fb79452d4ff3294d21f5626911cdcc56361d6c5e0ff2a78\",\"artifact:5475:1fe42434f3d2173461ccef7e83ed49523972768f3196aac75b3567ab8042c1f1\":\"1fe42434f3d2173461ccef7e83ed49523972768f3196aac75b3567ab8042c1f1\",\"artifact:5475:4896cb56cb1aaec18726488f07d3dfbd695d04f05066d7c90a02c542b97887d7\":\"4896cb56cb1aaec18726488f07d3dfbd695d04f05066d7c90a02c542b97887d7\",\"artifact:5475:48e44f456bdc718acc7737c1eda587bc83fdb73c2d106fcb1d51468de1b33b60\":\"48e44f456bdc718acc7737c1eda587bc83fdb73c2d106fcb1d51468de1b33b60\",\"artifact:5475:51bacd2e9588d702b4212b913a4068ff7c68cac209997858c81022ea0743fbda\":\"51bacd2e9588d702b4212b913a4068ff7c68cac209997858c81022ea0743fbda\",\"artifact:5475:6580aba7b998bb4f7ecdd7b454af367ac6f622301eaaf1e21f151b59a91e2720\":\"6580aba7b998bb4f7ecdd7b454af367ac6f622301eaaf1e21f151b59a91e2720\",\"artifact:5475:65a471442f104ec82a898ff40c505f9d3705c4517a643b2a2b86990655121a3b\":\"65a471442f104ec82a898ff40c505f9d3705c4517a643b2a2b86990655121a3b\",\"artifact:5475:76c68cc1552f86181b4bd9255d8c1bb4f5870cbaee185e7598c97b8d68a0c37c\":\"76c68cc1552f86181b4bd9255d8c1bb4f5870cbaee185e7598c97b8d68a0c37c\",\"artifact:5475:7fe9561cd3ffdd7ace1f6075844d4737b549008b45d7f21cd0a11d1c367fb8af\":\"7fe9561cd3ffdd7ace1f6075844d4737b549008b45d7f21cd0a11d1c367fb8af\",\"artifact:5475:a5f3054f65d38cc404d74f9f679aaedebaad7597dc109123c5cdc10763e09ad7\":\"a5f3054f65d38cc404d74f9f679aaedebaad7597dc109123c5cdc10763e09ad7\",\"artifact:5475:ada18baa58555e2b725e820fac3bb9063f1e5ad1be5b74b382935a58329e70c9\":\"ada18baa58555e2b725e820fac3bb9063f1e5ad1be5b74b382935a58329e70c9\",\"artifact:5475:bb23e84747fd3526d63fd28bab96ef6b591aab41f4cd821bba66ee82cba962a4\":\"bb23e84747fd3526d63fd28bab96ef6b591aab41f4cd821bba66ee82cba962a4\",\"artifact:5475:bfb5bdc0ef805bded510ae72dd10f9afffbe60700180e5a3ea3a4b76bad7b725\":\"bfb5bdc0ef805bded510ae72dd10f9afffbe60700180e5a3ea3a4b76bad7b725\",\"artifact:5475:c3a4b5df1f976890dfcc207c18bc4eaa1e614225fe0bca450a4d9f107a8fa6fc\":\"c3a4b5df1f976890dfcc207c18bc4eaa1e614225fe0bca450a4d9f107a8fa6fc\",\"artifact:5475:c412aa2db7888b1749b5e215bf709d625f4b60b48f6a10ce82ce5fa8c7c60693\":\"c412aa2db7888b1749b5e215bf709d625f4b60b48f6a10ce82ce5fa8c7c60693\",\"artifact:5475:d2dbf5ba5ed35846d38b05e9114de3ba65de31fd6ba312822228a0789903e25e\":\"d2dbf5ba5ed35846d38b05e9114de3ba65de31fd6ba312822228a0789903e25e\",\"artifact:5475:d99222d04dd62296a0b85e5463e71f7f7869e76cbafc0d72093c754b9ce8f1e1\":\"d99222d04dd62296a0b85e5463e71f7f7869e76cbafc0d72093c754b9ce8f1e1\",\"artifact:5475:f8a507aec3fe6a10617c45b661e481733e29a46831388e437035db7c548a1529\":\"f8a507aec3fe6a10617c45b661e481733e29a46831388e437035db7c548a1529\"},\"sha256\":\"cb4c845e3ad050de5130dac2d150a3a6e7758378fe55f9b888be0ab88dab587c\"},\"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:5475,input_sha256:\"ccb964fb793660e8549d5002992a58347c757c55ec8cfd3a6f35ef426fd24da3\",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/5498/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"aa9a4848376321dd8040cff56e146715491a2d899e7f94cbe767a8a61c3c9f78","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}