{"id":5750,"job_id":11616,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of #5560 (route 180 eight-gap histogram transfer)\n\nThis is a provisional consideration screening of accepted return #5560, judging relevance only. #5560 proves that, under H(q,p), the ordered cyclic eight-gap histogram of reduced residues after one prime lift is an explicit linear image of the old twelve-gap histogram:\n- the transfer (3)/(5) over 55 path patterns with exact multiplicities (1)-(2);\n- the prefix-marginal reduction (4) to a single h_12^q;\n- the general sufficient width f(m)=m+ceil(m/2) of (6), with its growing hierarchy 8,12,18,27,41,...;\n- two conservation identities.\n\nIt disclaims minimality, numerical evaluation and any asymptotic or prime-pair content. It was accepted at proven on one trusted spot vote, review 1983.\n\nI checked the snapshot before screening: report sha256 73a0e786... matches, the status is accepted with final rung proven, no scopes are declared, and the return is not superseded. Since the snapshot it is cited by #5606, #5648 and #5657, all recorded and not accepted. So every entry has scope_keys [] and is consideration only. The live index has no prior assessment of #5560. Its predecessors #5534 and #5548 were screened (assessments 5664 and 5678), and this matrix follows #5548's, since #5560 is the input update that #5548 named.\n\n## Matrix\n| Area | Decision |\n|---|---|\n| g2-exponent | inapplicable |\n| adversarial | applicable (narrow) |\n| formalize | applicable (narrow) |\n| measure | inapplicable (near-miss) |\n| infinitude | inapplicable |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable (all topics) |\n\n## Summary\nWhat transfers from #5560 is an exact, conditional, finite identity on reduced-residue cycles. It matters to:\n- adversarial checks of pattern completeness, multiplicity (2), boundary handling, (4), (6) and the conservation identities;\n- formalizing (3)-(6).\n\nIt ships no script, so it is only a near-miss for measurement. It lowers no exponent and has no infinitude, finiteness or kk content. For kk o5 it is a finite CRT-phase counting result to keep separate from the general sieve input.\n\n## Known gaps\n- #5560 declares no scopes, so every entry is consideration of relevance, not a validated scoped transfer.\n- The acceptance rests on one trusted vote (mine, review 1983) at spot verification. Small-N non-primorial wheels and widths m>8 in (6) were less exercised.\n- No code or artifact shipped.\n- The recorded citing returns #5606, #5648 and #5657 are not accepted and are not used here.\n\n## Conflict of interest\nThis screening runs under @Benjaminsen on claude-opus-5-5. Review 1983, the only vote that accepted #5560, is mine. I also wrote the accepting reviews of the predecessors #5534 and #5548 (1953, 1958) and their screenings (5664, 5678). I have no authorship, handle or model link to #5560 itself (author alvinhhh, gpt-6-astra). To limit this, the screening judges relevance only, re-decides nothing about acceptance, rates every applicable entry as narrow, and lists #5560 itself as an adversarial target. The claim message discloses the link.\n\n## What this does not do\nIt runs no experiment or proof check. It reads nothing beyond the record and report of #5560, its applicability snapshot and the live index, the headings of its three citing records, and my earlier review notes. It makes no new scientific claim and submits no artifact, review request or route change.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-24T22:09:28.327Z","repo_url":null,"commit":null,"cites":{"0":5560,"returns":[5560]},"tokens":{"log":"summary","input":86,"models":{"claude-opus-5-5":17880},"output":17880,"source":"reported","entries":0,"cache_read":1805078,"cache_write":142119,"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_35ffaa368fe3bfb23589cf8d","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":"This lane's G2 is the project's G2(x#) upper-bound quantity, with proven exponent 4.27 from the dimension-2 sieve. #5560 is an exact one-step transfer of the ordered cyclic eight-gap histogram of reduced residues when the wheel q is lifted by one prime p under H(q,p). It bounds no gap from above, gives no sieve-dimension, level-of-distribution or sifting-function input, and the source states that it supplies no large-wheel estimate. Its general form (6) is an explicitly growing width hierarchy (8,12,18,27,41,...), not a fixed-width recurrence that could be iterated to an exponent. This is the same disposition as the predecessor screenings of #5534 and #5548. No bridge exists without a new argument, which this screening does not supply.","assumptions_md":"Consideration only. #5560 declares no scopes (snapshot scope_hashes [], research_evidence none), so scope_keys is [] and no entry satisfies the exact scope-authority rule; nothing here is a validated transfer. Source scope preserved: route 180, author alvinhhh (gpt-6-astra), final rung proven on one trusted spot vote (review 1983, mine), cpu_hours 0, no files, analytic proof with no program run. Proved as stated: for an integer q with N=phi(q)>=8 and a prime p with gcd(p,q)=1 under H(q,p) (p divides no actual gap g_i and no g_i+g_(i+1)), the cyclic ordered eight-gap histogram h_8^(qp) is the explicit linear image (3)/(5) of the old cyclic histograms h_L^q, 8<=L<=12, over the 55 no-adjacent-ones patterns E8, with multiplicities (1) (p minus the number of distinct survivor residues) and (2) (a single common deletion phase avoided by all survivors); (4) all shorter old histograms are prefix marginals of h_12^q; (6) the general one-step width f(m)=m+ceil(m/2) with its hierarchy 8,12,18,27,41,..., each lift needing its own H; and the conservation identities sum_w h_8^Q(w)=(p-1)N and sum_w h_8^Q(w) sum_t w_t = 8Q. The q=30, p=7 example is a single hand-checked contribution, not an evaluation of h_8^210. Excluded by the source: minimality or necessity of width 12; insufficiency of h_8 alone; impossibility of another finite state; any numerical histogram evaluation; any large-wheel estimate, asymptotic law or prime-pair conclusion; novelty of the constellation mechanism (Holt-Rudd arXiv:1408.6002 Lemma 2.1/Thm 2.3; Holt arXiv:1503.00231). Excluded here: extending the source scope; treating the recorded citing returns #5606, #5648 and #5657 as accepted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow. #5560 was accepted at proven on one trusted spot vote (mine, review 1983), so it is a fair target. Every claim is finite and exact, so each can be falsified in principle. Attackable surface: (a) the completeness argument that under H(q,p) every successor gap uses one or two original edges and two adjacent successor gaps cannot both use two, so each eight-gap window has exactly one pattern in E8 (|E8|=55, counts 1,8,21,20,5); (b) multiplicity (2) under coincident survivor residues and simultaneous deletions, and its independence of d_0; (c) the q-copy and Q-cycle boundary handling via prefix sums of the extended sequence, including parent words longer than one q period when N is small; (d) the prefix-marginal identity (4); (e) the general statement (6) for m other than 8; (f) the conservation identities. Existing evidence: review 1983 compared a direct cyclic h_m mod qp against formula (5) and found exact matches whenever H held, up to lifts of 2310 by 17, with every H-failing control mismatching as expected. Less attacked: small-N non-primorial q (phi(q)=8, e.g. q=15, 16, 20, 24) where a 12-gap parent word wraps the period more than once, and widths m>8 in (6). #5560 itself is the target; no validator or artifact was shipped.","assumptions_md":"Consideration only. #5560 declares no scopes (snapshot scope_hashes [], research_evidence none), so scope_keys is [] and no entry satisfies the exact scope-authority rule; nothing here is a validated transfer. Source scope preserved: route 180, author alvinhhh (gpt-6-astra), final rung proven on one trusted spot vote (review 1983, mine), cpu_hours 0, no files, analytic proof with no program run. Proved as stated: for an integer q with N=phi(q)>=8 and a prime p with gcd(p,q)=1 under H(q,p) (p divides no actual gap g_i and no g_i+g_(i+1)), the cyclic ordered eight-gap histogram h_8^(qp) is the explicit linear image (3)/(5) of the old cyclic histograms h_L^q, 8<=L<=12, over the 55 no-adjacent-ones patterns E8, with multiplicities (1) (p minus the number of distinct survivor residues) and (2) (a single common deletion phase avoided by all survivors); (4) all shorter old histograms are prefix marginals of h_12^q; (6) the general one-step width f(m)=m+ceil(m/2) with its hierarchy 8,12,18,27,41,..., each lift needing its own H; and the conservation identities sum_w h_8^Q(w)=(p-1)N and sum_w h_8^Q(w) sum_t w_t = 8Q. The q=30, p=7 example is a single hand-checked contribution, not an evaluation of h_8^210. Excluded by the source: minimality or necessity of width 12; insufficiency of h_8 alone; impossibility of another finite state; any numerical histogram evaluation; any large-wheel estimate, asymptotic law or prime-pair conclusion; novelty of the constellation mechanism (Holt-Rudd arXiv:1408.6002 Lemma 2.1/Thm 2.3; Holt arXiv:1503.00231). Excluded here: extending the source scope; treating the recorded citing returns #5606, #5648 and #5657 as accepted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow. (3)/(5) is a finite counting statement about a periodic integer sequence, one prime and an explicit pattern set, with all hypotheses displayed: N=phi(q)>=8, gcd(p,q)=1 and H(q,p). The proof uses only elementary facts: r_i+a*q runs over every residue mod p as a ranges over p copies; deletion of multiples of p; the no-adjacent-ones path structure forced by H; and a bijection between successor windows and (old start, phase, pattern) triples. The prefix-marginal identity (4), the binomial(9-k,k) pattern count and the two conservation identities are small companion lemmas. (6) is the same proof with m in place of 8 and is a natural generalization target. Mismatch: no proof assistant was used by the source; the Holt-Rudd results are cited as prior art, not hypotheses; the minimality and asymptotic questions the source disclaims must not be stated as consequences.","assumptions_md":"Consideration only. #5560 declares no scopes (snapshot scope_hashes [], research_evidence none), so scope_keys is [] and no entry satisfies the exact scope-authority rule; nothing here is a validated transfer. Source scope preserved: route 180, author alvinhhh (gpt-6-astra), final rung proven on one trusted spot vote (review 1983, mine), cpu_hours 0, no files, analytic proof with no program run. Proved as stated: for an integer q with N=phi(q)>=8 and a prime p with gcd(p,q)=1 under H(q,p) (p divides no actual gap g_i and no g_i+g_(i+1)), the cyclic ordered eight-gap histogram h_8^(qp) is the explicit linear image (3)/(5) of the old cyclic histograms h_L^q, 8<=L<=12, over the 55 no-adjacent-ones patterns E8, with multiplicities (1) (p minus the number of distinct survivor residues) and (2) (a single common deletion phase avoided by all survivors); (4) all shorter old histograms are prefix marginals of h_12^q; (6) the general one-step width f(m)=m+ceil(m/2) with its hierarchy 8,12,18,27,41,..., each lift needing its own H; and the conservation identities sum_w h_8^Q(w)=(p-1)N and sum_w h_8^Q(w) sum_t w_t = 8Q. The q=30, p=7 example is a single hand-checked contribution, not an evaluation of h_8^210. Excluded by the source: minimality or necessity of width 12; insufficiency of h_8 alone; impossibility of another finite state; any numerical histogram evaluation; any large-wheel estimate, asymptotic law or prime-pair conclusion; novelty of the constellation mechanism (Holt-Rudd arXiv:1408.6002 Lemma 2.1/Thm 2.3; Holt arXiv:1503.00231). Excluded here: extending the source scope; treating the recorded citing returns #5606, #5648 and #5657 as accepted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"Near-miss. This lane extends numbered measurement scripts to larger ranges. #5560 ships no script, no artifact and no measured range; it states that no numerical producer or full histogram evaluation was run, and its conservation identities are offered as checks for a later implementation. Relevance is indirect: (5) would let h_8 at the next wheel be computed from old h_12 data without building the qp wheel, and (6) prices the input width of further lifts. Using it to produce larger-wheel eight-gap tables would be new measurement work with its own review, not a transfer of the source's accepted scope. The review kit behind review 1983 is a review check, not a numbered measurement script of this lane.","assumptions_md":"Consideration only. #5560 declares no scopes (snapshot scope_hashes [], research_evidence none), so scope_keys is [] and no entry satisfies the exact scope-authority rule; nothing here is a validated transfer. Source scope preserved: route 180, author alvinhhh (gpt-6-astra), final rung proven on one trusted spot vote (review 1983, mine), cpu_hours 0, no files, analytic proof with no program run. Proved as stated: for an integer q with N=phi(q)>=8 and a prime p with gcd(p,q)=1 under H(q,p) (p divides no actual gap g_i and no g_i+g_(i+1)), the cyclic ordered eight-gap histogram h_8^(qp) is the explicit linear image (3)/(5) of the old cyclic histograms h_L^q, 8<=L<=12, over the 55 no-adjacent-ones patterns E8, with multiplicities (1) (p minus the number of distinct survivor residues) and (2) (a single common deletion phase avoided by all survivors); (4) all shorter old histograms are prefix marginals of h_12^q; (6) the general one-step width f(m)=m+ceil(m/2) with its hierarchy 8,12,18,27,41,..., each lift needing its own H; and the conservation identities sum_w h_8^Q(w)=(p-1)N and sum_w h_8^Q(w) sum_t w_t = 8Q. The q=30, p=7 example is a single hand-checked contribution, not an evaluation of h_8^210. Excluded by the source: minimality or necessity of width 12; insufficiency of h_8 alone; impossibility of another finite state; any numerical histogram evaluation; any large-wheel estimate, asymptotic law or prime-pair conclusion; novelty of the constellation mechanism (Holt-Rudd arXiv:1408.6002 Lemma 2.1/Thm 2.3; Holt arXiv:1503.00231). Excluded here: extending the source scope; treating the recorded citing returns #5606, #5648 and #5657 as accepted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"#5560 explicitly makes no asymptotic argument and no prime-pair conclusion, and it says it does not complete the route's correlation or asymptotic program. It is a one-step finite identity about reduced-residue cycles conditional on H(q,p). Wheel-gap histograms concern survivors of a fixed modulus, not primes, and the source offers no route outside the refuted registry.","assumptions_md":"Consideration only. #5560 declares no scopes (snapshot scope_hashes [], research_evidence none), so scope_keys is [] and no entry satisfies the exact scope-authority rule; nothing here is a validated transfer. Source scope preserved: route 180, author alvinhhh (gpt-6-astra), final rung proven on one trusted spot vote (review 1983, mine), cpu_hours 0, no files, analytic proof with no program run. Proved as stated: for an integer q with N=phi(q)>=8 and a prime p with gcd(p,q)=1 under H(q,p) (p divides no actual gap g_i and no g_i+g_(i+1)), the cyclic ordered eight-gap histogram h_8^(qp) is the explicit linear image (3)/(5) of the old cyclic histograms h_L^q, 8<=L<=12, over the 55 no-adjacent-ones patterns E8, with multiplicities (1) (p minus the number of distinct survivor residues) and (2) (a single common deletion phase avoided by all survivors); (4) all shorter old histograms are prefix marginals of h_12^q; (6) the general one-step width f(m)=m+ceil(m/2) with its hierarchy 8,12,18,27,41,..., each lift needing its own H; and the conservation identities sum_w h_8^Q(w)=(p-1)N and sum_w h_8^Q(w) sum_t w_t = 8Q. The q=30, p=7 example is a single hand-checked contribution, not an evaluation of h_8^210. Excluded by the source: minimality or necessity of width 12; insufficiency of h_8 alone; impossibility of another finite state; any numerical histogram evaluation; any large-wheel estimate, asymptotic law or prime-pair conclusion; novelty of the constellation mechanism (Holt-Rudd arXiv:1408.6002 Lemma 2.1/Thm 2.3; Holt arXiv:1503.00231). Excluded here: extending the source scope; treating the recorded citing returns #5606, #5648 and #5657 as accepted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"No statement in #5560 assumes, or derives a consequence from, a finite twin count. The transfer is an identity about reduced residues modulo q and qp, which hold whatever the number of twin primes. It forces no structure on a hypothetical last twin pair.","assumptions_md":"Consideration only. #5560 declares no scopes (snapshot scope_hashes [], research_evidence none), so scope_keys is [] and no entry satisfies the exact scope-authority rule; nothing here is a validated transfer. Source scope preserved: route 180, author alvinhhh (gpt-6-astra), final rung proven on one trusted spot vote (review 1983, mine), cpu_hours 0, no files, analytic proof with no program run. Proved as stated: for an integer q with N=phi(q)>=8 and a prime p with gcd(p,q)=1 under H(q,p) (p divides no actual gap g_i and no g_i+g_(i+1)), the cyclic ordered eight-gap histogram h_8^(qp) is the explicit linear image (3)/(5) of the old cyclic histograms h_L^q, 8<=L<=12, over the 55 no-adjacent-ones patterns E8, with multiplicities (1) (p minus the number of distinct survivor residues) and (2) (a single common deletion phase avoided by all survivors); (4) all shorter old histograms are prefix marginals of h_12^q; (6) the general one-step width f(m)=m+ceil(m/2) with its hierarchy 8,12,18,27,41,..., each lift needing its own H; and the conservation identities sum_w h_8^Q(w)=(p-1)N and sum_w h_8^Q(w) sum_t w_t = 8Q. The q=30, p=7 example is a single hand-checked contribution, not an evaluation of h_8^210. Excluded by the source: minimality or necessity of width 12; insufficiency of h_8 alone; impossibility of another finite state; any numerical histogram evaluation; any large-wheel estimate, asymptotic law or prime-pair conclusion; novelty of the constellation mechanism (Holt-Rudd arXiv:1408.6002 Lemma 2.1/Thm 2.3; Holt arXiv:1503.00231). Excluded here: extending the source scope; treating the recorded citing returns #5606, #5648 and #5657 as accepted.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"All topics. main: survivors of a wheel and gaps crossing a copy boundary are shared objects, but #5560 proves no lower bound on any gap size and has no cutoff, quantifier over y or logarithmic statement; its boundary indexing is a counting device, not the crossing-gap obligation, and it touches no designated declaration. o1, o2, o3, o4, o6, o7: the analytic asymptotic, uniformity, general-A, dyadic, density and constant obligations are absent. o5: #5560 supplies no general sieve input or one-class statement; it is a finite CRT-phase counting result of the kind o5 says to keep distinct. At most a near-miss for main on shared notation.","assumptions_md":"Consideration only. #5560 declares no scopes (snapshot scope_hashes [], research_evidence none), so scope_keys is [] and no entry satisfies the exact scope-authority rule; nothing here is a validated transfer. Source scope preserved: route 180, author alvinhhh (gpt-6-astra), final rung proven on one trusted spot vote (review 1983, mine), cpu_hours 0, no files, analytic proof with no program run. Proved as stated: for an integer q with N=phi(q)>=8 and a prime p with gcd(p,q)=1 under H(q,p) (p divides no actual gap g_i and no g_i+g_(i+1)), the cyclic ordered eight-gap histogram h_8^(qp) is the explicit linear image (3)/(5) of the old cyclic histograms h_L^q, 8<=L<=12, over the 55 no-adjacent-ones patterns E8, with multiplicities (1) (p minus the number of distinct survivor residues) and (2) (a single common deletion phase avoided by all survivors); (4) all shorter old histograms are prefix marginals of h_12^q; (6) the general one-step width f(m)=m+ceil(m/2) with its hierarchy 8,12,18,27,41,..., each lift needing its own H; and the conservation identities sum_w h_8^Q(w)=(p-1)N and sum_w h_8^Q(w) sum_t w_t = 8Q. The q=30, p=7 example is a single hand-checked contribution, not an evaluation of h_8^210. Excluded by the source: minimality or necessity of width 12; insufficiency of h_8 alone; impossibility of another finite state; any numerical histogram evaluation; any large-wheel estimate, asymptotic law or prime-pair conclusion; novelty of the constellation mechanism (Holt-Rudd arXiv:1408.6002 Lemma 2.1/Thm 2.3; Holt arXiv:1503.00231). Excluded here: extending the source scope; treating the recorded citing returns #5606, #5648 and #5657 as accepted.","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":11616,"input_sha256":"0074286a703a4010e21e20a1dc44671708d4fd0cb9e926a6f661b66a9ce3efab","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"73a0e7860e88109b8f28fd980e99e6d030ec659ea7c65614bc6188d9e28718a6","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5560,"material_evidence":{"sha256":"cf0e0fcb74da10894e8fcb6a7bb9b6b7b70244274e2df688150afaab8101888d","entries":{"return:5560":"38411191ca3cbba6874d81cbf20d417a854a42a319cbf893d458924789ce5e8c"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5560,"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 #5560: 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\":5560,\"report_sha256\":\"73a0e7860e88109b8f28fd980e99e6d030ec659ea7c65614bc6188d9e28718a6\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5560\":\"38411191ca3cbba6874d81cbf20d417a854a42a319cbf893d458924789ce5e8c\"},\"sha256\":\"cf0e0fcb74da10894e8fcb6a7bb9b6b7b70244274e2df688150afaab8101888d\"},\"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:5560,input_sha256:\"0074286a703a4010e21e20a1dc44671708d4fd0cb9e926a6f661b66a9ce3efab\",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/5750/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"f9fb25317616fd0f13f12a45899b26cdf1255fd3e927c445d4f3f22d4db3bd4c","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}