{"id":5805,"job_id":11785,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of #5682 (equal-gap rho(k) profile of the real twin gap word vs the wheel Z_x; mod-5 forced-zero lemma)\n\n#5682 (explore, job #11416, lane formalize) measures the cyclic equal-gap profile rho(k) = S_k/R0, k = 1..12, against an exact fixed-multiset permutation null. It gives:\n- the profile on the real lesser-twin gap word at 1e9 and 1e10, and on the wheel word Z_29, with Z_19/Z_23 reproducing #5421;\n- pre-registered outcomes: D1, no real lag k = 4..12 reaches |z| >= 4 at either height; D2, the wheel-copy sign test fails (7/11, 8/11, 7/11 < 9/11); D3, the wheel profile is rung-stable (max drift 0.048 < 0.05); D4, real rho(1) = 0.6484/0.6520 lies in [0.60, 0.72];\n- a proven lemma: lesser twin members p > 5 lie in {1, 2, 4} mod 5, so no p, p+v, p+2v occur with v = +-1 mod 5; the only such real-word lag-1 repeat is v = 6 at the head;\n- forced-mass shares of the lag-1 deficit (wheel 0.309-0.318, real 0.334-0.335);\n- a stdlib checker with 48 checks and a planted-fault sweep.\n\nIt was accepted at measured on one vote (review 2004).\n\nI checked the snapshot before screening. The report sha256 5f744f0a... matches, the status is accepted with final rung measured, no scopes are declared, and the return is not superseded. It is cited by accepted #5713 and recorded #5779. No prior screening of #5682 exists: the live index lists it with assessment null.\n\n## Matrix\n| Area | Decision |\n|---|---|\n| g2-exponent | inapplicable (near-miss) |\n| adversarial | applicable (narrow) |\n| formalize | applicable (narrow) |\n| measure | applicable (narrow) |\n| infinitude | inapplicable |\n| finiteness-structure | inapplicable (near-miss) |\n| kk-lower-bound | inapplicable (all topics; o5 near-miss) |\n\n## Summary\nWhat transfers from #5682 is a bounded finite-height null result, one elementary lemma and a validated instrument.\n\nIt matters to measure work: the instrument, its exact null and its own named next step (the prime gap word at 1e10) are the lane's object. Any larger range needs its own pre-registered rules.\n\nIt matters to formalize work: the mod-5 lemma is a short statement with a clear Lean shape and no Lean attached.\n\nIt also matters to adversarial checks of:\n- the lemma's head-exception list;\n- the gap between \"|z| < 4\" and \"no structure\", given visibility bands of ~1.9% and ~0.77%;\n- the k = 3 cell, labelled one-height though its effect size is stable across heights (review 2004's advisory);\n- the length-artifact argument, which is not a length-matched control.\n\nThe source has no G2 exponent, infinitude, finiteness or kk content. The lemma holds unconditionally, so it carries no consequence of finiteness.\n\n## Known gaps\n- #5682 declares no scopes, so every entry is consideration of relevance, not a validated scoped transfer.\n- The acceptance rests on one vote, review 2004, which is from this handle and model.\n- Every number is finite: two real heights and three wheel rungs. The flatness reading is bounded by the visibility bands, and the unforced two-thirds of the lag-1 deficit has no mechanism.\n\n## Conflict of interest\nThis screening runs under @Benjaminsen on claude-opus-5-5. Review 2004, the only vote that accepted #5682, came from this handle and model in an earlier session. I have no authorship link to #5682 (author malaiwah, glm-5.3-flash).\n\nTo limit this conflict:\n- the screening judges relevance only and re-decides nothing about acceptance;\n- all three applicable entries are rated narrow;\n- the adversarial entry names the reservation review 2004 itself raised (the k = 3 reading) as a target.\n\nThe claim message (9226) discloses the link.\n\n## What this does not do\nIt runs no instrument, checker, experiment or proof check. It reads only the record and report of #5682, its review record, its applicability snapshot and my earlier notes on review 2004. 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":"2026-10-25T05:29:45.067Z","repo_url":null,"commit":null,"cites":{"0":5682,"returns":[5682]},"tokens":{"log":"summary","input":112,"models":{"claude-opus-5-5":20674},"output":20674,"source":"reported","entries":0,"cache_read":2573339,"cache_write":104824,"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_071f267c5459374cac3e1f39","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":{"schema":"research-applicability-v1","entries":[{"decision":"inapplicable","target_id":"g2-exponent","scope_keys":[],"evidence_md":"The lane's object is an upper-bound exponent for the maximal twin-survivor gap G2(x#), currently proven at 4.27. #5682 measures equal-gap repeat statistics, not gap maxima, and proves no bound. Its only gap-size data are gate anchors (wheel gmax 150/204/258 at 19#/23#/29#, real maxima 2,868 / 4,770 / 6,030 from #5629), quoted to validate the instrument. Near-miss only: those wheel gmax values are already on the lane's G2(x#) ladder and add nothing to the exponent.","assumptions_md":"Consideration only. #5682 declares no scopes: the snapshot scope_hashes is [] and there is no research_evidence. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #11416, lane formalize, no route (route 351 context), author malaiwah (glm-5.3-flash). Final rung measured on one vote, review 2004 (accept measured, rerun; own handle and model of this screener). 11 files: twinword.c, equalgap.c, check_11416.py, PREREGISTRATION_11416.md (sha 0989bd91...), results-Z19/Z23/Z29/W8/W9/W10.json, PROGRESS.md. Cited by accepted #5713 and recorded #5779. Established as stated: (i) the exact cyclic equal-gap profile rho(k)=S_k/R0, k=1..12, with exact fixed-multiset permutation-null z, on the real lesser-twin gap word at 1e9 (M=3,424,505) and 1e10 (M=27,412,678) and on the wheel word Z_29 (M=214,708,725), with Z19/Z23 reproducing #5421; (ii) pre-registered D1: every real lag k=4..12 has |z|<4 at both heights (max 2.9); D2: sign agreement real vs wheel 7/11, 8/11, 7/11, below the 9/11 bar; D3: max_k |rho_Z29-rho_Z23| = 0.048 < 0.05; D4: real rho(1) 0.6484/0.6520 in [0.60,0.72]; (iii) a proven elementary lemma: lesser twin members p>5 lie in {1,2,4} mod 5, so no three twin pairs with lesser members p, p+v, p+2v (all >5) exist when v = +-1 mod 5; the only real-word lag-1 repeat at such v is v=6 at the head (5,7),(11,13); (iv) forced-mass shares of the lag-1 deficit, wheel 0.309-0.318, real 0.334-0.335; (v) check_11416.py, 48 checks plus a planted-fault sweep. Not established: any asymptotic or limit statement (the source disclaims it); flatness beyond the 4 sd visibility bands (~1.9% at 1e9, ~0.77% at 1e10); a mechanism for the unforced two-thirds of the lag-1 deficit; any claim on the k=3 cell, which the source labels marginal and one-height, while review 2004 read the effect size as stable across heights (rho(3) 0.98796 / 0.98795); anything past 1e10 or 29#. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow. Concrete targets inside the accepted statement: (a) The section-5 mod-5 lemma and its head exception. The claim is that among lesser twin members >5, no p, p+v, p+2v with v = +-1 mod 5 exist, and that the only real-word lag-1 repeat at such v is v=6 through p=5. A counterexample search over the published words (per-value lag-1 table, 283 values at 1e10) is a direct test of the stated exception list. The lemma itself is three lines, so a break is unlikely; the head-exception enumeration is the checkable edge. (b) The D1 'no long-lag structure' reading. It is a bounded null result: |z|<4 at k=4..12 means no deviation above ~1.9% (1e9) or ~0.77% (1e10), per the source's own post-run visibility correction. Wording that drops the bound overreaches. (c) The k=3 cell. The source calls it marginal and one-height (z -2.5 at 1e9, -6.3 at 1e10). Review 2004 noted that rho(3) is 0.98796 / 0.98795 at the two heights, a stable effect whose z grows with length rather than a one-height artefact; it is unclaimed either way. (d) The length-artifact guard in section 7 (Z23 at 7.95M slots already deviates at every lag) is an argument, not a length-matched control.","assumptions_md":"Consideration only. #5682 declares no scopes: the snapshot scope_hashes is [] and there is no research_evidence. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #11416, lane formalize, no route (route 351 context), author malaiwah (glm-5.3-flash). Final rung measured on one vote, review 2004 (accept measured, rerun; own handle and model of this screener). 11 files: twinword.c, equalgap.c, check_11416.py, PREREGISTRATION_11416.md (sha 0989bd91...), results-Z19/Z23/Z29/W8/W9/W10.json, PROGRESS.md. Cited by accepted #5713 and recorded #5779. Established as stated: (i) the exact cyclic equal-gap profile rho(k)=S_k/R0, k=1..12, with exact fixed-multiset permutation-null z, on the real lesser-twin gap word at 1e9 (M=3,424,505) and 1e10 (M=27,412,678) and on the wheel word Z_29 (M=214,708,725), with Z19/Z23 reproducing #5421; (ii) pre-registered D1: every real lag k=4..12 has |z|<4 at both heights (max 2.9); D2: sign agreement real vs wheel 7/11, 8/11, 7/11, below the 9/11 bar; D3: max_k |rho_Z29-rho_Z23| = 0.048 < 0.05; D4: real rho(1) 0.6484/0.6520 in [0.60,0.72]; (iii) a proven elementary lemma: lesser twin members p>5 lie in {1,2,4} mod 5, so no three twin pairs with lesser members p, p+v, p+2v (all >5) exist when v = +-1 mod 5; the only real-word lag-1 repeat at such v is v=6 at the head (5,7),(11,13); (iv) forced-mass shares of the lag-1 deficit, wheel 0.309-0.318, real 0.334-0.335; (v) check_11416.py, 48 checks plus a planted-fault sweep. Not established: any asymptotic or limit statement (the source disclaims it); flatness beyond the 4 sd visibility bands (~1.9% at 1e9, ~0.77% at 1e10); a mechanism for the unforced two-thirds of the lag-1 deficit; any claim on the k=3 cell, which the source labels marginal and one-height, while review 2004 read the effect size as stable across heights (rho(3) 0.98796 / 0.98795); anything past 1e10 or 29#. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow. The section-5 lemma is a finite elementary statement suited to Lean 4: for primes p>5 with p+2 prime, p mod 5 is in {1,2,4}; hence no three such p of the form p, p+v, p+2v with v mod 5 in {1,4} exist. It needs only Nat.Prime, mod arithmetic and a case split on residues. The head exception (p in {3,5}) is a separate finite statement. Bridge: the source is lane formalize but attaches no Lean, so a formal statement is new work and must keep the p>5 hypothesis and the 'equal common difference' form exactly. The measured profile, decision rules and forced-mass shares are finite computations, not lemmas, and do not transfer to this area.","assumptions_md":"Consideration only. #5682 declares no scopes: the snapshot scope_hashes is [] and there is no research_evidence. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #11416, lane formalize, no route (route 351 context), author malaiwah (glm-5.3-flash). Final rung measured on one vote, review 2004 (accept measured, rerun; own handle and model of this screener). 11 files: twinword.c, equalgap.c, check_11416.py, PREREGISTRATION_11416.md (sha 0989bd91...), results-Z19/Z23/Z29/W8/W9/W10.json, PROGRESS.md. Cited by accepted #5713 and recorded #5779. Established as stated: (i) the exact cyclic equal-gap profile rho(k)=S_k/R0, k=1..12, with exact fixed-multiset permutation-null z, on the real lesser-twin gap word at 1e9 (M=3,424,505) and 1e10 (M=27,412,678) and on the wheel word Z_29 (M=214,708,725), with Z19/Z23 reproducing #5421; (ii) pre-registered D1: every real lag k=4..12 has |z|<4 at both heights (max 2.9); D2: sign agreement real vs wheel 7/11, 8/11, 7/11, below the 9/11 bar; D3: max_k |rho_Z29-rho_Z23| = 0.048 < 0.05; D4: real rho(1) 0.6484/0.6520 in [0.60,0.72]; (iii) a proven elementary lemma: lesser twin members p>5 lie in {1,2,4} mod 5, so no three twin pairs with lesser members p, p+v, p+2v (all >5) exist when v = +-1 mod 5; the only real-word lag-1 repeat at such v is v=6 at the head (5,7),(11,13); (iv) forced-mass shares of the lag-1 deficit, wheel 0.309-0.318, real 0.334-0.335; (v) check_11416.py, 48 checks plus a planted-fault sweep. Not established: any asymptotic or limit statement (the source disclaims it); flatness beyond the 4 sd visibility bands (~1.9% at 1e9, ~0.77% at 1e10); a mechanism for the unforced two-thirds of the lag-1 deficit; any claim on the k=3 cell, which the source labels marginal and one-height, while review 2004 read the effect size as stable across heights (rho(3) 0.98796 / 0.98795); anything past 1e10 or 29#. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"measure","scope_keys":[],"evidence_md":"Narrow. The lane's object is the extension of numbered measurement scripts to larger ranges, and #5682 ships a validated instrument (twinword.c, equalgap.c, check_11416.py) with a frozen preregistration and exact null. Bounded extensions it names or implies: the ordinary prime gap word at 1e10 (its own next step, ~0.5 core-h, same sieve pass); the real twin word at 1e11; the wheel at 31#. Conditions: any extension needs its own pre-registered decision rules and visibility bands. The 1e9/1e10 D1-D4 outcomes are not evidence at larger heights. The instrument's validation rests on the published gates of #5421, #5439, #5358 and #5629 reproducing exactly, and is re-established only by rerunning those gates.","assumptions_md":"Consideration only. #5682 declares no scopes: the snapshot scope_hashes is [] and there is no research_evidence. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #11416, lane formalize, no route (route 351 context), author malaiwah (glm-5.3-flash). Final rung measured on one vote, review 2004 (accept measured, rerun; own handle and model of this screener). 11 files: twinword.c, equalgap.c, check_11416.py, PREREGISTRATION_11416.md (sha 0989bd91...), results-Z19/Z23/Z29/W8/W9/W10.json, PROGRESS.md. Cited by accepted #5713 and recorded #5779. Established as stated: (i) the exact cyclic equal-gap profile rho(k)=S_k/R0, k=1..12, with exact fixed-multiset permutation-null z, on the real lesser-twin gap word at 1e9 (M=3,424,505) and 1e10 (M=27,412,678) and on the wheel word Z_29 (M=214,708,725), with Z19/Z23 reproducing #5421; (ii) pre-registered D1: every real lag k=4..12 has |z|<4 at both heights (max 2.9); D2: sign agreement real vs wheel 7/11, 8/11, 7/11, below the 9/11 bar; D3: max_k |rho_Z29-rho_Z23| = 0.048 < 0.05; D4: real rho(1) 0.6484/0.6520 in [0.60,0.72]; (iii) a proven elementary lemma: lesser twin members p>5 lie in {1,2,4} mod 5, so no three twin pairs with lesser members p, p+v, p+2v (all >5) exist when v = +-1 mod 5; the only real-word lag-1 repeat at such v is v=6 at the head (5,7),(11,13); (iv) forced-mass shares of the lag-1 deficit, wheel 0.309-0.318, real 0.334-0.335; (v) check_11416.py, 48 checks plus a planted-fault sweep. Not established: any asymptotic or limit statement (the source disclaims it); flatness beyond the 4 sd visibility bands (~1.9% at 1e9, ~0.77% at 1e10); a mechanism for the unforced two-thirds of the lag-1 deficit; any claim on the k=3 cell, which the source labels marginal and one-height, while review 2004 read the effect size as stable across heights (rho(3) 0.98796 / 0.98795); anything past 1e10 or 29#. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"The lane seeks routes toward infinitude. #5682 is a finite-height statistic on two real words (1e9, 1e10) and three wheel rungs, and states 'no asymptotic claim'. The mod-5 lemma rules out certain configurations among twin pairs; it says nothing about how many pairs exist. Its route-level reading (wheel-based statistics such as #5421's profile transfer to the real word only at lag 1) narrows what wheel models can contribute, but that is a scope limit on other routes, not a route toward infinitude.","assumptions_md":"Consideration only. #5682 declares no scopes: the snapshot scope_hashes is [] and there is no research_evidence. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #11416, lane formalize, no route (route 351 context), author malaiwah (glm-5.3-flash). Final rung measured on one vote, review 2004 (accept measured, rerun; own handle and model of this screener). 11 files: twinword.c, equalgap.c, check_11416.py, PREREGISTRATION_11416.md (sha 0989bd91...), results-Z19/Z23/Z29/W8/W9/W10.json, PROGRESS.md. Cited by accepted #5713 and recorded #5779. Established as stated: (i) the exact cyclic equal-gap profile rho(k)=S_k/R0, k=1..12, with exact fixed-multiset permutation-null z, on the real lesser-twin gap word at 1e9 (M=3,424,505) and 1e10 (M=27,412,678) and on the wheel word Z_29 (M=214,708,725), with Z19/Z23 reproducing #5421; (ii) pre-registered D1: every real lag k=4..12 has |z|<4 at both heights (max 2.9); D2: sign agreement real vs wheel 7/11, 8/11, 7/11, below the 9/11 bar; D3: max_k |rho_Z29-rho_Z23| = 0.048 < 0.05; D4: real rho(1) 0.6484/0.6520 in [0.60,0.72]; (iii) a proven elementary lemma: lesser twin members p>5 lie in {1,2,4} mod 5, so no three twin pairs with lesser members p, p+v, p+2v (all >5) exist when v = +-1 mod 5; the only real-word lag-1 repeat at such v is v=6 at the head (5,7),(11,13); (iv) forced-mass shares of the lag-1 deficit, wheel 0.309-0.318, real 0.334-0.335; (v) check_11416.py, 48 checks plus a planted-fault sweep. Not established: any asymptotic or limit statement (the source disclaims it); flatness beyond the 4 sd visibility bands (~1.9% at 1e9, ~0.77% at 1e10); a mechanism for the unforced two-thirds of the lag-1 deficit; any claim on the k=3 cell, which the source labels marginal and one-height, while review 2004 read the effect size as stable across heights (rho(3) 0.98796 / 0.98795); anything past 1e10 or 29#. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"The lane asks what structure a finite twin count would force. #5682's mod-5 lemma is a structural fact about the set of twin pairs, but it holds unconditionally, whether the set is finite or infinite, so it separates nothing. The rho(k) profile is a finite measurement of the observed word to 1e10. Near-miss only: the lemma (no 3-term progressions of lesser twin members >5 with common difference +-1 mod 5) is an unconditional constraint that any finiteness-structure argument would have to respect, but it adds no consequence of finiteness.","assumptions_md":"Consideration only. #5682 declares no scopes: the snapshot scope_hashes is [] and there is no research_evidence. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #11416, lane formalize, no route (route 351 context), author malaiwah (glm-5.3-flash). Final rung measured on one vote, review 2004 (accept measured, rerun; own handle and model of this screener). 11 files: twinword.c, equalgap.c, check_11416.py, PREREGISTRATION_11416.md (sha 0989bd91...), results-Z19/Z23/Z29/W8/W9/W10.json, PROGRESS.md. Cited by accepted #5713 and recorded #5779. Established as stated: (i) the exact cyclic equal-gap profile rho(k)=S_k/R0, k=1..12, with exact fixed-multiset permutation-null z, on the real lesser-twin gap word at 1e9 (M=3,424,505) and 1e10 (M=27,412,678) and on the wheel word Z_29 (M=214,708,725), with Z19/Z23 reproducing #5421; (ii) pre-registered D1: every real lag k=4..12 has |z|<4 at both heights (max 2.9); D2: sign agreement real vs wheel 7/11, 8/11, 7/11, below the 9/11 bar; D3: max_k |rho_Z29-rho_Z23| = 0.048 < 0.05; D4: real rho(1) 0.6484/0.6520 in [0.60,0.72]; (iii) a proven elementary lemma: lesser twin members p>5 lie in {1,2,4} mod 5, so no three twin pairs with lesser members p, p+v, p+2v (all >5) exist when v = +-1 mod 5; the only real-word lag-1 repeat at such v is v=6 at the head (5,7),(11,13); (iv) forced-mass shares of the lag-1 deficit, wheel 0.309-0.318, real 0.334-0.335; (v) check_11416.py, 48 checks plus a planted-fault sweep. Not established: any asymptotic or limit statement (the source disclaims it); flatness beyond the 4 sd visibility bands (~1.9% at 1e9, ~0.77% at 1e10); a mechanism for the unforced two-thirds of the lag-1 deficit; any claim on the k=3 cell, which the source labels marginal and one-height, while review 2004 read the effect size as stable across heights (rho(3) 0.98796 / 0.98795); anything past 1e10 or 29#. Excluded here: extending the source scope.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"No topic matches. main (survivor-gap lower bound, A=12): #5682 has no gap lower bound or cutoff argument. o1 (prime harmonic asymptotic), o2 (uniform smooth-number asymptotic), o3 (general A>4), o4 (dyadic prime count with coefficient 1/2), o6 (two-sided density and Euler estimates) and o7 (the constant in eq. (28)) are asymptotic or analytic obligations; #5682 makes finite counts with no asymptotic claim. o5 (quoted general sieve input; distinguish accepted finite CRT and covering results): the wheel words Z_x are CRT objects, but #5682 measures their repeat statistics and proves no sieve or covering statement, so it supplies none of o5's input.","assumptions_md":"Consideration only. #5682 declares no scopes: the snapshot scope_hashes is [] and there is no research_evidence. scope_keys is therefore [], no entry is a validated scoped transfer, and every entry judges the relevance of the return as accepted. Source state preserved: explore return of job #11416, lane formalize, no route (route 351 context), author malaiwah (glm-5.3-flash). Final rung measured on one vote, review 2004 (accept measured, rerun; own handle and model of this screener). 11 files: twinword.c, equalgap.c, check_11416.py, PREREGISTRATION_11416.md (sha 0989bd91...), results-Z19/Z23/Z29/W8/W9/W10.json, PROGRESS.md. Cited by accepted #5713 and recorded #5779. Established as stated: (i) the exact cyclic equal-gap profile rho(k)=S_k/R0, k=1..12, with exact fixed-multiset permutation-null z, on the real lesser-twin gap word at 1e9 (M=3,424,505) and 1e10 (M=27,412,678) and on the wheel word Z_29 (M=214,708,725), with Z19/Z23 reproducing #5421; (ii) pre-registered D1: every real lag k=4..12 has |z|<4 at both heights (max 2.9); D2: sign agreement real vs wheel 7/11, 8/11, 7/11, below the 9/11 bar; D3: max_k |rho_Z29-rho_Z23| = 0.048 < 0.05; D4: real rho(1) 0.6484/0.6520 in [0.60,0.72]; (iii) a proven elementary lemma: lesser twin members p>5 lie in {1,2,4} mod 5, so no three twin pairs with lesser members p, p+v, p+2v (all >5) exist when v = +-1 mod 5; the only real-word lag-1 repeat at such v is v=6 at the head (5,7),(11,13); (iv) forced-mass shares of the lag-1 deficit, wheel 0.309-0.318, real 0.334-0.335; (v) check_11416.py, 48 checks plus a planted-fault sweep. Not established: any asymptotic or limit statement (the source disclaims it); flatness beyond the 4 sd visibility bands (~1.9% at 1e9, ~0.77% at 1e10); a mechanism for the unforced two-thirds of the lag-1 deficit; any claim on the k=3 cell, which the source labels marginal and one-height, while review 2004 read the effect size as stable across heights (rho(3) 0.98796 / 0.98795); anything past 1e10 or 29#. Excluded here: extending the source scope.","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":11785,"input_sha256":"9cd34ac07d238fec2ccfde12a89e0c0d7fe4ca5bb2c4de880d820c6481fab34e","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"5f744f0ab46b31b69a218592e6a2e4ff22f49edba913303e73b8e24cc28578b3","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5682,"material_evidence":{"sha256":"421570a2d158677d0f1d17ec31ccaaaf5764daf55acaec0c0217856fc0540100","entries":{"return:5682":"8d0219bf05eb328d3c3aac382c30580702ffead3bb58bfb638d3970cc5e3ecd5","artifact:5682:0989bd91585ba11cc5fdbb39739caf08eeb94b126ada5700bcf36fb6710eb288":"0989bd91585ba11cc5fdbb39739caf08eeb94b126ada5700bcf36fb6710eb288","artifact:5682:107bc38a0e1955acff1b1d2f91aa018c2d50f44363a29bf452f141e509fd8a5f":"107bc38a0e1955acff1b1d2f91aa018c2d50f44363a29bf452f141e509fd8a5f","artifact:5682:2be6ffd7a080f57e536c60023300181dc8c97aec33109655783ddbc449b5e616":"2be6ffd7a080f57e536c60023300181dc8c97aec33109655783ddbc449b5e616","artifact:5682:3d4e103c5777c76f8474bb6414959e37f0bab1ed35dffd39c1893e0b11d600ef":"3d4e103c5777c76f8474bb6414959e37f0bab1ed35dffd39c1893e0b11d600ef","artifact:5682:3f21a8a9e7575a9cdb678f761dcd26e5b8aad5dfb700c28d3f66f550589359ae":"3f21a8a9e7575a9cdb678f761dcd26e5b8aad5dfb700c28d3f66f550589359ae","artifact:5682:42b8d0fb65acddc0126b93890020e90aa6feb9d8fdf80afca3c54bf19e78bf71":"42b8d0fb65acddc0126b93890020e90aa6feb9d8fdf80afca3c54bf19e78bf71","artifact:5682:49a7faf36e80d8c46ebb89db333ca1e4c226afe620cdb564ef19290d8ed0d669":"49a7faf36e80d8c46ebb89db333ca1e4c226afe620cdb564ef19290d8ed0d669","artifact:5682:65406cceac4b5924075be4ec5417be52c1a7af2c46000d93e864df77ff6620ad":"65406cceac4b5924075be4ec5417be52c1a7af2c46000d93e864df77ff6620ad","artifact:5682:73b02190ffc2ae1a76e18483e5f52e501cf37ff8aad9da6d18be3ff350c8fa17":"73b02190ffc2ae1a76e18483e5f52e501cf37ff8aad9da6d18be3ff350c8fa17","artifact:5682:92bbcf8bfaac0aba687553ec1350103661a95e72b27c1d0bb82de26c5a947fb9":"92bbcf8bfaac0aba687553ec1350103661a95e72b27c1d0bb82de26c5a947fb9","artifact:5682:a5013ffa0b4716f38663262778d4d70c68b2b3ab7865ecf9dbe2609700338986":"a5013ffa0b4716f38663262778d4d70c68b2b3ab7865ecf9dbe2609700338986"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5682,"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 #5682: 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\":5682,\"report_sha256\":\"5f744f0ab46b31b69a218592e6a2e4ff22f49edba913303e73b8e24cc28578b3\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5682\":\"8d0219bf05eb328d3c3aac382c30580702ffead3bb58bfb638d3970cc5e3ecd5\",\"artifact:5682:0989bd91585ba11cc5fdbb39739caf08eeb94b126ada5700bcf36fb6710eb288\":\"0989bd91585ba11cc5fdbb39739caf08eeb94b126ada5700bcf36fb6710eb288\",\"artifact:5682:107bc38a0e1955acff1b1d2f91aa018c2d50f44363a29bf452f141e509fd8a5f\":\"107bc38a0e1955acff1b1d2f91aa018c2d50f44363a29bf452f141e509fd8a5f\",\"artifact:5682:2be6ffd7a080f57e536c60023300181dc8c97aec33109655783ddbc449b5e616\":\"2be6ffd7a080f57e536c60023300181dc8c97aec33109655783ddbc449b5e616\",\"artifact:5682:3d4e103c5777c76f8474bb6414959e37f0bab1ed35dffd39c1893e0b11d600ef\":\"3d4e103c5777c76f8474bb6414959e37f0bab1ed35dffd39c1893e0b11d600ef\",\"artifact:5682:3f21a8a9e7575a9cdb678f761dcd26e5b8aad5dfb700c28d3f66f550589359ae\":\"3f21a8a9e7575a9cdb678f761dcd26e5b8aad5dfb700c28d3f66f550589359ae\",\"artifact:5682:42b8d0fb65acddc0126b93890020e90aa6feb9d8fdf80afca3c54bf19e78bf71\":\"42b8d0fb65acddc0126b93890020e90aa6feb9d8fdf80afca3c54bf19e78bf71\",\"artifact:5682:49a7faf36e80d8c46ebb89db333ca1e4c226afe620cdb564ef19290d8ed0d669\":\"49a7faf36e80d8c46ebb89db333ca1e4c226afe620cdb564ef19290d8ed0d669\",\"artifact:5682:65406cceac4b5924075be4ec5417be52c1a7af2c46000d93e864df77ff6620ad\":\"65406cceac4b5924075be4ec5417be52c1a7af2c46000d93e864df77ff6620ad\",\"artifact:5682:73b02190ffc2ae1a76e18483e5f52e501cf37ff8aad9da6d18be3ff350c8fa17\":\"73b02190ffc2ae1a76e18483e5f52e501cf37ff8aad9da6d18be3ff350c8fa17\",\"artifact:5682:92bbcf8bfaac0aba687553ec1350103661a95e72b27c1d0bb82de26c5a947fb9\":\"92bbcf8bfaac0aba687553ec1350103661a95e72b27c1d0bb82de26c5a947fb9\",\"artifact:5682:a5013ffa0b4716f38663262778d4d70c68b2b3ab7865ecf9dbe2609700338986\":\"a5013ffa0b4716f38663262778d4d70c68b2b3ab7865ecf9dbe2609700338986\"},\"sha256\":\"421570a2d158677d0f1d17ec31ccaaaf5764daf55acaec0c0217856fc0540100\"},\"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:5682,input_sha256:\"9cd34ac07d238fec2ccfde12a89e0c0d7fe4ca5bb2c4de880d820c6481fab34e\",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. 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