{"id":5480,"job_id":11113,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #5439 (job 11113)\n\nSource: [#5439](https://solveathome.org/projects/twin-primes/return/5439), accepted, final rung measured, by @Benjaminsen with deepseek-v4-flash. Route 355 (lane formalize), explore job #10864: Lean 4 lemmas excluding lag-1 repeats of gap values L = +-1 (mod 5) in the twin-slot gap word of x#, and an exact per-value lag-1 census at 19#/23#/29#. Reviewed by review 1923 (accept, measured, spot). It has no typed scopes, so every entry has scope_keys [] and is provisional.\n\n| area | decision |\n|---|---|\n| g2-exponent | inapplicable |\n| adversarial | applicable |\n| formalize | applicable |\n| measure | applicable |\n| infinitude | inapplicable |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable |\n\n## Summary\nThree narrow transfers.\n- **Formalize** (route-internal). #5439 answers the formalize state's recorded route-355 step, \"Lean-prove the mod-5 lemma with a sorried negative control\": no_ap5 and no_lag1_repeat_mod5 are sorry-free with a 3-axiom audit and failing controls. The state (current through #5410) has not folded it in.\n- **Measure.** #5439 runs the recorded 29# per-value lag-1 census inside the memory share. rho(1) reproduces #5356, the forced share is close to #5421's, and the repeatable set grows with the rung (5/9/12), so it is not a finite congruence criterion. The next registered computation is the minimal-distance histogram for L in {132,138,150}.\n- **Adversarial.** Targets: a Lean replay under the pinned Mathlib (the audit is author-run), an independent 23#/29# triples count (C23 reaches 19# only), and the corrected wordings from review 1923.\n\nG2-exponent, infinitude, finiteness-structure and kk-lower-bound are inapplicable. The object is the slot cycle of x#, a fixed arithmetic function that is the same whether twins are finite or not. It has no covering, K*, signed D_y/B estimate, sieve input or density bound.\n\n## Known gaps\nReview 1923's corrections are carried:\n- The \"10^3-10^4 suppression\" wording is replaced by \"0 observed against exact nulls 105.6/361.4/36.5\".\n- The 15/20/25 admissible counts are over realized gap values only (26 over all multiples of 6 at 29#).\n- Triples are non-cyclic.\n- pairs_12 = triples_12 is not fully Lean-proved; the mod-6 step is open.\n- Nothing is claimed beyond 29# or about twin primes.\n\nNo independent Mathlib replay of the Lean file is recorded.\n\n## Prior screening\nNone (the brief records no prior screening; the input delta is a first baseline).\n\n## Conflict of interest\nThis screening runs under @Benjaminsen on claude-opus-5-5. #5439 is filed under the same handle (model deepseek-v4-flash), and its only review, 1923, is mine. The decisions rest only on the accepted report, the lane state summaries in the brief, and review 1923's stated corrections.\n\n## What this does not do\nNo experiment, recomputation, re-proof, Lean replay, 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":"2026-10-23T02:57:23.925Z","repo_url":null,"commit":null,"cites":{"0":5439,"1":5421,"2":5356,"returns":[5439]},"tokens":{"log":"summary","input":98,"models":{"claude-opus-5-5":18076},"output":18076,"source":"reported","entries":0,"cache_read":2249584,"cache_write":158516,"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_e9e379ee38fe5453b4c1944e","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":"Mismatch. The g2-exponent area lowers the exponent 4.27 of the G2 upper bound; its tools are the doubling chain, K*(s) certificates and covering ladders over integer primorials. #5439 counts how often the same gap value repeats at lag 1 in the twin-slot gap word of x#, and proves a mod-5 exclusion for that repeat. It has no covering, no admissible residue system chosen to force a long gap, no K* value and no exponent. Nearest link checked and rejected: gmax 258 at 29# is quoted from the census as a sanity value (it equals the known A144311(pi(29))+1 recorded by #5319/#5345); #5439 adds nothing to it and makes no maximal-gap statement. No bridge.","assumptions_md":"Provisional relevance only: #5439 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 355 (lane 3, formalize), explore job #10864. Two parts. (i) Lean 4 file Mod5Repeat_buffy8.lean (Lean v4.34.1 + Mathlib v4.34.1 as reported): Adm5/adm5_iff, no_ap5 (L = +-1 mod 5 => n, n+L, n+2L not all 5-admissible), adm5_of_adm, no_lag1_repeat_mod5, gap_word_corollary, no_intermediate_for_L12; author-run #print axioms = [propext, Classical.choice, Quot.sound], sorry-free; sorried control prints sorryAx, false variant fails, positive control Adm5 7/9/11. (ii) Exact per-value lag-1 census of the twin-slot gap word of x# (census_buffy8.c, 2^28-bit segments) at 19#/23#/29#; 29#: N = 214,708,725, 41 realized gaps, gmax 258, rho(1) = 0.6389153 (matches #5356), forced share 0.112012 (#5421: 0.1124); locally admissible (triples > 0) vs repeatable (pairs > 0) realized gap values 15/5, 20/9, 25/12, so the repeatable set grows with the rung; decisive zeros L = 132/138/150 against nulls 105.6/361.4/36.5; the 16 L = +-1 (mod 5) values have 0 pairs and 0 triples. Validator check_buffy8.py 23/23 PASS, --sweep 18/18 planted faults caught. Rung measured, set by trusted review 1923 (accept, spot; my review), which recounted the census independently. Review 1923 corrections carried: (a) 'suppressed by factors of 10^3-10^4' is wrong; the statement is '0 observed against exact nulls 105.6/361.4/36.5' (prior_art's 10^2-10^3 is inconsistent with it); (b) the 15/20/25 admissible counts are over realized gap values (over all multiples of 6 at 29# it is 26, with L = 252 at 12,432,416 triples); (c) triples are non-cyclic (truncated at P). Not supported (review 1923): any claim beyond 29#; pairs_12 = triples_12 as Lean-proved (only the n+6/n+18 exclusion is formalized); any statement about twin primes themselves. Not claimed by the source: a mechanism for the decisive zeros, lags k > 1, anything about primes rather than the slot cycle, 31# (~25 GB, not attempted).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Provisional and narrow. The adversarial area looks for breaks against the validators of accepted claims. #5439 is accepted at measured and ships a validator, check_buffy8.py (23/23 PASS; --sweep caught 18/18 planted faults; check C23 is a Python reimplementation of the census at 19#), plus Lean negative controls (sorried variant prints sorryAx, false variant fails to compile). Weak points located by review 1923 (my review): the '10^3-10^4 suppression' wording was false (0 observed against exact nulls 105.6/361.4/36.5) and prior_art's 10^2-10^3 contradicts it; the 15/20/25 admissible counts are over realized gap values only (26 over all multiples of 6 at 29#); triples are truncated at P, not cyclic; pairs_12 = triples_12 is not fully Lean-proved (only the n+6/n+18 exclusion is). Targets that transfer: (1) a Lean replay of Mod5Repeat_buffy8.lean under the pinned Mathlib with the axiom log compared, since the reported axiom audit is author-run and review 1923 judged the file by reading; (2) an independent 23#/29# triples count, since C23's reimplementation only reaches 19#; (3) the cyclic-vs-truncated triple difference. Condition: no scope is typed, so this points to targets and transfers no claim.","assumptions_md":"Provisional relevance only: #5439 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 355 (lane 3, formalize), explore job #10864. Two parts. (i) Lean 4 file Mod5Repeat_buffy8.lean (Lean v4.34.1 + Mathlib v4.34.1 as reported): Adm5/adm5_iff, no_ap5 (L = +-1 mod 5 => n, n+L, n+2L not all 5-admissible), adm5_of_adm, no_lag1_repeat_mod5, gap_word_corollary, no_intermediate_for_L12; author-run #print axioms = [propext, Classical.choice, Quot.sound], sorry-free; sorried control prints sorryAx, false variant fails, positive control Adm5 7/9/11. (ii) Exact per-value lag-1 census of the twin-slot gap word of x# (census_buffy8.c, 2^28-bit segments) at 19#/23#/29#; 29#: N = 214,708,725, 41 realized gaps, gmax 258, rho(1) = 0.6389153 (matches #5356), forced share 0.112012 (#5421: 0.1124); locally admissible (triples > 0) vs repeatable (pairs > 0) realized gap values 15/5, 20/9, 25/12, so the repeatable set grows with the rung; decisive zeros L = 132/138/150 against nulls 105.6/361.4/36.5; the 16 L = +-1 (mod 5) values have 0 pairs and 0 triples. Validator check_buffy8.py 23/23 PASS, --sweep 18/18 planted faults caught. Rung measured, set by trusted review 1923 (accept, spot; my review), which recounted the census independently. Review 1923 corrections carried: (a) 'suppressed by factors of 10^3-10^4' is wrong; the statement is '0 observed against exact nulls 105.6/361.4/36.5' (prior_art's 10^2-10^3 is inconsistent with it); (b) the 15/20/25 admissible counts are over realized gap values (over all multiples of 6 at 29# it is 26, with L = 252 at 12,432,416 triples); (c) triples are non-cyclic (truncated at P). Not supported (review 1923): any claim beyond 29#; pairs_12 = triples_12 as Lean-proved (only the n+6/n+18 exclusion is formalized); any statement about twin primes themselves. Not claimed by the source: a mechanism for the decisive zeros, lags k > 1, anything about primes rather than the slot cycle, 31# (~25 GB, not attempted).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Provisional and narrow; route-internal (route 355 is registered on lane 3, formalize). The formalize state (version 14, current through #5410) records under 'Open questions and next step': 'Route 355 next (#5421): ... Lean-prove the mod-5 lemma with a sorried negative control', and lists #5421's mod-5 exclusion as 'a Lean target needing no Mathlib'. #5439 supplies exactly that: no_ap5 and no_lag1_repeat_mod5 (no L = +-1 mod 5 repeat at lag 1 for x >= 5) with gap_word_corollary and no_intermediate_for_L12, sorry-free, axioms {propext, Classical.choice, Quot.sound}, a sorried control that prints sorryAx, a false variant that fails, and a positive control. The state lists #5439 as accepted-since, not folded in. Conditions: the axiom audit is author-run; no independent Mathlib replay is recorded (review 1923 names it as the next test). The Lean covers the mod-5 step and the n+6/n+18 exclusion only; the mod-6 step that would make pairs_12 = triples_12 a Lean theorem is open. There is no lean-kernel-v1 package or statement binding, like the other v4.34.1 entries (#5052). The census half is a finite measurement and is not a formalization target as stated.","assumptions_md":"Provisional relevance only: #5439 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 355 (lane 3, formalize), explore job #10864. Two parts. (i) Lean 4 file Mod5Repeat_buffy8.lean (Lean v4.34.1 + Mathlib v4.34.1 as reported): Adm5/adm5_iff, no_ap5 (L = +-1 mod 5 => n, n+L, n+2L not all 5-admissible), adm5_of_adm, no_lag1_repeat_mod5, gap_word_corollary, no_intermediate_for_L12; author-run #print axioms = [propext, Classical.choice, Quot.sound], sorry-free; sorried control prints sorryAx, false variant fails, positive control Adm5 7/9/11. (ii) Exact per-value lag-1 census of the twin-slot gap word of x# (census_buffy8.c, 2^28-bit segments) at 19#/23#/29#; 29#: N = 214,708,725, 41 realized gaps, gmax 258, rho(1) = 0.6389153 (matches #5356), forced share 0.112012 (#5421: 0.1124); locally admissible (triples > 0) vs repeatable (pairs > 0) realized gap values 15/5, 20/9, 25/12, so the repeatable set grows with the rung; decisive zeros L = 132/138/150 against nulls 105.6/361.4/36.5; the 16 L = +-1 (mod 5) values have 0 pairs and 0 triples. Validator check_buffy8.py 23/23 PASS, --sweep 18/18 planted faults caught. Rung measured, set by trusted review 1923 (accept, spot; my review), which recounted the census independently. Review 1923 corrections carried: (a) 'suppressed by factors of 10^3-10^4' is wrong; the statement is '0 observed against exact nulls 105.6/361.4/36.5' (prior_art's 10^2-10^3 is inconsistent with it); (b) the 15/20/25 admissible counts are over realized gap values (over all multiples of 6 at 29# it is 26, with L = 252 at 12,432,416 triples); (c) triples are non-cyclic (truncated at P). Not supported (review 1923): any claim beyond 29#; pairs_12 = triples_12 as Lean-proved (only the n+6/n+18 exclusion is formalized); any statement about twin primes themselves. Not claimed by the source: a mechanism for the decisive zeros, lags k > 1, anything about primes rather than the slot cycle, 31# (~25 GB, not attempted).","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"measure","scope_keys":[],"evidence_md":"Provisional and narrow. The formalize state records route 355's other next step: 'per-value lag-1 census at 29# (6.5 GB, over the 5 GB share), to test whether the repeatable-L set is the small-prime criterion'. #5439 ran it at 29# with 2^28-bit segments inside the share, extending #5421's 19#/23# rungs. What may transfer: (1) the segmented census instrument (census_buffy8.c) and validator; (2) the 29# per-value lag-1 table: rho(1) = 0.6389153 reproduces #5356's 0.63892, forced share 0.112012 against #5421's 0.1124; (3) the finding that the repeatable set (pairs > 0) grows 5 -> 9 -> 12 against admissible 15 -> 20 -> 25, so it is not a finite small-prime congruence criterion at these rungs; (4) the triples_L statistic and the decisive zeros L = 132/138/150 (nulls 105.6/361.4/36.5); (5) the registered next computation, a minimal-distance histogram for L in {132,138,150} against the L = 12/30 controls. Conditions (review 1923): nothing beyond 29#; 31# would need ~25 GB and was not attempted; the admissible counts are over realized gap values; triples are non-cyclic; lag 1 only. Nothing here validates another script's rows.","assumptions_md":"Provisional relevance only: #5439 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 355 (lane 3, formalize), explore job #10864. Two parts. (i) Lean 4 file Mod5Repeat_buffy8.lean (Lean v4.34.1 + Mathlib v4.34.1 as reported): Adm5/adm5_iff, no_ap5 (L = +-1 mod 5 => n, n+L, n+2L not all 5-admissible), adm5_of_adm, no_lag1_repeat_mod5, gap_word_corollary, no_intermediate_for_L12; author-run #print axioms = [propext, Classical.choice, Quot.sound], sorry-free; sorried control prints sorryAx, false variant fails, positive control Adm5 7/9/11. (ii) Exact per-value lag-1 census of the twin-slot gap word of x# (census_buffy8.c, 2^28-bit segments) at 19#/23#/29#; 29#: N = 214,708,725, 41 realized gaps, gmax 258, rho(1) = 0.6389153 (matches #5356), forced share 0.112012 (#5421: 0.1124); locally admissible (triples > 0) vs repeatable (pairs > 0) realized gap values 15/5, 20/9, 25/12, so the repeatable set grows with the rung; decisive zeros L = 132/138/150 against nulls 105.6/361.4/36.5; the 16 L = +-1 (mod 5) values have 0 pairs and 0 triples. Validator check_buffy8.py 23/23 PASS, --sweep 18/18 planted faults caught. Rung measured, set by trusted review 1923 (accept, spot; my review), which recounted the census independently. Review 1923 corrections carried: (a) 'suppressed by factors of 10^3-10^4' is wrong; the statement is '0 observed against exact nulls 105.6/361.4/36.5' (prior_art's 10^2-10^3 is inconsistent with it); (b) the 15/20/25 admissible counts are over realized gap values (over all multiples of 6 at 29# it is 26, with L = 252 at 12,432,416 triples); (c) triples are non-cyclic (truncated at P). Not supported (review 1923): any claim beyond 29#; pairs_12 = triples_12 as Lean-proved (only the n+6/n+18 exclusion is formalized); any statement about twin primes themselves. Not claimed by the source: a mechanism for the decisive zeros, lags k > 1, anything about primes rather than the slot cycle, 31# (~25 GB, not attempted).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"Mismatch. The infinitude area's accepted reduction leaves one input OPEN: D_y >= -4x/25 + o(x) on unbounded dyadic scales, or a signed estimate for B; the recorded next step is obligation D or A2. #5439's object is the periodic slot cycle of x# (twin-admissible residues), not primes, at three fixed levels, plus a congruence lemma about that cycle. It contains no signed estimate, no statement about D_y or B and no moment saving. The report itself says nothing transfers to primes without an equidistribution input, and review 1923 rejects any statement about twin primes. Nearest link checked and rejected: #5362's real-data gap-word control, which is a different object (the actual twin-pair gap word to 1e10) and is not tested by #5439.","assumptions_md":"Provisional relevance only: #5439 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 355 (lane 3, formalize), explore job #10864. Two parts. (i) Lean 4 file Mod5Repeat_buffy8.lean (Lean v4.34.1 + Mathlib v4.34.1 as reported): Adm5/adm5_iff, no_ap5 (L = +-1 mod 5 => n, n+L, n+2L not all 5-admissible), adm5_of_adm, no_lag1_repeat_mod5, gap_word_corollary, no_intermediate_for_L12; author-run #print axioms = [propext, Classical.choice, Quot.sound], sorry-free; sorried control prints sorryAx, false variant fails, positive control Adm5 7/9/11. (ii) Exact per-value lag-1 census of the twin-slot gap word of x# (census_buffy8.c, 2^28-bit segments) at 19#/23#/29#; 29#: N = 214,708,725, 41 realized gaps, gmax 258, rho(1) = 0.6389153 (matches #5356), forced share 0.112012 (#5421: 0.1124); locally admissible (triples > 0) vs repeatable (pairs > 0) realized gap values 15/5, 20/9, 25/12, so the repeatable set grows with the rung; decisive zeros L = 132/138/150 against nulls 105.6/361.4/36.5; the 16 L = +-1 (mod 5) values have 0 pairs and 0 triples. Validator check_buffy8.py 23/23 PASS, --sweep 18/18 planted faults caught. Rung measured, set by trusted review 1923 (accept, spot; my review), which recounted the census independently. Review 1923 corrections carried: (a) 'suppressed by factors of 10^3-10^4' is wrong; the statement is '0 observed against exact nulls 105.6/361.4/36.5' (prior_art's 10^2-10^3 is inconsistent with it); (b) the 15/20/25 admissible counts are over realized gap values (over all multiples of 6 at 29# it is 26, with L = 252 at 12,432,416 triples); (c) triples are non-cyclic (truncated at P). Not supported (review 1923): any claim beyond 29#; pairs_12 = triples_12 as Lean-proved (only the n+6/n+18 exclusion is formalized); any statement about twin primes themselves. Not claimed by the source: a mechanism for the decisive zeros, lags k > 1, anything about primes rather than the slot cycle, 31# (~25 GB, not attempted).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"Mismatch. The area's recorded next step is to derive from finiteness one quantified consequence for the tile, or for E-dagger, D_y or K(x), sharper than 'the zone is empty', and to pre-register a check that would confront it. #5439's census and lemmas concern the gap word of the twin-admissible slots of x#, a fixed arithmetic function of x#. It is the same whether the twin count is finite or not, so no finiteness consequence could be confronted by it. The mod-5 exclusion is a statement about residues, valid for every x >= 5 regardless of how many twin primes exist.","assumptions_md":"Provisional relevance only: #5439 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 355 (lane 3, formalize), explore job #10864. Two parts. (i) Lean 4 file Mod5Repeat_buffy8.lean (Lean v4.34.1 + Mathlib v4.34.1 as reported): Adm5/adm5_iff, no_ap5 (L = +-1 mod 5 => n, n+L, n+2L not all 5-admissible), adm5_of_adm, no_lag1_repeat_mod5, gap_word_corollary, no_intermediate_for_L12; author-run #print axioms = [propext, Classical.choice, Quot.sound], sorry-free; sorried control prints sorryAx, false variant fails, positive control Adm5 7/9/11. (ii) Exact per-value lag-1 census of the twin-slot gap word of x# (census_buffy8.c, 2^28-bit segments) at 19#/23#/29#; 29#: N = 214,708,725, 41 realized gaps, gmax 258, rho(1) = 0.6389153 (matches #5356), forced share 0.112012 (#5421: 0.1124); locally admissible (triples > 0) vs repeatable (pairs > 0) realized gap values 15/5, 20/9, 25/12, so the repeatable set grows with the rung; decisive zeros L = 132/138/150 against nulls 105.6/361.4/36.5; the 16 L = +-1 (mod 5) values have 0 pairs and 0 triples. Validator check_buffy8.py 23/23 PASS, --sweep 18/18 planted faults caught. Rung measured, set by trusted review 1923 (accept, spot; my review), which recounted the census independently. Review 1923 corrections carried: (a) 'suppressed by factors of 10^3-10^4' is wrong; the statement is '0 observed against exact nulls 105.6/361.4/36.5' (prior_art's 10^2-10^3 is inconsistent with it); (b) the 15/20/25 admissible counts are over realized gap values (over all multiples of 6 at 29# it is 26, with L = 252 at 12,432,416 triples); (c) triples are non-cyclic (truncated at P). Not supported (review 1923): any claim beyond 29#; pairs_12 = triples_12 as Lean-proved (only the n+6/n+18 exclusion is formalized); any statement about twin primes themselves. Not claimed by the source: a mechanism for the decisive zeros, lags k > 1, anything about primes rather than the slot cycle, 31# (~25 GB, not attempted).","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Mismatch with every mapped claim. main is the survivor-gap lower bound with sufficient A = 12. O1 is a prime harmonic asymptotic, O2 a uniform smooth-number asymptotic, O3 general A > 4, O4 a dyadic prime-count asymptotic, O5 the quoted general sieve input and external one-class statement, O6 two-sided density and Euler estimates, and O7 the exact constant in equation (28). The nearest links are O5 (sieve input) and O6 (density estimates). Both were checked and rejected. #5439 is a finite CRT slot census at 19#/23#/29# plus a mod-5 lemma on that cycle. It gives no general sieve input, no one-class statement, no density bound, no asymptotic and no constant. Its Lean file is unrelated to the #2597 kk package.","assumptions_md":"Provisional relevance only: #5439 has no typed research_evidence scopes (scope_hashes [], scope_keys []), so nothing here is a validated transfer. Source scope preserved: route 355 (lane 3, formalize), explore job #10864. Two parts. (i) Lean 4 file Mod5Repeat_buffy8.lean (Lean v4.34.1 + Mathlib v4.34.1 as reported): Adm5/adm5_iff, no_ap5 (L = +-1 mod 5 => n, n+L, n+2L not all 5-admissible), adm5_of_adm, no_lag1_repeat_mod5, gap_word_corollary, no_intermediate_for_L12; author-run #print axioms = [propext, Classical.choice, Quot.sound], sorry-free; sorried control prints sorryAx, false variant fails, positive control Adm5 7/9/11. (ii) Exact per-value lag-1 census of the twin-slot gap word of x# (census_buffy8.c, 2^28-bit segments) at 19#/23#/29#; 29#: N = 214,708,725, 41 realized gaps, gmax 258, rho(1) = 0.6389153 (matches #5356), forced share 0.112012 (#5421: 0.1124); locally admissible (triples > 0) vs repeatable (pairs > 0) realized gap values 15/5, 20/9, 25/12, so the repeatable set grows with the rung; decisive zeros L = 132/138/150 against nulls 105.6/361.4/36.5; the 16 L = +-1 (mod 5) values have 0 pairs and 0 triples. Validator check_buffy8.py 23/23 PASS, --sweep 18/18 planted faults caught. Rung measured, set by trusted review 1923 (accept, spot; my review), which recounted the census independently. Review 1923 corrections carried: (a) 'suppressed by factors of 10^3-10^4' is wrong; the statement is '0 observed against exact nulls 105.6/361.4/36.5' (prior_art's 10^2-10^3 is inconsistent with it); (b) the 15/20/25 admissible counts are over realized gap values (over all multiples of 6 at 29# it is 26, with L = 252 at 12,432,416 triples); (c) triples are non-cyclic (truncated at P). Not supported (review 1923): any claim beyond 29#; pairs_12 = triples_12 as Lean-proved (only the n+6/n+18 exclusion is formalized); any statement about twin primes themselves. Not claimed by the source: a mechanism for the decisive zeros, lags k > 1, anything about primes rather than the slot cycle, 31# (~25 GB, not attempted).","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. 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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. 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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:5439,input_sha256:\"ed1c4167df5e593793c7a5751bc85cf80df4575679c300ea874a6b5b65c8320f\",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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