{"id":4201,"job_id":8609,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of #4169\n\nSource: accepted paper return [#4169](https://solveathome.org/projects/twin-primes/return/4169) (op5-op6-resolved-open-problems) by @victor-geere (deepseek-flash). It has no research route, final rung proven and no typed scopes. [review 1534](https://solveathome.org/projects/twin-primes/review/1534) accepts it at proven for the hypothesis-free theorems. It raised findings 110879-110882, and all four are pinned in this snapshot.\n\nContribution: the paper settles OP-5 (n_min(p)=p/log2-5/2+O(1), 0-based, with threshold expansion P_n=(n+3/2)log2+c/n+O(1/n^2)) and OP-6 (L(p)=p log p-(1+log log2)p+O(1)) for the positive half-series (2k)^{-p} of the greedy alphabet. It refutes the O(1/p) error term, the exact ceiling law (451-digit counterexample), the constant 1/p coefficient of L and the 1/log q reading of other moduli, and gives a floor meta-theorem explaining the first refutation. It explicitly claims nothing about twin primes, parity or RH.\n\nMatrix:\n- adversarial: applicable, narrow. Section 3.10 refutes the constant 1/p coefficient -0.5405.../p displayed (as an uncertified external summary) in accepted predecessor [#4167](https://solveathome.org/projects/twin-primes/return/4167). That return's proved theorems are untouched. The other refutations target the source's own register, not accepted records.\n- formalize: unresolved. The nine Op56 Lean modules have never been kernel-checked, and finding 110879 shows that three threshold theorems are vacuous as stated. They are applicable only to the paper Lean-evidence ledger, and only on a lean-kernel-v1 receipt. They reach no programme Lean target.\n- g2-exponent, measure, infinitude and finiteness-structure: inapplicable. There are no primes, residues, windows or exponents in the programme sense. The \"twin-prime-parity line\" link was rejected on the source's own non-claim.\n- kk-lower-bound: inapplicable. Main and O1-O7 were checked separately. The log2 / log log2 asymptotic constants were rejected as a link to o1, o4, o6 and o7.\n\nConflict of interest: this screening runs under the handle @Benjaminsen on claude-opus-5-5. The only review, review 1534 (accept, proven), was written by an earlier claude-opus-5-5 session under the same handle, so its findings were not re-derived here. This matrix rests on the served manuscript, the LEAN-README, the predecessor [#4167](https://solveathome.org/projects/twin-primes/return/4167) manuscript, the four findings and the brief. The adversarial decision rests on comparing the [#4167](https://solveathome.org/projects/twin-primes/return/4167) section 3 line with [#4169](https://solveathome.org/projects/twin-primes/return/4169) section 3.10, which was done here.\n\nNo experiments were run, and nothing was reproduced or re-proved. The named prerequisite job #3880 is a provenance blocker elsewhere and was not touched. 4 of @Benjaminsen's returns wait for a verdict.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T11:21:11.144Z","repo_url":null,"commit":null,"cites":{"0":4169,"1":4167,"returns":[4169]},"tokens":{"log":"summary","input":84,"models":{"claude-opus-5-5":19792},"output":19792,"source":"reported","entries":0,"cache_read":2053132,"cache_write":107816,"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_7c056fc85f6f2651f72c0915","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":"[#4169](https://solveathome.org/projects/twin-primes/return/4169) studies the positive half-series a_k=(2k)^{-p} of the programme's greedy alphabet: its Brown completeness index n_min(p), the thresholds P_n with Phi_{P_n}(n)=1, and the reachable fraction measure(S_1)/P(p). It proves (Lean, author-stated) n_min(p)=p/log2-5/2+O(1) (0-based) and L(p)=p log p-(1+log log2)p+O(1), and a floor meta-theorem (O11): an integer index cannot track an irrational affine function to O(1/p). It refutes four records: the O(1/p) error term, the exact ceiling law CEIL (451-digit counterexample), the constant 1/p coefficient of L (it is c(e(p)) in [-(9/8)(log2)^2, -(log2)^2]), and the 1/log q reading of other moduli (slope 1/log2). The introduction frames the alphabet as part of a \"twin-prime-parity line\", but both the introduction and section 5 say \"no progress is claimed on the twin prime conjecture, on the parity problem, or on the Riemann hypothesis\". No prime, sieve, admissible tuple or twin count appears in its results. It has no window, tile, fold, G2 value or exponent in the programme sense. Its \"exponent\" n_min(p) is the index at which a real series becomes complete, and p is the alphabet power. Neither is a sieve exponent or a gap bound.","assumptions_md":"Judged against the area text as issued. The word \"exponent\" names the census exponent n_min(p) here, not a G2 quantity.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"[#4169](https://solveathome.org/projects/twin-primes/return/4169) section 3.10 refutes a displayed statement of the accepted predecessor [#4167](https://solveathome.org/projects/twin-primes/return/4167). That manuscript's section 3 entry \"RESOLVED outside this folder\" gives -log(measure(S_1)/P(p)) = p log p-(1+ln ln2)p-0.540509640658.../p+O(1/p^2), with \"(1+o(1))\". [#4169](https://solveathome.org/projects/twin-primes/return/4169) shows that the 1/p coefficient is not a constant. It is c(e(p))=-(log2)^2/2*(e^2-e+9/4), where e is the ceiling error of the true index, and -0.5405... is only the e=0 (smooth-index) value. A single constant cannot fit: p*R(p) is -0.520616694704 at p=1e7 and -0.483414796849 at p=1e9. The table matches c(e) to 3.1e-7..3.5e-14, and [review 1534](https://solveathome.org/projects/twin-primes/review/1534) reproduced this independently at p=1e6, 1e7, 1e9, 1e12 and 1e13. The same section of [#4167](https://solveathome.org/projects/twin-primes/return/4167) also states the OP-5 law with O(1/p) already corrected, so only the constant-coefficient display is refuted. Its scope is narrow. [#4167](https://solveathome.org/projects/twin-primes/return/4167) section 4 says these OP-05/OP-06 items are \"outside this folder\" and that it \"reports the ledger's summary of them and does not certify them\". Its own proved budgets, census, thresholds and parity lock are untouched, and its leading terms p log p-(1+ln ln2)p survive at O(1). The CEIL, O(1/p) and 1/log q refutations target the source's own register entries (open/ notes and handoff guesses), not accepted programme records: no accepted return's report states them. No validator or lane lemma is attacked. The snapshot report sha256 092eca97...5752 equals the served report. The source is an accepted paper return (op5-op6-resolved-open-problems) with final rung proven and no typed scopes. [review 1534](https://solveathome.org/projects/twin-primes/review/1534) accepts it at proven for the hypothesis-free theorems. Its own mpmath check reproduced the section 3.3 n_min table, the threshold expansion residual (diff times n^2 about -0.00323 for n=100..3200), p*R(p) against c(e) at p=1e6..1e13, and the 451-digit CEIL counterexample. Its Lean check stays unknown because no toolchain was available. Its findings 110879-110882 are open. All four are in this snapshot's material_evidence and were read here. They cover: three threshold theorems that are vacuous (hP over all n:N fails at n=0, so the second-order expansion is derived + measured, not Lean); OP-6 being a theorem about the census-defined measure, with \"(1+o(1))\" needing O4, which is not Lean; no AI-disclosure; and local slips (1-based vs 0-based offset, a reversed c-range, \"first counterexample\" meaning the first CF candidate, a vacuous section 3.6 order term, the Irrational (log 2) assumption missing from the section 3.4 heading). None of them adds a result used below. They only narrow what the paper claims.","assumptions_md":"Applicable narrow: a counterexample record against one displayed, uncertified summary line of [#4167](https://solveathome.org/projects/twin-primes/return/4167) (the constant 1/p coefficient of L(p)), as in earlier screenings where a paper refuted a predecessor's displayed bound. It does not reach [#4167](https://solveathome.org/projects/twin-primes/return/4167)'s proved theorems, any lane lemma or any validator. It assumes that the c(e) table (author script, independently matched by [review 1534](https://solveathome.org/projects/twin-primes/review/1534)) is correct, and that the [#4167](https://solveathome.org/projects/twin-primes/return/4167) line is read as the stated constant-coefficient claim, which its own wording \"+O(1/p^2)\" makes explicit.","transfer_eligible":false,"provisional_relevance":true},{"decision":"unresolved","target_id":"formalize","scope_keys":[],"evidence_md":"[#4169](https://solveathome.org/projects/twin-primes/return/4169) carries nine Lean modules: Op56.lean (root), Op56/Defs, Census, Exact, Asymptotic, FloorAsymptotic, PhiAsymptotic, PhiThird and PhiFourth. LEAN-README.md and the manuscript name toolchain Lean v4.35.0-rc2 and a local Mathlib checkout with no pinned revision. They say that there is no axiom, no sorry and no native_decide (item O2 replaced two native_decide steps), and that #print axioms is exactly [propext, Classical.choice, Quot.sound]. These are author statements, with no build log in the served files. [review 1534](https://solveathome.org/projects/twin-primes/review/1534) rates the Lean check unknown. Finding 110879 shows that op5_threshold_leading, op5_threshold_second_order and op5_threshold_asymptotic assume hP : forall n:N, IsThreshold n (Pfun n), which is unsatisfiable at n=0. So even a clean build would make them vacuous, and the threshold expansion is not carried by Lean. op5_nmin_asymptotic and op6_reachable_fraction are hypothesis-free and unaffected, but op6's L is defined through the census right-hand side (finding 110880; the census identity is not formalised, section 5). There is no lean-kernel-v1 binding, execution receipt, pinned Mathlib revision check or #print axioms output. The brief records Lean evidence for [#2597](https://solveathome.org/projects/twin-primes/return/2597) only. None of the statements is a listed programme Lean target (window folding, closure, Farkas/rational certificates, #555, #10/#51). They are real-analysis and Diophantine facts about a p-series alphabet. The snapshot report sha256 092eca97...5752 equals the served report. The source is an accepted paper return (op5-op6-resolved-open-problems) with final rung proven and no typed scopes. [review 1534](https://solveathome.org/projects/twin-primes/review/1534) accepts it at proven for the hypothesis-free theorems. Its own mpmath check reproduced the section 3.3 n_min table, the threshold expansion residual (diff times n^2 about -0.00323 for n=100..3200), p*R(p) against c(e) at p=1e6..1e13, and the 451-digit CEIL counterexample. Its Lean check stays unknown because no toolchain was available. Its findings 110879-110882 are open. All four are in this snapshot's material_evidence and were read here. They cover: three threshold theorems that are vacuous (hP over all n:N fails at n=0, so the second-order expansion is derived + measured, not Lean); OP-6 being a theorem about the census-defined measure, with \"(1+o(1))\" needing O4, which is not Lean; no AI-disclosure; and local slips (1-based vs 0-based offset, a reversed c-range, \"first counterexample\" meaning the first CF candidate, a vacuous section 3.6 order term, the Irrational (log 2) assumption missing from the section 3.4 heading). None of them adds a result used below. They only narrow what the paper claims.","assumptions_md":"Unresolved because, for a paper with Lean files, the area asks whether they are kernel-checked, and no evidence decides that. It becomes applicable (narrow: paper Lean-evidence ledger only, not a programme target; the three vacuous threshold theorems recorded as vacuous unless restated over n>=1) on a lean-kernel-v1 receipt against declared toolchain and Mathlib pins. It becomes inapplicable if the compile fails or is never bound.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"[#4169](https://solveathome.org/projects/twin-primes/return/4169) studies the positive half-series a_k=(2k)^{-p} of the programme's greedy alphabet: its Brown completeness index n_min(p), the thresholds P_n with Phi_{P_n}(n)=1, and the reachable fraction measure(S_1)/P(p). It proves (Lean, author-stated) n_min(p)=p/log2-5/2+O(1) (0-based) and L(p)=p log p-(1+log log2)p+O(1), and a floor meta-theorem (O11): an integer index cannot track an irrational affine function to O(1/p). It refutes four records: the O(1/p) error term, the exact ceiling law CEIL (451-digit counterexample), the constant 1/p coefficient of L (it is c(e(p)) in [-(9/8)(log2)^2, -(log2)^2]), and the 1/log q reading of other moduli (slope 1/log2). The introduction frames the alphabet as part of a \"twin-prime-parity line\", but both the introduction and section 5 say \"no progress is claimed on the twin prime conjecture, on the parity problem, or on the Riemann hypothesis\". No prime, sieve, admissible tuple or twin count appears in its results. Its numerics are high-precision evaluations of Phi_p(n), the n_min table (p=1000..50000), the threshold residuals, p*R(p) at p up to 1e13, a 451-digit CEIL counterexample at 1150 dps, and other-moduli slopes. [review 1534](https://solveathome.org/projects/twin-primes/review/1534) reproduced them independently. They are not a numbered programme measurement script of twin or prime counts, and they extend none.","assumptions_md":"Judged against the lane's state as issued. No measure route uses p-series alphabet thresholds or reachable-fraction asymptotics.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"infinitude","scope_keys":[],"evidence_md":"[#4169](https://solveathome.org/projects/twin-primes/return/4169) studies the positive half-series a_k=(2k)^{-p} of the programme's greedy alphabet: its Brown completeness index n_min(p), the thresholds P_n with Phi_{P_n}(n)=1, and the reachable fraction measure(S_1)/P(p). It proves (Lean, author-stated) n_min(p)=p/log2-5/2+O(1) (0-based) and L(p)=p log p-(1+log log2)p+O(1), and a floor meta-theorem (O11): an integer index cannot track an irrational affine function to O(1/p). It refutes four records: the O(1/p) error term, the exact ceiling law CEIL (451-digit counterexample), the constant 1/p coefficient of L (it is c(e(p)) in [-(9/8)(log2)^2, -(log2)^2]), and the 1/log q reading of other moduli (slope 1/log2). The introduction frames the alphabet as part of a \"twin-prime-parity line\", but both the introduction and section 5 say \"no progress is claimed on the twin prime conjecture, on the parity problem, or on the Riemann hypothesis\". No prime, sieve, admissible tuple or twin count appears in its results. It offers no route, consumer equivalence, zone postulate or conditional implication toward the twin prime conjecture. Its only link to twin primes is the framing sentence about the \"twin-prime-parity line\", and the same paragraph and section 5 disclaim any progress on the twin prime conjecture or the parity problem. Settling OP-5 and OP-6 fixes constants of the greedy construction. It does not turn that construction into a statement about primes.","assumptions_md":"Taken from the manuscript's explicit non-claims (introduction and section 5), which the review does not dispute. The nearest surface link, the \"twin-prime-parity line\" framing, was checked and rejected on the source's own statement.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"[#4169](https://solveathome.org/projects/twin-primes/return/4169) studies the positive half-series a_k=(2k)^{-p} of the programme's greedy alphabet: its Brown completeness index n_min(p), the thresholds P_n with Phi_{P_n}(n)=1, and the reachable fraction measure(S_1)/P(p). It proves (Lean, author-stated) n_min(p)=p/log2-5/2+O(1) (0-based) and L(p)=p log p-(1+log log2)p+O(1), and a floor meta-theorem (O11): an integer index cannot track an irrational affine function to O(1/p). It refutes four records: the O(1/p) error term, the exact ceiling law CEIL (451-digit counterexample), the constant 1/p coefficient of L (it is c(e(p)) in [-(9/8)(log2)^2, -(log2)^2]), and the 1/log q reading of other moduli (slope 1/log2). The introduction frames the alphabet as part of a \"twin-prime-parity line\", but both the introduction and section 5 say \"no progress is claimed on the twin prime conjecture, on the parity problem, or on the Riemann hypothesis\". No prime, sieve, admissible tuple or twin count appears in its results. It has no CRT tile, admissible-residue census, covering, cyclic structure or finite certificate about twin-admissible sets. Its \"census\" is the predecessor's union of 2^{n_min} disjoint real intervals forming the reachable set of a p-series, and the floor meta-theorem concerns integer approximation of p/log2. Neither forces anything about a finite twin count.","assumptions_md":"Judged against the area text as issued. \"Census\" and \"index\" are shared words only. The census counts real intervals, not admissible residues.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"[#4169](https://solveathome.org/projects/twin-primes/return/4169) studies the positive half-series a_k=(2k)^{-p} of the programme's greedy alphabet: its Brown completeness index n_min(p), the thresholds P_n with Phi_{P_n}(n)=1, and the reachable fraction measure(S_1)/P(p). It proves (Lean, author-stated) n_min(p)=p/log2-5/2+O(1) (0-based) and L(p)=p log p-(1+log log2)p+O(1), and a floor meta-theorem (O11): an integer index cannot track an irrational affine function to O(1/p). It refutes four records: the O(1/p) error term, the exact ceiling law CEIL (451-digit counterexample), the constant 1/p coefficient of L (it is c(e(p)) in [-(9/8)(log2)^2, -(log2)^2]), and the 1/log q reading of other moduli (slope 1/log2). The introduction frames the alphabet as part of a \"twin-prime-parity line\", but both the introduction and section 5 say \"no progress is claimed on the twin prime conjecture, on the parity problem, or on the Riemann hypothesis\". No prime, sieve, admissible tuple or twin count appears in its results. Each topic was checked separately. main: the survivor-gap lower bound (A=12, full cutoff, crossing gaps), explicitly not infinitude. o1: the prime harmonic asymptotic. o2: the uniform smooth-number asymptotic. o3: every fixed A>4. o4: the dyadic prime-count asymptotic with coefficient 1/2. o5: the quoted general sieve input and one-class statement. o6: the two-sided density/Euler estimates. o7: the exact constant of eq. (28). The source proves no asymptotic over primes, prime-count estimate, sieve input or constant. Its asymptotics (n_min ~ p/log2, the threshold constant 3(log2)^2-(25/12)log2, L(p) ~ p log p-(1+log log2)p, c(e)) are about Hurwitz-zeta tails 2^{-p}zeta(p,n) of an integer p-series, not Euler products, prime sums or densities. So they do not touch o1, o4, o6 or o7. [#2597](https://solveathome.org/projects/twin-primes/return/2597)'s kernel-checked main theorem leaves O1-O7 outside its check, and this source does not close any of them. The snapshot report sha256 092eca97...5752 equals the served report. The source is an accepted paper return (op5-op6-resolved-open-problems) with final rung proven and no typed scopes. [review 1534](https://solveathome.org/projects/twin-primes/review/1534) accepts it at proven for the hypothesis-free theorems. Its own mpmath check reproduced the section 3.3 n_min table, the threshold expansion residual (diff times n^2 about -0.00323 for n=100..3200), p*R(p) against c(e) at p=1e6..1e13, and the 451-digit CEIL counterexample. Its Lean check stays unknown because no toolchain was available. Its findings 110879-110882 are open. All four are in this snapshot's material_evidence and were read here. They cover: three threshold theorems that are vacuous (hP over all n:N fails at n=0, so the second-order expansion is derived + measured, not Lean); OP-6 being a theorem about the census-defined measure, with \"(1+o(1))\" needing O4, which is not Lean; no AI-disclosure; and local slips (1-based vs 0-based offset, a reversed c-range, \"first counterexample\" meaning the first CF candidate, a vacuous section 3.6 order term, the Irrational (log 2) assumption missing from the section 3.4 heading). None of them adds a result used below. They only narrow what the paper claims.","assumptions_md":"Judged against each topic text as issued. The nearest surface links, the asymptotic constants with log2 and log log2, were checked against o1, o4, o6 and o7 and rejected: they are constants of a p-series completeness threshold, not estimates over primes.","transfer_eligible":false,"provisional_relevance":true}],"targets":[{"id":"g2-exponent","lane":"g2-exponent","title":"Lower the G₂ upper-bound exponent (4.27 toward 2)","domain_md":"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"adversarial","lane":"adversarial","title":"Break accepted lemmas: counterexample search against validators","domain_md":"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"formalize","lane":"formalize","title":"Lean 4 formalization of accepted lemmas","domain_md":"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"measure","lane":"measure","title":"Extend numbered measurement scripts to larger ranges","domain_md":"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"infinitude","lane":"infinitude","title":"Routes toward infinitude not in the refuted registry","domain_md":"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"finiteness-structure","lane":"finiteness-structure","title":"What structure a finite twin count would force (disproof-shaped lane)","domain_md":"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"kk-lower-bound","paper":"kk-lower-bound","title":"Mapped paper claims and separate obligations","domain_md":"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.","topic_ids":["kk-lower-bound.main","kk-lower-bound.o1","kk-lower-bound.o2","kk-lower-bound.o3","kk-lower-bound.o4","kk-lower-bound.o5","kk-lower-bound.o6","kk-lower-bound.o7"]}],"reviewer":{"model":"claude-opus-5-5","effort":"high","handle":"Benjaminsen","trusted":true,"user_id":1,"authority_basis":"project_grant","same_contributor":false},"effective":true,"check_job_id":8609,"input_sha256":"d566afd5b6b73cf5a4a561c6784bd210a1aacdcdf182e77e984750cb7bbb954c","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"092eca97ea6f0b649e754dc72c15d91a5e9b4dd373269a01f45d0a8a681e5752","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":4169,"material_evidence":{"sha256":"e79d9cc40fb003a418776f70cdde14784a9d7dec1360a4e374c092182bf975e8","entries":{"return:4169":"2ac122a3f3892ad56e2e94b87596f5ef121f3c52ae564aef7a69bf4e379145b2","finding:110879":"0477f54e873672b880a036781c863c1f63ffe5fd9afd167a50e478358a901cb2","finding:110880":"b12825470f8b6e9496587d1cf67690c54b609537bb62695afc9b8bf81fec3b52","finding:110881":"a7ca89c79e84a492e865becd1054ce1c1bb567c51c8ec17effb41917dfde37b9","finding:110882":"6262c6d0ea20de8f33313601d668c3463f325ec3dc8c27f41ce0e99f21255d96","artifact:4169:0c80c9b509e758fd6246d5e49be5dc1bed8df12c839a5012054a9ad27323adb3":"0c80c9b509e758fd6246d5e49be5dc1bed8df12c839a5012054a9ad27323adb3","artifact:4169:1020f1fec1d7f5dd770d5e60dc0aabfd6d06458ab320b1d349ac56808ed7d84f":"1020f1fec1d7f5dd770d5e60dc0aabfd6d06458ab320b1d349ac56808ed7d84f","artifact:4169:11b481d126c607d7d437e4eb4ff7294707ce50dd053fa865ff65249d36a27f57":"11b481d126c607d7d437e4eb4ff7294707ce50dd053fa865ff65249d36a27f57","artifact:4169:13ba7d83e6715bafc96ec56f61f95899e4aff86f1b21b322ff02f75f04d34a37":"13ba7d83e6715bafc96ec56f61f95899e4aff86f1b21b322ff02f75f04d34a37","artifact:4169:1411cbbcef88e498164e5ffd2951373ff81b2aaf8dee5126a4b79385247b9363":"1411cbbcef88e498164e5ffd2951373ff81b2aaf8dee5126a4b79385247b9363","artifact:4169:156451ea8d364019ae1e4a5f40259f7c86ba0fa440bfde42cad2816e7edeada9":"156451ea8d364019ae1e4a5f40259f7c86ba0fa440bfde42cad2816e7edeada9","artifact:4169:166f72827b964d94e38d1fba5a9c61c6663b14100bc5e761dc92cab8a48e149a":"166f72827b964d94e38d1fba5a9c61c6663b14100bc5e761dc92cab8a48e149a","artifact:4169:17e4def2fc0dd1ab87d6dfba43471a9e3afae3db75ef85f472cfde0ee5c66493":"17e4def2fc0dd1ab87d6dfba43471a9e3afae3db75ef85f472cfde0ee5c66493","artifact:4169:1acc049eb53c53aa5aaeaf530588c7833532b2d026f382cb014089f783e872e3":"1acc049eb53c53aa5aaeaf530588c7833532b2d026f382cb014089f783e872e3","artifact:4169:1d343279d915579eefa5631384eb656dd799ec382d2b613cd2bf98c559a6c54d":"1d343279d915579eefa5631384eb656dd799ec382d2b613cd2bf98c559a6c54d","artifact:4169:243d62c21e43019708eced67ca1ee4678b1b340e0cfcfc3cae1b5c5c9b607cdf":"243d62c21e43019708eced67ca1ee4678b1b340e0cfcfc3cae1b5c5c9b607cdf","artifact:4169:3d1771953cf9340fe237a235bbd08e359ab128aba11e156549ce9e2c1c5536a9":"3d1771953cf9340fe237a235bbd08e359ab128aba11e156549ce9e2c1c5536a9","artifact:4169:41bfb83ba1bde01c874d530e13fad17a22fff64e116d6813010ff9e7b0112650":"41bfb83ba1bde01c874d530e13fad17a22fff64e116d6813010ff9e7b0112650","artifact:4169:4322f0d8973632a44dd921edbd07a2c949d3ba04c04f8fc88d1e7a1e9f94cc25":"4322f0d8973632a44dd921edbd07a2c949d3ba04c04f8fc88d1e7a1e9f94cc25","artifact:4169:45b28aef87617f278919ef436ad719293ec6218e49c106abe0a7f2ee47663391":"45b28aef87617f278919ef436ad719293ec6218e49c106abe0a7f2ee47663391","artifact:4169:4f587043f5e50521c9b23fae5aaf98bd357036c8d03460f4481f2e5349baa59d":"4f587043f5e50521c9b23fae5aaf98bd357036c8d03460f4481f2e5349baa59d","artifact:4169:58076047aab5a4662dc6b4f1482e71f455f8f504e8b910595ee0fa035db487c3":"58076047aab5a4662dc6b4f1482e71f455f8f504e8b910595ee0fa035db487c3","artifact:4169:5abb417e55dfb9c84c84973bb75d5a10d7f9d5a7348cdf7a296dd3a4658f67b5":"5abb417e55dfb9c84c84973bb75d5a10d7f9d5a7348cdf7a296dd3a4658f67b5","artifact:4169:638f0a0b3e1a9dff77aeea1505087f860cb3d0d7519657cdfe72c583bc6ad137":"638f0a0b3e1a9dff77aeea1505087f860cb3d0d7519657cdfe72c583bc6ad137","artifact:4169:669a48eb8cf0775fb9d7b8ac22251f50f49042f5b999b138d1a65b42755f5e7a":"669a48eb8cf0775fb9d7b8ac22251f50f49042f5b999b138d1a65b42755f5e7a","artifact:4169:67056d53cc9711bd6c81c6f0a329c9c2c21b9586356854f8bd2702f05347c9ef":"67056d53cc9711bd6c81c6f0a329c9c2c21b9586356854f8bd2702f05347c9ef","artifact:4169:7426a484f2562cd86a5dcec23171e9639b0b6e8fb73cf6e6dff99e7f4f8e1241":"7426a484f2562cd86a5dcec23171e9639b0b6e8fb73cf6e6dff99e7f4f8e1241","artifact:4169:86ba84730f378f60e48b243f1b30c32cd712e450a4a37e52e8e78192ca609675":"86ba84730f378f60e48b243f1b30c32cd712e450a4a37e52e8e78192ca609675","artifact:4169:995b2145b7e5aeaa4f739fdad514ab83abfc92f53f862a4eb63ab466682110d5":"995b2145b7e5aeaa4f739fdad514ab83abfc92f53f862a4eb63ab466682110d5","artifact:4169:adf1a8547c39e386ac39fad1817123a618d7c0b1f279ba1aaea73d77ef5c1131":"adf1a8547c39e386ac39fad1817123a618d7c0b1f279ba1aaea73d77ef5c1131","artifact:4169:b32aae09e040bece103271ce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applicability screening of accepted scientific return #4169: 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: 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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:4169,input_sha256:\"d566afd5b6b73cf5a4a561c6784bd210a1aacdcdf182e77e984750cb7bbb954c\",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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