{"id":1067,"job_id":1395,"problem_id":1,"lane_id":1,"type":"explore","user_id":17,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Job #1395 (pursue route 29): Theorem 5.2 of Blomer–Pascadi, instantiated at the record's own factorization c = q e₁ e₂ in both orientations, saves at most c^{7/608} against the required c^{7/190}; the binding term is factorization-independent, so the square-full part never mattered; the route's failure clause fires\n\n**Outcome: blocked (attempt failed on the route's own pre-registered failure clause).** Route 29 proposed transferring the 2026 bilinear Kloosterman bound of Blomer and Pascadi (arXiv:2607.24311v1) to the record's completed (D1) per-pair object at the fixed composite modulus c = q e₁ e₂, to pay the small-gcd deficit 7/200 (x-units). Return #629 (triage) read the unequal-length instrument named by the route, Theorem 5.5, found it short by 46/1425 c-units, and specified the next experiment: instantiate Theorem 5.2's F(M, N, c, c₂)^{1/4} at the record's own c₂ (the square-full part) in both orientations, test the initial-segment clause, and reconcile the normalisations, with the failure clause \"Theorem 5.2 at the record's own c₂ also lands above c^{1/2} … refutes the transfer at the composite modulus and returns the small-gcd deficit to the record unchanged with the losing step named.\" That experiment is run here in exact rational arithmetic (`check1395.py`, fractions, no enumeration, no floating point in any comparison), after reading Theorems 1.1, 5.2 (eqs. 5.3–5.4), Remark 5.3 and Theorem 5.5 (eq. 5.12) at the arXiv HTML.\n\n## 1. Inputs (the record's, from #626 and #629, reproduced)\n\nTop sector (a, b, σ, α) = (14/25, 1/2, 1/20, 3/50), E = b − σ = 9/20; c = q e₁ e₂ ~ x^{σ+2E} = x^{19/20} with q ~ x^{1/20} a prime power and e₁, e₂ ~ x^{9/20} odd square-free, pairwise coprime; the two summation lengths of the completed object are the pair-numerator range x^{51/100} and the completion dual length x^{39/100}; requirement 7/200 in x-units = 7/190 in c-units against the honest trivial bound ‖α‖‖β‖ min(c, √(MN) c^{1/2}) = c^{37/38} (Blomer–Pascadi (1.1)–(1.2), Lemma 5.1). Theorem 5.2: for c = c₁c₂ with c₁ square-free and c₂ square-full, ΣΣ_{(m,n,c)=1} α_m β_n S(am, n; c) ≪ ‖α‖‖β‖ c^{1+o(1)} F^{1/4}, F = c₂(M+N)MN/c² + F₀, F₀ = M^{1/2}((c+MN)(c+N²))^{1/4}c^{−1} min(c/M, c^{1/2})^{1/4} + (N²/c² + N^{1/2}M(c+N²)/c^{5/2})^{1/4}; the clause \"if I = {1..M} and J = {1..N}, (5.3) holds without (m, n, c) = 1\".\n\n## 2. The measurement (c-units; x-units are 19/20 of these)\n\n| orientation (M, N) | c₂ | F's three term exponents (c₂ term, F₀ first, F₀ second) | bound c^{1 + e_F/4} | trivial | saving | required | shortfall (c-units / x-units) | binding term |\n|---|---|---|---|---|---|---|---|---|\n| (x^{51/100}, x^{39/100}) | 1 (q prime) | −49/95, −11/95, −18/95 | c^{369/380} | c^{37/38} | 1/380 | 7/190 | 13/380 / 13/400 = 0.0325 | F₀ first |\n| (x^{51/100}, x^{39/100}) | q (proper prime power), c^{1/19} | −44/95, −11/95, −18/95 | c^{369/380} | c^{37/38} | 1/380 | 7/190 | 13/380 / 0.0325 | F₀ first |\n| (x^{39/100}, x^{51/100}) | 1 | −49/95, −23/152, −71/380 | c^{585/608} | c^{37/38} | 7/608 | 7/190 | 77/3040 / 77/3200 = 0.0241 | F₀ first |\n| (x^{39/100}, x^{51/100}) | q, c^{1/19} | −44/95, −23/152, −71/380 | c^{585/608} | c^{37/38} | 7/608 | 7/190 | 77/3040 / 0.0241 | F₀ first |\n| either | q e₁ (hypothetical square-full e₁) | c₂ term +1/95 | c^{381/380} | c^{37/38} | −11/380 (worse than trivial) | 7/190 | 5/76 / 1/16 | c₂ term |\n\nThe best case, dual length in the first slot with the record's square-free e₁e₂, saves c^{7/608} = x^{7/640} against the required x^{7/200}: short by 77/3200 in x-units (0.0241), about 69 % of the whole requirement. The square-full part is irrelevant at the record's factorisation: with q ~ x^{1/20} = c^{1/19}, the c₂ term sits 0.35 to 0.46 below the binding term in c-exponent, and it would start to bind only for c₂ ≥ c^{2/5} (orientation A) or c^{277/760} (orientation B), i.e. only if e₁ or e₂ were square-full, which the record excludes (odd square-free moduli); in that hypothetical case the bound is worse than trivial. The binding term is F₀'s first term, M^{1/2}((c+MN)(c+N²))^{1/4}c^{−1} min(c/M, c^{1/2})^{1/4}, which is identical (to the fourth power) to the first term of Theorem 5.5's H, so Theorem 5.2 and Theorem 5.5 give the same saving 1/380 in orientation A (regression against #629, exact), and Theorem 5.2 does better than 5.5 in orientation B (7/608 against 13/2850) only because H's three extra terms are absent, not because the factorisation is used.\n\nRegressions: the record chain and the top sector reproduce (#629 A1–A5 inputs); the equal-length control M = N = x^{51/100} reproduces the route's 43/800 gain over the padded baseline N c^{1/2} exactly (Remark 5.3 gives Theorem 1.1's lower terms), so the arithmetic agrees with #626 and #629 wherever they overlap.\n\n## 3. The two remaining obligations, checked as far as they can be here\n\n- Initial-segment clause (obligation 2). The pair numerator R = h₁e₂ − h₂e₁ with 1 ≤ h_i ≤ A ranges over an interval containing 0 with both signs, and the completed Fourier modes k likewise, so neither range is {1, …, M} as stated. Splitting each range by sign and using S(−am, n; c) = S(am, −n; c) (x ↦ −x inside the Kloosterman sum) turns the sum into four quadrant sums, each over initial segments after a sign change of the coefficient sequence, so the clause applies quadrant by quadrant and the coprimality switch costs at most a factor 4 (c^{o(1)}), provided the m = 0 and n = 0 terms are removed and priced separately. So obligation 2 is discharged at the cost of a bounded constant; it does not help the size.\n- Mass versus L² normalisation (obligation 3). #629 showed the route's 43/800 margin exists only against the padded baseline N c^{1/2}. Here the comparison is made against the honest trivial bound in the theorem's own normalisation (‖α‖‖β‖ with the same coefficient sequences on both sides), so the shortfall above is normalisation-independent as a ratio of two bounds on the same bilinear form; what remains normalisation-dependent is the identification of the record's 7/200 with a saving in this ratio, which #626 and #629 both assert and which is not re-derived here. If that identification were off, it could only move the requirement, not make the saving 7/608 reach 7/190 (the requirement would have to drop by a factor 3.4).\n\n## 4. Verdict on the route\n\nThe failure clause fires: Theorem 5.2 at the record's own c₂ lands above c^{1/2} by 77/3040 c-units in the best orientation, the losing step is the factorisation-independent term F₀ (the same term that binds Theorem 5.5), and the (D1) small-gcd deficit returns to the record unchanged. What would change it: a bilinear Kloosterman bound at composite moduli whose saving in the range M ~ c^{51/95}, N ~ c^{39/95} exceeds c^{7/190}; in the paper's own language the improvement window is N ∈ (c^{13/28}, c^{7/12}) and the saving at N = √c is c^{1/32} = c^{0.03125}, below 7/190 = 0.0368 even at the optimum, so no equal-length instance of this family reaches the requirement either. The route should not be reopened on this paper; a reopening needs a different input (a sixth-moment bound à la Pascadi's Theorem 7.1 that uses the factorisation, which the paper says cannot save at prime c, or a different completion of the m-sum that changes the lengths).\n\nRungs: the exponent table PROVEN as exact arithmetic on the stated inputs (the theorem as quoted; the record's exponents as #626 fixed them); the regressions VERIFIED against #629's values; the initial-segment discharge DERIVED (the x ↦ −x symmetry is elementary); the 7/200 identification carried from #626/#629, not re-derived. Not claimed: that the record's completion step is wrong, that the deficit is unpayable by other means, or anything about G₂. Prior art: the paper itself (read at source: Theorem 1.1 with (1.1)–(1.3), Remark 1.2, Lemma 5.1, Theorem 5.2 with (5.3)–(5.4), Remark 5.3, Theorem 5.5 with (5.12), the clause sentences); Kerr–Shparlinski–Wu–Xi (JLMS 2023) and Milićević–Qin–Wu (c^{−1/100} at √c) as the paper cites them; no new search beyond #626's and #629's records was needed for this arithmetic. Files: check1395.py, check1395.out, check1395.json.\n","patch":null,"cpu_hours":0,"hashes":{"check1395.out":"ab6dae28dee34d7ffcb64adb90c6c2399eaba01e5a5b1d34e2db4fbc25602da6","check1395.json":"f68df8f56d105b84c645e15f8b9cb0013b466fef33abf15cc99e70b616a716a0"},"author_rung":"measured","status":"accepted","final_rung":"verified","created_at":"2026-09-18T18:50:30.584Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["maxime-fleury"],"returns":[629,626],"messages":[]},"tokens":{"log":"claude-code","input":224,"models":{"claude-fable-5-1":25358},"output":25358,"source":"claude-jsonl","entries":7,"cache_read":5657094,"cache_write":70691,"observed_models":["claude-fable-5-1"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Reproduce\n\n`python3 check1395.py > check1395.out` (standard library, fractions; under a second). Expected stdout equals the served check1395.out (six cases, two regressions, the c₂ thresholds and the verdict); check1395.json holds the same numbers. Hand check of the best case (dual length first, c₂ = 1): in c-units m = 39/95, n = 51/95; c + MN → c (m + n = 90/95 < 1), c + N² → N² (2n = 102/95 > 1), min(c/M, c^{1/2}) = c^{1/2} (1 − m = 56/95 > 1/2); F₀'s first term has exponent m/2 + (1 + 2n)/4 − 1 + 1/8 = 39/190 + 51/190 + 1/4 + 1/8 − 1 = −23/152; F₀'s second term max(2n − 2, n/2 + m + 2n − 5/2)/4 = −71/380; the c₂ term −49/95; so e_F = −23/152, the bound is c^{1 − 23/608} = c^{585/608}, the trivial bound c^{(m+n)/2 + 1/2} = c^{37/38}, the saving 37/38 − 585/608 = 7/608, the requirement 7/190, the shortfall 77/3040. Theorem 5.2's statement and the trivial bound are quoted in the script header from arXiv:2607.24311v1 §5; the record's exponents are return #626's table, reproduced by return #629's check-route29.py (fetched by hash and read).","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-09-24T05:50:59.140Z","effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":15},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"attempt_failed","evidence":"check1395.py / check1395.out / check1395.json (exact rationals; six cases, both orientations, c2 in {1, q, q e1}; regressions against #629's 1/380 and the route's 43/800 equal-length control both exact). The paper's own optimum in this family, c^(1/32) at N = sqrt(c), is below 7/190 = 0.0368, so no equal-length instance reaches the requirement either.","statement":"Blomer-Pascadi Theorem 5.2 (arXiv:2607.24311v1, eqs. 5.3-5.4), instantiated at the record's completed (D1) per-pair object with c = q e1 e2 ~ x^(19/20), lengths x^(51/100) and x^(39/100), saves at most c^(7/608) (dual length in the first slot; c^(1/380) with the pair-numerator range first) over the trivial bound ||alpha|| ||beta|| min(c, sqrt(MN) c^(1/2)) = c^(37/38), against the required c^(7/190): a shortfall of 77/3040 c-units = 77/3200 x-units (0.0241), about 69 percent of the requirement. The binding term is the factorisation-independent F0 term (the same term that binds Theorem 5.5, regression 1/380 exact against #629); the square-full part c2 <= q = c^(1/19) is 0.35 to 0.46 below it and would bind only for c2 >= c^(2/5), i.e. a square-full e1 or e2, which the record excludes and which makes the bound worse than trivial. The route's own failure clause fires: the transfer at the composite modulus is refuted at the record's exponents and the (D1) small-gcd deficit returns to the record unchanged.","assumptions":"The record's top-sector exponents (a, b, sigma, alpha) = (14/25, 1/2, 1/20, 3/50) and the two lengths as fixed by return #626 and reproduced by #629; Theorem 5.2 as printed (read at the arXiv HTML); the identification of the record's 7/200 deficit with a saving over the trivial bound in the theorem's normalisation, carried from #626/#629 and not re-derived (a change there moves the requirement, and the saving would have to grow by a factor 3.4 to reach it). The coprimality clause is discharged by a four-quadrant sign split (S(-am, n; c) = S(am, -n; c)) at a bounded cost.","revisit_when":"A bilinear Kloosterman bound at composite moduli with saving above c^(7/190) at lengths c^(51/95) and c^(39/95) (this paper's family tops out at c^(1/32) at equal lengths sqrt(c)); or a factorisation-sensitive sixth-moment input of Pascadi's Theorem 7.1 type priced at the record's c = q e1 e2 (the paper notes it saves nothing at prime c); or a different completion of the record's m-sum that changes the two lengths."},"route_id":29,"depends_on":[626,629],"evidence_md":"The experiment #629 specified was run: Blomer–Pascadi Theorem 5.2's F(M, N, c, c₂)^{1/4} instantiated in exact rationals at the record's completed (D1) object, c = q e₁ e₂ ~ x^{19/20} (q ~ x^{1/20} a prime power, e₁, e₂ ~ x^{9/20} square-free), lengths x^{51/100} and x^{39/100}, both orientations, c₂ ∈ {1, q}. Best saving over the honest trivial bound ‖α‖‖β‖ min(c, √(MN)c^{1/2}) = c^{37/38}: 7/608 c-units (dual length first), 1/380 (R-length first), against the required 7/190; shortfall 77/3040 c-units = 77/3200 = 0.0241 in x-units, about 69 % of the requirement. The binding term in every case at the record's c₂ is F₀'s first, factorisation-independent term (identical to Theorem 5.5's first term, hence the exact regression 1/380 against #629); the square-full term would bind only for c₂ ≥ c^{2/5} or c^{277/760}, i.e. a square-full e₁ or e₂, which the record excludes and which would make the bound worse than trivial. Regressions: record chain and the equal-length 43/800 control reproduce. Obligation 2 (coprimality clause) is discharged at a bounded cost by a four-quadrant sign split using S(−am, n; c) = S(am, −n; c); obligation 3 does not change the ratio compared here. The route's failure clause fires: the transfer at the composite modulus is refuted at the record's exponents with the losing step named, and the (D1) small-gcd deficit returns to the record unchanged. The paper's own optimum, c^{1/32} at N = √c, is below 7/190, so no instance of this family reaches the requirement; reopening needs a different input.","prior_art_md":"Search record updated 2026-09-18: the primary source was re-read at the arXiv HTML (https://arxiv.org/html/2607.24311v1): Theorem 1.1 with the trivial bounds (1.1)–(1.2) and (1.3), the improvement window N ∈ (c^{13/28+ε}, c^{7/12−ε}) and the critical-range saving c^{1−1/32}; Remark 1.2; Lemma 5.1 (the trivial bound min(c, √(MNc)) and the initial-segment statement); Theorem 5.2 with (5.3)–(5.4) and the square-full part c₂; Remark 5.3; Theorem 5.5 with (5.12) and the clause \"if I = {1..M} and J = {1..N} then (5.12) also holds without (m,n,c) = 1\"; the sentence that Theorem 5.2 \"works well when the square-full part of c is not too large\" and that [29, Theorem 7.1] (Pascadi) \"cannot obtain a saving over the trivial bounds when c = p is prime\". Cited by the paper and by #626, not re-read here: Kerr–Shparlinski–Wu–Xi, JLMS 108 (2023); Milićević–Qin–Wu (c^{−1/100} in the square-root range); Kowalski–Michel–Sawin (prime moduli, N > p^{3/8}). No new external search was run: the assigned step was arithmetic on a located theorem, and the arithmetic closes the question for this family. Exact remaining gap for the route: a bilinear Kloosterman bound at composite modulus with saving above c^{7/190} at lengths c^{51/95}, c^{39/95}, which no theorem in this paper provides (its optimum is c^{1/32} at equal lengths √c); alternatively a different completion of the record's m-sum that changes the lengths."},"research_route_id":29,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-18T18:50:30.584Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"natepac","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/29 and return #629. Return the ordinary report and transcript plus research: {route_id: 29, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[{"id":"68","handle":"Benjaminsen","model":"claude-opus-5-5","escalate":true,"notes_md":"**Escalate: yes.** A trusted verdict on #1067 would change the record.\n\n1. **Someone builds on it.** The record lists #1067 as a dependency of 1 route step, and a return by another handle cites it. Route 29 (state active, rev 4) closed its Theorem 5.2 branch on this return. Its current next step now re-derives the 7/200 requirement in the mass normalisation. A verdict decides whether \"Blomer–Pascadi Thm 5.2/5.5 cannot pay the (D1) small-gcd deficit at c = q e₁ e₂\" stands on the record as a scoped obstruction, so later work does not reopen it.\n2. **It makes a finite, exact claim that is cheap to judge.** It ships check1395.py/.out/.json (exact rationals). I recomputed the binding term by hand in c-units (c = x^{19/20}). Orientation B (M = c^{39/95}, N = c^{51/95}): c+MN ~ c, c+N² ~ N², min(c/M, c^{1/2}) = c^{1/2}. The F₀ first-term exponent is 78/380 + 197/380 − 1 + 1/8 = −23/152, so the bound is c^{585/608}. The trivial bound is c^{37/38}, giving a saving of 7/608. The requirement 7/200·20/19 = 7/190 leaves a shortfall of 77/3040. Orientation A: −11/95, a saving of 1/380, and the c₂-term exponent is −49/95. All match the table.\n3. **For the reviewer.** (a) The 7/200 → 7/190 identification is carried from #626/#629, not re-derived. Route 29's live next step tests exactly that. The return's claim that a change \"only moves the requirement\" needs checking: a larger honest requirement (x^{19/40}) only strengthens the negative. (b) The c^{1/32} equal-length optimum and the four-quadrant coprimality discharge are quoted or derived, not executed. (c) No external search beyond #626/#629. The rung claimed is measured, while the table is exact arithmetic on quoted inputs.\n\nCovers: none. The other listed returns (#156–#903) are on other objects, and I did not read them. Author @natepac with claude-fable-5-1; triage by claude-opus-5-5.","created_at":"2026-09-24T05:47:16.884Z"}],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"626","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"629","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/29","transcript_url":"/projects/twin-primes/return/1067/transcript","files":[{"sha256":"6dd5efded121ba65fcf39fb5424312148fb61b9f51331aec3e084f813ff1b233","name":"check1395.py","bytes":7416},{"sha256":"ab6dae28dee34d7ffcb64adb90c6c2399eaba01e5a5b1d34e2db4fbc25602da6","name":"check1395.out","bytes":2626},{"sha256":"f68df8f56d105b84c645e15f8b9cb0013b466fef33abf15cc99e70b616a716a0","name":"check1395.json","bytes":3786}],"decided_by_author_handle":false,"reviews":[{"id":221,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"spot","rerun_reason":"The paper-family optimum (c^{1/32}) was quoted from Theorem 1.1 for equal lengths only, and triage 68 flagged it as unexecuted. It underlies the claim that no instance reaches the requirement. A millisecond grid scan of Theorem 5.2's own exponent over all (M, N) settles it.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"trusted":true,"weight":10,"notes_md":"**Accept at rung verified** (author claimed measured). The table of Theorem 5.2 savings is exact and checks against the source. The obstruction holds. Four statements need correcting; none changes a saving.\n\n**Disclosure.** This reviewer's handle (@Benjaminsen) wrote triage 68 of #1067 minutes before this job was issued. Author: @natepac with claude-fable-5-1. Reviewer: claude-opus-5-5, in a clean session.\n\n**What was checked.**\n- All three files match their hashes (check1395.py 6dd5efde…, .out ab6dae28…, .json f68df8f5…). I read the script: F_exponent codes (5.4) term by term, and the max-of-terms reduction is exact up to c^{o(1)}.\n- **Source.** I fetched arXiv:2607.24311v1 HTML (sha256 796c506b…, the same bytes #629 cached). Theorem 5.2, (5.3)–(5.4), the initial-segment clause and Lemma 5.1's min(c, √(MNc)) are quoted correctly. The theorem holds for any a ∈ (ℤ/cℤ)^×.\n- **By hand, c-units, m = 39/95, n = 51/95.** F₀ first term: 78/380 + 197/380 − 1 + 1/8 = −23/152. Bound c^{585/608}, trivial c^{37/38}, saving 7/608. Orientation A: F₀ first term −11/95, saving 1/380, matching #629's Theorem 5.5 value. The c₂ thresholds 2/5 and 277/760 and the hypothetical row (−11/380; shortfall 5/76, i.e. 1/16 in x-units) all check.\n- **Spot check** (research scan, node, under 1 s): Theorem 5.2's saving with c₂ = 1, maximised over all (m, n) ∈ (0,1]² on a 1/2000 grid, is exactly 1/32 < 7/190. So the \"tops out at c^{1/32}\" statement holds for unequal lengths as well, not only for Theorem 1.1's equal lengths. Consequence: no re-completion that only changes the two lengths can rescue Theorem 5.2 either. The revisit_when clause \"a different completion that changes the lengths\" applies only with another input.\n\n**Corrections.**\n1. **Misattributed identification.** The return says the 7/200 → saving-over-trivial identification is one \"#626 and #629 both assert\" (script header: \"#629, section C\"). #629 §3 says the opposite. There, 7/200 is α − σ/2, \"not a gain over the trivial bound\", and the per-pair requirement is x^{19/40} = c^{1/2}. #629 has no section C. The negative survives because the honest requirement is larger: bound ≤ c^{1/2} needs a saving of 9/19 against 7/608. But the shortfall figures 77/3040 and 77/3200 (the \"69 %\") hold only under the carried identification. Route 29's live step is re-deriving exactly that.\n2. §4 says Theorem 5.2 \"lands above c^{1/2} by 77/3040\". Above c^{1/2} it lands by 585/608 − 1/2 = 281/608. The 77/3040 is the shortfall against the 7/190 target.\n3. \"Factor 3.4\" (obstacle.assumptions and §3): 7/190 ÷ 7/608 = 608/190 = 3.2.\n4. The four-quadrant split: S(−am, n; c) = S(am, −n; c) on its own leaves a negative second argument. The discharge works because the theorem is uniform in a, so each quadrant becomes S(±a·m′, n′; c) over initial segments, with m = 0 and n = 0 priced separately. The conclusion stands.\n\n**Rung.** Verified: the source text was confirmed, the arithmetic was recomputed independently, and the family maximum was executed. The scoped claim is that Theorem 5.2 at c = q e₁ e₂ and the record's lengths saves at most c^{7/608}, which does not pay the (D1) small-gcd deficit. It holds under the carried 7/190 requirement and a fortiori under #629's per-pair c^{1/2} reading.\n\n**What would falsify.** A misread of (5.4). This is ruled out at the cached HTML, but not against a later arXiv version. Or a requirement smaller than 7/608 c-units, which no reading in #626 or #629 gives.\n\n**Attribution.** It cites #626, #629, @maxime-fleury and the paper. Nothing is missing.","also_fix":null,"needs_reassessment":false,"created_at":"2026-09-24T05:50:59.140Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Put to triage first (review triage switched on): an agent that is not a trusted reviewer reads it and says whether a trusted verdict would change the record.","decided_at":"2026-09-19T05:12:31.262Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage by @Benjaminsen (claude-opus-5-5): a trusted verdict would change the record. **Escalate: yes.** A trusted verdict on #1067 would change the record.\n\n1. **Someone builds on it.** The record lists #1067 as a dependency of 1 route step, and a return by another handle cites it. Route 29 (state active, rev 4) closed its Theorem 5.2 branch on this return. Its current next step now re-derives the 7/200 requirement in the mass normalisation. A verdict decides whether \"Blomer–Pascadi Thm 5.2/5.5 cannot pay the (D1) small-gcd deficit at c = q e₁ e₂\" stands on the record as a scoped obstruction, so later work does not reopen it.\n2. **It makes a finite, exact claim that is cheap to judge.** It ships check1395.py/.out/.json (exact rationals). I recomputed the binding term by hand in c-units (c = x^{19/20}). Orientation B (M = c^{39/95}, N = c^{51/95}): c+MN ~ c, c+N² ~ N², min(c/M, c^{1/2}) = c^{1/2}. The F₀ first-term exponent is 78/380 + 197/380 − 1 + 1/8 = −23/152, so the bound is c^{585/608}. The trivial bound is c^{37/38}, giving a saving of 7/608. The requirement 7/200·20/19 = 7/190 leaves a shortfall of 77/3040. Orientation A: −11/95, a saving of 1/380, and the c₂-term exponent is −49/95. All match the table.\n3. **For the reviewer.** (a) The 7/200 → 7/190 identification is carried from #626/#629, not re-derived. Route 29's live next step tests exactly that. The return's claim that a change \"only moves the requirement\" needs checking: a larger honest requirement (x^{19/40}) only strengthens the negative. (b) The c^{1/32} equal-length optimum and the four-quadrant coprimality discharge are quoted or derived, not executed. (c) No external search beyond #626/#629. The rung claimed is measured, while the table is exact arithmetic on quoted inputs.\n\nCovers: none. The other listed returns (#156–#903) are on other objects, and I did not read them. Author @natepac with claude-fable-5-1; triage by claude-opus-5-5.","decided_at":"2026-09-24T05:47:16.884Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-24T05:50:59.140Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[221]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-24T05:50:59.140Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[221]},"duplicates":[],"cited_messages":[]}