{"id":950,"job_id":1800,"problem_id":1,"lane_id":3,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 48 pursue: Pascadi reading answered — the modulus is good, the length below c^{5/12} is the hard gap\n\nCalibration: **heuristic** (a literature reading of Pascadi arXiv:2511.08445v2; the cited facts are from its abstract, §1.2 Theorems 1.1–1.2, and Example 1.3).\n\n## Result\n\nI read Pascadi 2511.08445v2 to answer the four decisive questions the next experiment posed.\n\n- **Q1 (where the length enters).** The method improves the trivial bound min(c, √(MNc)) only in the range M ≍ N ∈ [c^{5/12+ε}, c^{5/8−ε}] (Example 1.3, for c = p² or pq); Theorem 1.1 is stated for M,N ≪ c^{1/2+o(1)} with saving c^{−1/700}. The amplification is tuned to the M≈N≈√c balance point where the Weil and Fourier trivial bounds meet; below c^{5/12} the Weil bound √(MNc) is already the trivial bound and the method does not improve it. There is **no** sub-1/2 (or sub-5/12) extension in the paper.\n- **Q2 (near-prime exclusion).** \"Near-prime\" is a modulus with a prime factor larger than c^{1−ε} (Theorem 1.2's condition — then no factorization c = dd′e with d ∈ (c^{1/3+ε}, c^{1−ε}) exists). The cell's modulus n = de is y-smooth, so every prime factor is ≤ y = c^{o(1)}: it is **not** near-prime. This is a positive sharpening — the modulus is a *good* (factorable) case for Pascadi, so the modulus is not the obstruction.\n- **Q3 (trilinear extension).** The paper has no third-moment/trilinear extension; a trilinear second moment moving the exponent below 5/12 would be new work.\n- **Q4 (determinant-divisor coefficient).** Pascadi's object is the fixed-modulus S(m,n;c) (S-family). The #935 determinant-divisor coefficient comes from the F-family phase e(−2ν inv(d)/e) with the denominator a summation variable; the non-abelian Fourier expansion does not touch that moving-modulus coefficient.\n\n## Conclusion\n\nThe reading is now precise: the blocking cell's failure is a **pure length failure**. The modulus n = de (y-smooth, factorable) is inside Pascadi's good set, but the cell's length n^{0.21} lies far below the improving-range floor c^{5/12} (≈ c^{0.417}), which is the lowest length any 2026 S-family method reaches. The experiment's success criterion (a source-matched mechanism at N < c^{1/2}) is **not met**; the failure criterion is only half met (the length barrier is real, but the near-prime exclusion does not cover the modulus). The remaining unexamined candidate is the iterated q-van der Corput method for smooth square-free moduli (Pascadi refs [55],[53]; origin [20], Blomer–Milićević [6]) — the one S-family line tailored to smooth moduli, whose improving range has not yet been checked against n^{0.21}.\n","patch":null,"cpu_hours":0.01,"hashes":{"qa.md":"89f0a5e04cc3e912797c347fb58ee4bde93dab442bf1c9de31d3bf9cd061960e"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T20:35:27.059Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[934,935,948],"messages":[]},"tokens":{"log":"custom","input":14063,"models":{"deepseek-v4-pro":12417},"output":12417,"source":"custom-jsonl","entries":6,"cache_read":1924736,"cache_write":0,"observed_models":["deepseek-v4-pro"]},"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":{"outcome":"progress","route_id":48,"next_step":{"method":"Read [55],[53] (iterated q-van der Corput for smooth square-free moduli) and [6]; extract their improving length ranges for y-smooth moduli; compare against n^{0.21}; if any reaches below c^{5/12}, check the F-family->S-family completion cost (route's measured <1/3 log) makes it applicable.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Every known S-family improving range has floor >= c^{5/12} (and near-prime-like exclusions); then the cell requires a genuinely new F-family (moving-modulus) bound, and the route records a precise scoped obstruction.","success":"An S-family method (q-van der Corput or a modification) with improving range reaching below c^{5/12} for y-smooth moduli, giving c^{-delta} at the cell's length.","question":"What is the improving range of the iterated q-van der Corput method for smooth square-free moduli (Pascadi refs [55],[53]; Blomer-Milićević [6]) — does it reach below c^{5/12} for y-smooth moduli, or is c^{5/12} a universal S-family floor that forces a new F-family (moving-modulus) method?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[934,935],"evidence_md":"Pascadi 2511.08445v2 read. (1) Improving range M≍N in [c^{5/12+eps}, c^{5/8-eps}] (Ex 1.3); Thm 1.1 saving c^{-1/700} at M,N≪c^{1/2+o(1)}; below c^{5/12} the Weil bound is the trivial bound, so no sub-5/12 extension. (2) 'Near-prime' = a prime factor > c^{1-eps} (Thm 1.2); y-smooth n=de is NOT near-prime, so the modulus is a good factorable case. (3) No trilinear/third-moment extension in the paper. (4) Pascadi is fixed-modulus S(m,n;c) (S-family), so it does not touch the F-family determinant-divisor coefficient of #935. Net: pure length failure — n^{0.21} < c^{5/12} — with the modulus confirmed good.","prior_art_md":"2026-09-17, updated. Read Pascadi 2511.08445v2 (intro, Thm 1.1/1.2, Ex 1.3): composite moduli, improving range [c^{5/12}, c^{5/8}], saving c^{-1/12} for p^2/pq, near-primes (factor > c^{1-eps}) excluded; iterated q-van der Corput for smooth square-free moduli cited as refs [55],[53] (origin [20] Blomer-Milićević [6]). Fouvry-Shparlinski 2210.15761 remains prime-p, N>=p^{1/8+eps}. Exact remaining gap: a c^{-delta} saving at N=n^{o(1)}..n^{0.21} uniformly over y-smooth composite n=de; no S-family method reaches below c^{5/12}."},"research_route_id":48,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_fb22a1982ed97962998ff647","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","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/48 and return #948. Return the ordinary report and transcript plus research: {route_id: 48, 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":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"934","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"935","status":"accepted","final_rung":"proven","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/48","transcript_url":"/projects/twin-primes/return/950/transcript","files":[{"sha256":"89f0a5e04cc3e912797c347fb58ee4bde93dab442bf1c9de31d3bf9cd061960e","name":"qa.md","bytes":11068}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}