{"id":948,"job_id":1797,"problem_id":1,"lane_id":3,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 48 rescue: the \"modulus vs range\" reading resolves toward RANGE, via a missed composite-modulus source\n\nCalibration: **heuristic** (a literature reading, not a new estimate; the cited facts are from the arXiv abstracts below).\n\n## Result\n\nI performed the route's own stated next step — the \"reading\" of whether the blocking cell's failure is a *modulus* failure (prime-only methods) or a *range* failure (length far below c^{1/2}). The route's prior art (refreshed 2026-09-17) cited, among 2026 sources in this length regime, only a \"Shen\" paper scoped to LARGE PRIME q with N just below q^{1/2}, and left the modulus-vs-range question open.\n\nA focused search surfaced a 2026 source that the prior art does not cite and that directly bears on the composite-modulus side:\n\n- **A. Pascadi, \"Non-abelian amplification and bilinear forms with Kloosterman sums\", arXiv:2511.08445 (v2, 21 Jun 2026).** It bounds bilinear (Type II) sums of Kloosterman sums with **composite moduli c** via Fourier analysis on SL_2(Z/cZ) and non-abelian amplification. At length √c it is non-trivial for all moduli **except near-primes**, with saving c^{-1/12} for products of two primes of equal size, and (combined with prime-modulus results) savings beyond Pólya–Vinogradov for all moduli.\n\nReading. A composite-modulus bilinear Kloosterman bound now exists (at length √c). This is direct evidence that the composite modulus itself is not the fundamental obstruction: the cell's modulus n = de (y-smooth, (d,e)=1) is structurally the \"product of two comparable factors\" case closest to Pascadi's pq regime. The cell's decisive gap is therefore the LENGTH — N = n^{o(1)}..n^{0.21} is far below even the composite-modulus method's √c, and far below the S-family's c^{13/28} floor. This resolves the central uncertainty's two readings toward **range failure**, and it changes what to attack: not \"make the modulus prime\" but \"push a composite-modulus method to short length.\"\n\nCaveats (stated, not resolved). Pascadi's length is √c, not the cell's n^{0.21}; its \"near-prime\" exclusion and its smooth-modulus coverage must be checked against n = de; and a method at √c does not by itself give the needed c^{-δ} at n^{0.21}. Fouvry–Shparlinski arXiv:2210.15761 (character sums with determinants) is, consistently, PRIME p with N ≥ p^{1/8+ε}, so the determinant-divisor coefficient of #935 stays unmatched for composite y-smooth moduli — this reading does not remove that obstruction, it re-routes around it.\n\n## Conclusion\n\nA distinct next experiment exists: assess whether Pascadi's non-abelian SL_2 amplification can be pushed below length c^{1/2} (toward c^{δ}, δ<1/2) uniformly over y-smooth composite moduli, or whether a trilinear second-moment extension of it absorbs the determinant-divisor coefficient of #935. This avoids the #935 coefficient decomposition by attacking the length/composite-modulus axis instead.\n","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T20:31:12.484Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[934,935],"messages":[]},"tokens":{"log":"custom","input":12149,"models":{"deepseek-v4-pro":12128},"output":12128,"source":"custom-jsonl","entries":6,"cache_read":1738624,"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 Pascadi 2511.08445v2 in full; identify where the length sqrt(c) enters (amplification/spectral gap) and whether the near-prime exclusion covers n=de; check whether a third-moment/trilinear extension moves the length exponent below 1/2, and whether the determinant-divisor coefficient is handled by the non-abelian Fourier expansion.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The length sqrt(c) is essential to Pascadi's spectral gap (no sub-1/2 extension), and the near-prime exclusion covers the cell's modulus; then the range failure is confirmed as a hard gap.","success":"A source-matched mechanism (or a concrete modification of Pascadi's) that yields c^{-delta} for delta>0 at N < c^{1/2} for y-smooth composite n=de.","question":"Can Pascadi's non-abelian SL_2 amplification (arXiv:2511.08445) be pushed below length c^{1/2} to c^{delta} (delta < 1/2) uniformly over y-smooth composite moduli, or absorbed into a trilinear second moment that also handles the determinant-divisor coefficient of return #935?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[934,935],"evidence_md":"New 2026 source not in the route's prior art: Pascadi arXiv:2511.08445 (v2 21 Jun 2026) bounds bilinear (Type II) Kloosterman sums with COMPOSITE moduli c by SL_2(Z/cZ) Fourier analysis and non-abelian amplification — non-trivial at length sqrt(c) for all moduli except near-primes, saving c^{-1/12} for products of two equal primes, and (with prime results) beyond Polya-Vinogradov for all moduli. This is evidence the composite modulus is within reach, so the blocking cell's failure is primarily a RANGE failure (N = n^{o(1)}..n^{0.21} far below c^{1/2} and c^{13/28}), not a modulus failure. Fouvry-Shparlinski 2210.15761 is PRIME p with N >= p^{1/8+eps}, consistent with the determinant-divisor obstruction staying composite-only.","prior_art_md":"Updated 2026-09-17 search. Added: Pascadi 2511.08445v2 (composite moduli, length sqrt(c), c^{-1/12} for pq, near-primes excluded). Confirmed: Fouvry-Shparlinski 2210.15761v2 (determinant character sums, prime p, N >= p^{1/8+eps}); Bettin-Chandee 2018 (trilinear Kloosterman fractions). The route's prior art cited only a prime-modulus 'Shen' in the 2026 length regime and missed the composite-modulus Pascadi result. Exact remaining gap unchanged: a c^{-delta} (delta>0) saving at N = n^{o(1)}..n^{0.21} uniformly over composite y-smooth moduli; Pascadi is at N = c^{1/2}."},"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":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/48 and return #935. 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/948/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}