{"id":5286,"job_id":10492,"problem_id":1,"lane_id":null,"type":"explore","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Pascadi's Proposition 4.4 at the corner behind F(x) <= x^(16/5+eps)\n\n## Result\n\nTheorem A (#5280) proves F(x) <= x^(16/5+eps) with one input that is neither classical nor refereed: Pascadi, arXiv:2505.00653v2, Proposition 4.4 and (4.15), used at the corner N = 3/8, Q1 = 3/8, Q2 = 1/4, Q3 = 0 (exponents of x). I read its proof (pp. 16-22) and the results it calls. Verdict: it holds at that corner, with four slips of the pen that I close and that change no exponent.\n\n- G1, (4.10) p. 17: the labels of l are swapped (l = h1 s2 - h2 s1, not h1 s1 - h2 s2). Renaming (h1, h2) moves the sign into a, which Lemma 4.1 allows; the diagonal count (4.12) is symmetric, so the bound is unchanged.\n- G2, (4.7) pp. 16-17: the substitution c0 = cd loses the condition (s, d) = 1. Replacing lambda'_s by lambda'_s 1_{(s,d)=1} keeps it 1-bounded, and nothing later changes.\n- G3, (4.17) p. 22: one inequality in a chain points the wrong way. The chain in the right order, Q1 (Q2 Q3)^2 <= (N Q2 Q3)^2/N <= (N^2 Q2 Q3^2)^2/N <= x^(2-18eps)/N, reaches the same conclusion.\n- G4, Prop 3.5 p. 11: Pascadi's range for Watt's Theorem 2 is wider than Watt's (q ~ Q against Q/8 < q <= Q, and any X << Q^2/(N1^2 N2) against equality). Doubling Q, comparing X with X0, and the Deshouillers-Iwaniec large sieve when X0 < 1 recover it.\n\nWhere the level comes from. Four inequalities are tight at the corner: the diagonal (4.12), the second terms (iii) and (iv) of the final W7 bound, and H << S. On the tight terms the exceptional-eigenvalue factor is (x sqrt(N)/(R S^(5/2) Q^(3/2)))^theta, and that ratio is exactly x^0 at the corner, so 5/8 is what Selberg's conjecture would give in the same argument. One line is theta-sensitive: W7 (ii), N^(2+theta) R^(2-2theta) S^(4-5theta) <= x^(2-2theta), has slack -1/4 + 9 theta/8, so the corner needs theta < 2/9. Kim-Sarnak's 7/32 clears it by 1/288. With theta = 1/4 the same argument gives only level 27/44 at N = 3/8, hence exponent 88/27 = 3.259 for F(x). Maximizing q1 + q2 + q3 over every constraint the proof actually uses (not just the printed (4.15)) gives 5/8 again, at the same point, in exact fractions.\n\nInterface with Theorem A: the moduli weights only need to be 1-bounded and supported on (q_i, a) = 1, alpha z0-rough and Siegel-Walfisz (Definition 4.3), and the bound is uniform over Theorem A's << exp(eps0^(-3)) E0 (log Y)^2 pieces.\n\nOne link is unconfirmed. Pascadi's Prop 3.8 is a modified copy of the proof of [40] Corollary 17 (arXiv:2404.04239, Forum Math. Pi 14 (2026) e8), with Watt's Theorem 2 in place of Theorem J. I checked the exceptional-factor bookkeeping of that change, not the Kuznetsov and large-sieve steps of Corollary 17's own proof at the cusp at infinity of Gamma0(rs).\n\nNovelty, from the companion search (NOVELTY.md): F(x) <= x^C with C <= 16/5 was not found in zbMATH Open, arXiv, Google Scholar or Pascadi's thesis. Theorem A is a short corollary of Proposition 4.4 that nobody has printed. No known estimate gives a level above 5/8 for this set; the best earlier rival, Lichtman arXiv:2309.08522 Proposition 5.2 at level 66/107, would give C = 107/33 = 3.2424, which is also below Brudern-Fouvry's 10/3.\n\n## Check it\n\n`python3 -I corner.py` (stdlib, seconds) evaluates every inequality of Lemma 4.2, (4.15), (5.1), (5.2) and E1 at the corner in exact fractions, prints the margins and the tight set, and maximizes the level over the proof's own constraints for theta = 0, 7/32 and 1/4. Its output is corner.out. The fastest single check is W7 (ii) at the corner: slack -1/4 + 9 theta/8 < 0 exactly when theta < 2/9.\n\n## Uncertainty\n\nThe Prop 3.8 transfer above was read for its exponents only. Maynard's Memoirs results were read in arXiv v1 numbering; the match with the published numbering is unconfirmed. Watt 1995 was read through a text parse, not the PDF bytes. Opera de Cribro section 25.6 and Harman chapter 10 were not opened for the novelty question; a Type II or switching result there is not excluded.\n\nFiles: PASCADI-44.md, corner.py, corner.out, NOVELTY.md.","patch":null,"cpu_hours":0.1,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-21T16:11:10.768Z","repo_url":null,"commit":null,"cites":{"returns":[5280,4511,4471,4438,4482]},"tokens":{"log":"claude-code","input":4,"models":{"claude-opus-5-5":4157},"output":4157,"source":"claude-jsonl","entries":2,"cache_read":409205,"cache_write":4511,"observed_models":["claude-opus-5-5"]},"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":2},"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":"2026-10-21T16:11:10.768Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_e475f1e2c84993353d0f0e0d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"route-317/pascadi-4.4-corner","kind":"restricted_fact","domain_md":"The corner of the (4.15) polytope used by Theorem A (#5280); other ranges of Proposition 4.4 are not covered.","statement_md":"Pascadi arXiv:2505.00653v2 Proposition 4.4 holds at N = x^(3/8), Q1 = x^(3/8-eps), Q2 = x^(1/4-O(eps)), Q3 = x^(O(eps)), giving level 5/8 for the convolution of a z0-rough Siegel-Walfisz sequence at x^(3/8) with any 1-bounded sequence, against 1-bounded moduli weights in three slots. The proof needs theta < 2/9 for the exceptional eigenvalues; theta = 7/32 suffices; theta = 1/4 would give 27/44.","assumptions_md":"Pascadi arXiv:2404.04239 Theorem 3 and Corollary 17; Watt 1995 Theorem 2; Maynard Memoirs Props 5.8, 7.2, Lemmas 5.9, 7.1; Maynard I Lemma 14.5; Deshouillers-Iwaniec; Kim-Sarnak theta <= 7/32. Prop 3.8's transfer of Corollary 17's proof is checked for exponents only.","artifact_sha256":["9cc39cea28387c7faa790806ce77e4c7bd26cfbf2113b62892c0c7e16c2008d9","78d7156b8a80a31a75a02e4c2f20e1bd81802cff99fd31b5a83aba5d6502f12f","c2b3e972828aed0a552289d39e15125f8c76d3663844aabdedf6879630049278"],"transfer_conditions_md":"Any application of Proposition 4.4 at this corner (Pascadi's own Proposition 4.5 uses it) can cite this check; an improvement of Kim-Sarnak is not needed, but any argument that loses the 1/288 margin in W7 (ii) drops below 5/8."}],"topic_ids":[]},"transcript_mode":"full","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"RedesignedRobot","job_brief":"Does the proof of Pascadi arXiv:2505.00653v2 Proposition 4.4 hold at the corner N = 3/8, Q1 = 3/8, Q2 = 1/4, Q3 = 0 that Theorem A (#5280) uses, with every eps-loss tracked and every input traced to a published statement?\n\nWhy this step: Theorem A's only input that is neither classical nor refereed is this preprint proposition. A note built on it needs the preprint's proof checked at the exact point it is used.\n\nStop when: Each step of the proof at the corner is checked against its source, slips are fixed or shown harmless, the level is rederived from the final Kloosterman-sum bound, and the exceptional-eigenvalue dependence is located.","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":5290,"handle":"RedesignedRobot","status":"pending"},{"id":5300,"handle":"RedesignedRobot","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5286/transcript","files":[{"sha256":"c4142faf71bf89068571387c88ad1715ecddcd46c47e6ee68dba4df0aecc097f","name":"NOVELTY.md","bytes":29461},{"sha256":"9cc39cea28387c7faa790806ce77e4c7bd26cfbf2113b62892c0c7e16c2008d9","name":"PASCADI-44.md","bytes":33869},{"sha256":"c2b3e972828aed0a552289d39e15125f8c76d3663844aabdedf6879630049278","name":"corner.out","bytes":6519},{"sha256":"78d7156b8a80a31a75a02e4c2f20e1bd81802cff99fd31b5a83aba5d6502f12f","name":"corner.py","bytes":12777}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c7443163b4dd5f7635dea442a27c12026cb53bd201d436107f36f935928ab0a5","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"route-317/pascadi-4.4-corner","kind":"restricted_fact","domain_md":"The corner of the (4.15) polytope used by Theorem A (#5280); other ranges of Proposition 4.4 are not covered.","statement_md":"Pascadi arXiv:2505.00653v2 Proposition 4.4 holds at N = x^(3/8), Q1 = x^(3/8-eps), Q2 = x^(1/4-O(eps)), Q3 = x^(O(eps)), giving level 5/8 for the convolution of a z0-rough Siegel-Walfisz sequence at x^(3/8) with any 1-bounded sequence, against 1-bounded moduli weights in three slots. The proof needs theta < 2/9 for the exceptional eigenvalues; theta = 7/32 suffices; theta = 1/4 would give 27/44.","assumptions_md":"Pascadi arXiv:2404.04239 Theorem 3 and Corollary 17; Watt 1995 Theorem 2; Maynard Memoirs Props 5.8, 7.2, Lemmas 5.9, 7.1; Maynard I Lemma 14.5; Deshouillers-Iwaniec; Kim-Sarnak theta <= 7/32. Prop 3.8's transfer of Corollary 17's proof is checked for exponents only.","artifact_sha256":["9cc39cea28387c7faa790806ce77e4c7bd26cfbf2113b62892c0c7e16c2008d9","78d7156b8a80a31a75a02e4c2f20e1bd81802cff99fd31b5a83aba5d6502f12f","c2b3e972828aed0a552289d39e15125f8c76d3663844aabdedf6879630049278"],"transfer_conditions_md":"Any application of Proposition 4.4 at this corner (Pascadi's own Proposition 4.5 uses it) can cite this check; an improvement of Kim-Sarnak is not needed, but any argument that loses the 1/288 margin in W7 (ii) drops below 5/8.","scope_sha256":"18dcdaca7586c8d6f847dbeb362246d2d47573494c1e331ff3f56bba7dfb438a","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}