{"id":"d7c57ad3-f2b4-43c1-b3e1-7066663c17db","arxiv_id":"2607.24311","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Bilinear Kloosterman forms save c^{-1/32} at length √c for all moduli via a new link to quadratic character sums, improving prior prime and composite bounds.","lead":"The paper proves stronger bounds on bilinear sums of Kloosterman sums for every modulus, saving c to the minus 1/32 in the critical square-root range. Those bounds feed better error terms for twisted L-function moments and a cleaner large sieve for exceptional Maass forms.","discovery_kind":"new_method","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"No significant objection identified","rationale":"The reader identified the right place to look—the amplification/square-full reduction—but the paper’s own exponent optimization appears to close that range rather than merely assume it away. The core square-free argument is comparatively transparent: the fourth moment is reduced to a Jacobi-symbol sum, followed by Hölder, elementary or Fourier-analytic counting, and Weil bounds. The remaining dependence on [29, Theorem 7.1] is explicit and is used through stated bounds, not through an unquantified uniformity assumption. In the critical range, the first and second sixth-moment choices cover the region where the new fourth-moment method’s square-full diagonal term becomes too large, with overlap rather than a gap. This does not independently re-verify every lengthy counting estimate, especially Proposition 4.3, but none reveals a specific assumption on which the central \\(c^{-1/32}\\) claim is likely to fail. The proposed exponent audit is worthwhile because it checks the exact place where a uniform statement could silently degrade, but the present text does not justify moving the reader’s ACCEPT.","tokens_in":36523,"tokens_out":19061,"duration_ms":1049172,"concrete_test":"Parametrize \\(c_2=c^u\\) and \\(M=N=c^v\\), \\(0\\le u,v\\le1\\). From the unsimplified \\(F\\) in (5.4) and the \\(G\\)-bounds in (5.13)–(5.14), compute \\(e=\\min(1+f/4,1+g_1/6,1+g_2/6)\\) piecewise and compare it with the exponent of \\(cH(c^v,c^v,c)\\) in (5.12), checking all breakpoints and a fine grid. If \\(e\\) ever exceeds the displayed exponent, the uniform reduction loses more than claimed; otherwise the square-full hybrid is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern lands. The natural pressure point is the reader’s: Lemma 3.3 applies only after amplification to the square-free part, leaving the square-full part \\(c_2\\) to be controlled through the diagonal term in Theorem 5.2 and the two applications of [29, Theorem 7.1]. The exponent accounting does not show a gap. Write \\(c_2=c^u\\) and \\(N=\\sqrt c\\). The square-full term in Theorem 5.2 contributes \\(c^{7/8}c_2^{1/4}\\), which meets the target \\(c^{31/32}\\) for \\(u\\le 3/8\\). For \\(u>3/8\\), the two sixth-moment bounds overlap: (5.13) gives saving \\(\\min(1-u,u)/6\\), while (5.14) gives saving \\(\\min(1-4u/5,u/4)/6\\); their maximum is at least \\(1/32\\) throughout \\(3/8\\le u\\le1\\). Thus the uniform critical-range statement is not supported by an unchecked intermediate range. Lemma 3.3 itself is a direct fixed-point count on \\(\\mathbf P^1(\\mathbf F_p)\\), extended multiplicatively, and appears sound.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper proves new bounds for bilinear forms ∑_{m∈I,n∈J} α_m β_n S(am,n;c) with (m,n,c)=1, valid for all moduli c. Theorem 1.1 gives a saving factor with three terms (N^{1/8}c^{-3/32} + N^{5/16}c^{-3/16} + N^{2/3}c^{-7/18}), which in the critical range N=√c yields a saving of c^{-1/32} over the trivial bound ‖α‖‖β‖Nc^{1/2+o(1)} — doubling the previous p^{-1/64} saving of Kowalski–Michel–Sawin for prime moduli and improving on Milićević–Qin–Wu for general moduli. The method reduces the bilinear form, via the spectral norm and a fourth-moment trace bound, to a character sum of SL₂(ℤ/cℤ) (Proposition 3.1, with non-abelian amplification to strip the square-full part), identifies the resulting character χ°_c with a Jacobi symbol in the discriminant Tr(g)²−4 for odd square-free c (Lemma 3.3), and treats the resulting quadratic character sum by a Hölder reduction to two counting problems and one complete character sum (Propositions 3.6, 4.1–4.5, using Weil's bound). For moduli with large square-full part the result is combined with the sixth-moment bound from the second author's prior work [29] (Theorems 5.2–5.5). Applications give an error term q^{-1/90+ε} in the second moment of twisted L-functions (Theorem 1.3) and a uniform large sieve inequality for exceptional Maass forms (Theorem 1.6).","tokens_in":36743,"tokens_out":4900,"duration_ms":147386,"significance":"If the estimates hold — and I found no gap — this is a strong result in the analytic theory of exponential sums: it doubles the best previous saving in the critical range even for prime moduli, is the first bound beating the trivial estimate in the range c^{13/28+ε} < N < c^{7/12−ε} for arbitrary c, and yields the best known error terms in the second moment of twisted L-functions (q^{-1/90+ε}) and a level-uniform improvement of the Deshouillers–Iwaniec exceptional-spectrum large sieve. Particular strengths: the argument is self-contained modulo clearly cited theorems (Weil bounds, [29]); it introduces a genuinely new mechanism (the trace χ°_c equals a Jacobi symbol of Tr(g)²−4); the exponents are explicit and hence falsifiable; and the proof uses no unproved hypotheses. The auxiliary bound of Lemma 5.6/Theorem 5.7 for arbitrary moduli is of independent interest.","major_comments":[{"comment":"The passage from (5.7) to (5.9) is the place where the exponents of Theorem 1.1 are actually fixed, and it rests on two monotonicity assertions that are not fully substantiated: (i) that every term in the expansion of (5.7) grows at most like d^{-1}, so the maximum over d in [c_2, c] is attained at d = c_2; and (ii) that at d = c_2 the expression is non-increasing in c_2, justifying the replacement c_2, d \\mapsto 1. Because (5.7) contains a minimum of two expressions with different d-dependence (the elementary bound of Prop. 4.2 and the Fourier bound of Prop. 4.3), the d^{-1} claim is not immediate from the displayed formula; a term-by-term expansion or a small lemma tabulating the d-exponents would make this checkable. As written, the reader must redo a nontrivial optimization to verify the load-bearing step of the paper.","section":"§5, proof of Theorem 5.2, (5.7)-(5.9)"},{"comment":"The error term q^{-1/90} in Theorem 1.3 is one of the headline results, but its derivation is outsourced to a Mathematica Maximize computation whose printed output is the only verification. The combination of (6.2), (6.3), (6.4) under (6.1) is a finite piecewise-linear optimization, so a human-checkable certificate should be easy to give: please identify the active constraints at the optimum (m, n) = (43/90, 43/30), state which of the three bounds is attained there, and confirm the value -1/90 by exact rational arithmetic. Also, in the derivation of (6.4), the claim that 'the smallest power of r is positive' (so that r = q may be substituted) is asserted after substituting N^* = q^2/N in all but one factor; one line identifying the two extremal r-exponents would remove any ambiguity.","section":"§6, Theorem 1.3 (end of section, In[1]/Out[1])"},{"comment":"Proposition 3.1 is imported almost entirely from [29] (Corollary 4.11, and the amplification of Proposition 5.1), and the manuscript's uniformity over all moduli depends on the amplified estimate (3.3) holding with the indicated error O_epsilon(c^{-100}) and on the nonnegativity of S used in the amplification step. The argument given is convincing, but since this is the only place where the square-full part c_2 enters before the hybrid combination of Theorem 5.5, I ask the authors to state explicitly which hypotheses of [29, Corollary 4.11] are being invoked (in particular the role of the weight nu_{(m,n,c_1)} and the indicator 1_{(m,n,c_2)=1}) and to confirm that the error term survives the 2k-th root in (3.1) unchanged. A short remark would suffice; no new mathematics is needed.","section":"§3.1, Proposition 3.1"}],"minor_comments":[{"comment":"'Here prove the following result.' — missing 'we'.","section":"§1.2, paragraph before Theorem 1.3"},{"comment":"In the displayed estimate there appears to be a stray bracket: 'H_1^3 H^{3/2}]_2 / (cd^{5/4})'. Please check the source; the same display also has heavy exponent bookkeeping that would benefit from one more intermediate line.","section":"§4.2, proof of Proposition 4.3"},{"comment":"For complex matrices the variational characterization should read |w^* A v| (conjugate transpose), not w^T A v.","section":"§2.1, Eq. (2.5)"},{"comment":"The claim that E_{c/d} is 'essentially optimal' is plausible (diagonal vs. generic pairing), but for composite c/d the interpolating terms depend on the factorization; a half-sentence clarifying in what sense optimality is meant (e.g., matching the two displayed extremes G=1 and G=c/d) would help.","section":"§4.3, Remark 4.6"},{"comment":"The third member of the minimum in Theorem 5.5 is used silently in the proof of Theorem 1.1 when N is small (the N^{1/15}/c^{1/15} branch); a forward reference from Theorem 1.1 to the case analysis H(N,N,c) would make the range N <= c^{13/28} easier to follow.","section":"§5, Remark 1.2 / Theorem 5.5"},{"comment":"Two references central to the comparison are currently preprints ([27], [13]); if they have appeared or been accepted by revision time, please update. The acknowledgements' mention of AI assistance could be moved to a footnote describing its scope, per emerging journal conventions.","section":"References / acknowledgements"}],"recommendation":"minor_revision","confidential_remarks":"Two points for the editor only. (1) The manuscript leans substantially on the second author's own paper [29] (cited as accepted in GAFA), using its Theorem 7.1, Corollary 4.11, Proposition 5.1, Corollary A.2 and Proposition 9.2 as black boxes; this is appropriate given the hybrid method, but [29] should be available to the handling editor if any question arises. (2) The acknowledgements state that ChatGPT Pro was used to check an earlier version for errors; the journal may wish to confirm this conforms to its AI-assistance policy, though it has no bearing on the mathematics. The result is strong and, if the two verification requests in my major comments are met, I recommend publication essentially as is."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The headline result is a uniform bilinear bound for Kloosterman sums that saves c^{-1/32} when the lengths are square-root of the modulus, for every c, and doubles the best prime-modulus saving. That is the number people will quote.\n\nWhat is actually new is the explicit identification of the special character χ°_c with a Jacobi symbol once the square-full part is amplified away (Lemma 3.3 / Prop 3.4), followed by a tailored Hölder step and elementary/Fourier counts for the resulting four-variable character sum. The rest of the skeleton (Fourier analysis on SL_2, amplification, combination with the sixth-moment bound from Pascadi [29]) is standard, but the quadratic-character bridge is the genuine increment over KMS, Blomer–Milićević, MQW, and [29] itself. The write-up is complete and readable; the exponent bookkeeping in the hybrid (Thms 5.2–5.5) closes the square-full ranges without a gap, as the stress-test confirms. Applications are concrete: twisted second moment error q^{-1/90} and a factorization-free exceptional large sieve with an explicit X.\n\nSoft spots are minor and proportional. The method is hybrid by design, so the deepest square-full moduli still lean on [29]; that is acknowledged and controlled, not hidden. The computer-algebra optimization for the moment exponent is printed and checkable. No circularity, no free parameters, citations look honest.\n\nThis is for people who actually use bilinear Kloosterman forms in moments, sieves, or primes in AP. It deserves a serious referee and belongs in a top analytic-number-theory venue. I would bring it to reading group and expect to cite the main bound and the large-sieve form within the year. Send it out.","headline":"Uniform c^{-1/32} bilinear Kloosterman bound via a clean SL_2-to-Jacobi link; hybrid with [29] checks out and the applications are real.","tokens_in":37902,"tokens_out":486,"would_cite":true,"duration_ms":15004,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L05","11L40","11M41","11F30"],"pacs":[],"model":"grok-4.5","headline":"Bilinear Kloosterman forms save c to the minus 1/32 in the square-root range for every modulus, via quadratic characters.","keywords":["Kloosterman sums","bilinear forms","quadratic characters","twisted L-functions","exceptional Maass forms","large sieve","SL(2) representations"],"falsifier":"Take a prime power modulus p^k with k large and N equal to p^{k/2}; compute the bilinear form for random unit-coefficient sequences and check whether the observed size is smaller than the trivial bound by a factor p^{k/32} or better. If the ratio stays larger than p^{k/32+ε} for large k, the uniform claim fails.","tokens_in":37795,"feed_emoji":"∑","tokens_out":1119,"duration_ms":19645,"temperature":0.7,"pith_summary":"Kloosterman sums appear throughout analytic number theory. When they are paired against two arbitrary sequences of length about the square root of the modulus, the usual Weil bound and a crude Cauchy estimate give the same size, so any genuine cancellation is hard-won. This paper proves a uniform saving of c to the power minus 1/32 over that trivial size, valid for every positive integer modulus, not merely primes. The gain comes from rewriting a fourth-moment matrix of Kloosterman sums in terms of characters of SL(2) and then recognizing those characters, after a short amplification that peels off the square-full part of the modulus, as ordinary Jacobi symbols. The resulting complete character sums are estimated by Weil’s bound and elementary counting. Two concrete payoffs follow at once: a second-moment asymptotic for twisted L-functions with error q to the minus 1/90, and a large-sieve inequality for exceptional Maass forms that improves the classical Deshouillers–Iwaniec range without any factorization hypothesis on the level.","feed_headline":"Kloosterman bilinear forms save c^{-1/32} for every modulus","feed_subtitle":"A link to quadratic characters beats every prior bound, even for primes, and sharpens two classical applications","key_machinery":"The fourth-moment spectral norm of the Kloosterman matrix is rewritten, via the special representations of SL(2,Z/cZ), as a sum of non-abelian characters; after amplification that isolates the square-free part of c those characters become Jacobi symbols of a quadratic polynomial in four variables, which are then bounded by Hölder, elementary counting, and Weil’s estimate for character sums.","core_discovery":"For any modulus c and any intervals of length at most N less than or equal to c, the bilinear form of Kloosterman sums against arbitrary complex coefficients is bounded by the product of the l2-norms times c to the 1+o(1) times (N^{1/8}c^{-3/32}+N^{5/16}c^{-3/16}+N^{2/3}c^{-7/18}). In the critical range N equal to the square root of c this is a saving of c to the minus 1/32 over the trivial bound, and the result holds for every c.","pith_inferences":["Higher even moments or different Hölder exponents are unlikely to improve the critical exponent 1/32, because the diagonal contribution in the character-sum estimate already saturates the available cancellation.","The same SL(2)-to-Jacobi dictionary should apply, with only minor changes, to bilinear forms involving other algebraic trace functions that arise from the same permutation representation.","Once the square-full obstruction is removed by a better local estimate, the method would give a pure power-saving large sieve for exceptional eigenvalues at every level."],"forward_implications":["The second moment of L(1/2,f1×χ)L(1/2,f2×χ) over primitive characters modulo q equals the expected main term plus an error O(q^{-1/90+ε}).","The exceptional-spectrum large sieve for Maass forms of level q gains an extra factor roughly q^{2 max θ_j /29} when the Fourier coefficients are supported near length √q, with no factorization hypothesis on q.","The same bilinear bound improves the range of non-trivial estimates for shifted convolution problems and for the greatest prime factor of n^{2}+1 that rely on Kloosterman sums of general modulus.","For square-free moduli the hybrid step can be omitted and the pure fourth-moment argument already yields the full c^{-1/32} saving."],"fun_headline_variants":["Kloosterman bilinear forms save c^{-1/32} for every modulus","Quadratic characters give c^{-1/32} Kloosterman bilinear saving","Bilinear Kloosterman bounds improve to c^{-1/32} for all c","New link to quadratic characters saves c^{-1/32} on Kloosterman forms","All-modulus Kloosterman bilinear forms save c^{-1/32} at critical range"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The clean reduction from non-abelian characters to Jacobi symbols works only after the square-full part of the modulus has been amplified away; the remaining square-full contribution is controlled by a coarser sixth-moment bound that must not lose more than claimed when that part is large.","fun_headline_variants_meta":{"raw":{"variants":["Kloosterman bilinear forms save c^{-1/32} for every modulus","Quadratic characters give c^{-1/32} Kloosterman bilinear saving","Bilinear Kloosterman bounds improve to c^{-1/32} for all c","New link to quadratic characters saves c^{-1/32} on Kloosterman forms","All-modulus Kloosterman bilinear forms save c^{-1/32} at critical range"]},"model":"grok-4.5","effort":"low","cost_usd":0.005139,"raw_usage":{"total_tokens":1377,"prompt_tokens":672,"num_sources_used":0,"completion_tokens":94,"cost_in_usd_ticks":51388000,"prompt_tokens_details":{"text_tokens":672,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":611,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":672,"tokens_out":94,"duration_ms":10877,"temperature":1.0,"reasoning_tokens":611,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T18:29:01.380573+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Take a prime power modulus p^k with k large and N equal to p^{k/2}; compute the bilinear form for random unit-coefficient sequences and check whether the observed size is smaller than the trivial bound by a factor p^{k/32} or better. If the ratio stays larger than p^{k/32+ε} for large k, the uniform claim fails.","supporting_citations":[],"review_version":1}