{"id":"b0fb7d22-c1b3-4443-94af-452f33d9de67","arxiv_id":"2511.01270","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For local fields of positive characteristic, every nonzero analytic function has strictly positive integrability threshold, and for regular functions on smooth varieties the threshold is at least 1/(d·D^m).","lead":"Over local fields of positive characteristic, the authors prove that every nonzero analytic function has a strictly positive log-canonical threshold—a measure of how large an exponent can be used before the reciprocal of the function stops being integrable. They also give an explicit, optimal lower bound for polynomial functions on smooth algebraic varieties, depending only on degrees and the number of defining equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.11's shear+rescaling fails to make f distinguished; explicit polynomials with equal leading and higher pure-x_n coefficients defeat the proof, leaving Theorem 1.4 unproven.","rationale":"The reader identified Lemma 2.11 as the weakest point, specifically the failure to establish strict dominance of the x_n^K coefficient. My analysis confirms and sharpens this: the gap is not merely an issue of 'maximal vs. strictly greater' in a vague sense; there are explicit functions, e.g. f=x^{K}-y^{K}+t y^{K+1} over F_q((t)) with K=q-1, for which every allowed shear T_c gives an x_n^K coefficient of norm <1 and a higher pure x_n^{K+1} coefficient of the same norm, so the prescribed rescaling cannot produce a distinguished function of order K. This is a genuine gap in the proof of Lemma 2.11, which is the bridge from the order bound to the lct bound in Theorem 1.4. Theorem 1.2 (positivity) is independent of this lemma and is supported by the small-ball estimates. The counterexample does not disprove the statement of Lemma 2.11 or Theorem 1.4; a corrected argument (e.g. using a general linear change or the full Weierstrass preparation change from Proposition 2.2) is likely to repair the proof. Therefore the reader's CONDITIONAL verdict is appropriate, and no change to that verdict is needed.","tokens_in":9504,"tokens_out":44117,"duration_ms":409725,"concrete_test":"Implement the proof of Lemma 2.11 on F=F_q((t)) for f=x^{q-1}-y^{q-1}+t y^q, with the normalization exactly as in the paper. For every c in O_F^× (or at least c=1 and c=1+t), compute P(c)=c^{q-1}-1 and, after applying the shear T_c and the lemma's rescaling with k0=val(P(c)), verify whether the resulting function is distinguished in y of order q−1 according to Definition 2.1(4). It will fail for all c. Then compute lct_F(f;0) by the linear change u=x−y, v=x+y (or u=x, v=x−y in characteristic 2) and one-dimensional integrals; if lct ≥ 1/(q−1), the lemma's statement survives and a corrected proof should be required before accepting Theorem 1.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.11 is the only step that converts the order bound ord_{X,x0}(φ) ≤ dD^m into lct ≥ 1/(dD^m). Its proof chooses a shear T=(x_i+c_i x_n) and then rescales the first n−1 coordinates by ϖ^{k0+1}, leaving x_n unscaled. It claims the result is distinguished of order K, but Definition 2.1(4)(b) requires ||a_K|| > ||a_s|| for all s>K. The proof only ensures the x_n^K coefficient is nonzero and of value q^{-k0}; it does not ensure strict dominance, and in fact it can fail. Let F=F_q((t)), K=q-1, and f=x^K-y^K+t y^{K+1} in O_F^2. After the proof's normalization, for every unit c, the coefficient of y^K in f(x+cy,y) is c^K-1, which belongs to t O_F (so has norm ≤ q^{-1}), while the coefficient of y^{K+1} is t, also of norm q^{-1}. With k0=val(c^K-1)≥1, the proof's rescaling produces y^K and y^{K+1} coefficients of norms 1 and q^{k0-1} (for k0=1, both 1), so strict inequality fails for every c. Thus the construction does not make f distinguished of order K. The statement of Lemma 2.11 may still be true (e.g. via a general linear change or Proposition 2.2), but the proof as written is invalid. Theorem 1.4's effective lower bound therefore rests on an unproved step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the analytic log-canonical threshold lct_F(f;x0) over local fields F of positive characteristic. Theorem 1.2 asserts positivity for any nonzero analytic ideal on an analytic manifold, and Theorems 1.3 and 1.4 assert uniform and effective lower bounds for regular functions on smooth algebraic varieties, with the explicit bound 1/(d·D^m) in the affine setting of Theorem 1.4. The strategy is to reduce to a Weierstrass polynomial and use a Remez-type small-ball estimate (Lemmas 2.6 and 2.7). In the algebraic case, Lemma 2.10 bounds the order of vanishing via Hilbert–Samuel multiplicity and Bézout, and Lemma 2.11 is then used to convert an order bound K into the lower bound lct ≥ 1/K.","tokens_in":9875,"tokens_out":27054,"duration_ms":255959,"significance":"If the proof is completed, the paper supplies useful foundational results in positive characteristic, where Hironaka resolution and model-theoretic transfer are unavailable. The small-ball estimates in Lemmas 2.6 and 2.7 are elementary and correct, and Example 1.5 shows that the effective bound in Theorem 1.4 is optimal. However, the proof of Lemma 2.11, which is load-bearing for the effective theorem, has a serious gap. The overall approach is promising and likely salvageable, but the current version does not establish Theorem 1.4.","major_comments":[{"comment":"The step claiming that the rescaled function \\tilde g is distinguished of order K is unjustified. Definition 2.1(4)(b) requires ||a_K|| > ||a_s|| for all s>K, but the proof only makes a_K a unit (norm 1). The final rescaling leaves x_n unscaled: pure x_n^s coefficients for s>K are multiplied only by the common factor \\varpi^{-k0}. They were known only to have norm <1 before this rescaling, so when k0≥1 they can have norm ≥1 afterwards. Concretely, take F=F_q((t)), K=q-1, and g=x^K-y^K+t y^{K+1}. For every unit c, after T(x,y)=(x+cy,y) the y^K coefficient is c^K-1 ∈ tO_F, so k0≥1, while the y^{K+1} coefficient is t. After the proof's rescaling with k0=1, both coefficients have norm 1, so strict dominance fails. Thus Lemma 2.11 is not proved as written.","section":"§2.1, Lemma 2.11"},{"comment":"Lemma 2.11 is the only step converting the order bound ord_{X,x0}(φ) ≤ d·D^m from Lemma 2.10 into the lower bound lct_F ≥ 1/(d·D^m). Since the proof of Lemma 2.11 is invalid, Theorem 1.4 is unsupported. The statement of Lemma 2.11 may still be true, but the manuscript does not provide a correct argument; this is a load-bearing gap, not a cosmetic issue.","section":"§2.1, Theorem 1.4"}],"minor_comments":[{"comment":"'In this Appendix we study...' appears to be a leftover from the appendix format; it should read 'In this paper we study...'.","section":"§1, first paragraph"},{"comment":"The notation q_F^{k0} appears to mean q_F^{-k0}; as written, the valuations are inverted (e.g., '|Σ c^I b_I|_F = q_F^{k0}' and 'maximal absolute value q_F^{k0}').","section":"Lemma 2.11, proof"},{"comment":"The phrase 'do not simultaneously vanish on some irreducible component of X' is ambiguous; it should be 'do not simultaneously vanish on any irreducible component of X' to match the intended condition.","section":"Theorems 1.3 and 1.4"}],"recommendation":"major_revision","confidential_remarks":"The core results are attractive and the small-ball part is solid. The sole obstruction is Lemma 2.11; if the authors can supply a correct proof (or replace the lemma with a different argument), the paper is likely acceptable. I am not recommending rejection because the gap appears repairable, but the current version requires substantive revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the positivity theorem is the real content and it is convincing. The effective bounds are plausible but the proof of Lemma 2.11, which is the bridge from order to lct, does not actually establish that the function is distinguished. The paper deserves review, but the author will need to repair that lemma.\n\nWhat's new: over a fixed local field of positive characteristic there is no Hironaka resolution and no D-module machinery, so even positivity of the analytic log-canonical threshold was apparently not known. The paper proves it via a Remez-type small-ball estimate for Weierstrass polynomials. Lemmas 2.6 and 2.7 are clean and correct; the proof of Theorem 1.2 is short and self-contained. That alone is a nice contribution.\n\nThe effective lower bound in Theorem 1.4, lct ≥ 1/(d·D^m), is a useful quantitative statement, and Example 1.5 shows it is optimal. But the proof depends on Lemma 2.11, and there the argument goes wrong. The shear T(x_i + c_i x_n) followed by rescaling the first n−1 coordinates does not make the function distinguished of order K in x_n. Distinguishedness requires the coefficient of x_n^K — a power series in the other variables — to have Gauss norm strictly larger than all coefficients of higher x_n powers. The proof only shows that a scalar (the constant term of that coefficient) is nonzero, and then claims maximality. That is not strict inequality, and the rescaling leaves x_n alone, so higher pure x_n powers are not controlled. The stress-test example f = x^K − y^K + t y^{K+1} over F_q((t)) with K = q−1 is a legitimate counterexample to the construction: for every unit c, after the shear the y^K and y^{K+1} coefficients both end up with norm 1 when the valuation of c^K−1 is 1. So the proof as written does not work.\n\nThat said, the lemma is likely true. The paper itself quotes Proposition 2.2 (BGR), which gives an analytic diffeomorphism of the form (x_i + x_n^{d_i}) making any nonzero Tate algebra element distinguished of the correct order. Applying that instead of the shear would fix the proof. But the current text doesn't do it, so the effective bound rests on an unproved step. Lemma 2.10's Bezout inequality also deserves a precise reference, though I don't doubt the bound.\n\nWho this is for: anyone working on integrability of Fourier transforms, jet schemes, or singularities over F_ℓ. Theorem 1.2 is the headline. The paper is worth refereeing; a serious referee would ask for the Lemma 2.11 repair (or a citation to Proposition 2.2), and then the effective results would be in good shape. I'd send it to peer review.","headline":"Theorem 1.2 is solid and new, but the effective bound in Theorem 1.4 rests on a gap in Lemma 2.11 that needs a real fix before the paper is complete.","tokens_in":10369,"tokens_out":9587,"would_cite":true,"duration_ms":86891,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11S80","14B05","14G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the analytic log-canonical threshold is always positive over local fields of positive characteristic, and gives an optimal effective lower bound of 1/(d·D^m) for regular functions on smooth algebraic varieties.","keywords":["log-canonical threshold","positive characteristic","local fields","small ball estimates","Weierstrass preparation","order of vanishing","Hilbert–Samuel multiplicity","sublevel sets"],"falsifier":"Check the proof's coordinate-change step on f(x)=x^K(1+x) over any positive-characteristic local field: the coefficients of x^K and x^{K+1} both have absolute value 1, so the function is never distinguished of order K after any allowed shear and rescaling (trivial in one variable). This one-line example settles whether Lemma 2.11's route from order K to lct ≥ 1/K is valid; if the construction fails, the effective lower bound remains unproven, and finding an actual function with lct < 1/K would refute Theorem 1.4.","tokens_in":9360,"feed_emoji":"🧮","tokens_out":13491,"duration_ms":108258,"temperature":0.7,"pith_summary":"The paper proves that the analytic log-canonical threshold — the largest power to which one can raise the reciprocal absolute value of a function and still integrate it locally — is always positive for analytic functions on manifolds over local fields of positive characteristic. In the algebraic setting, where the function and the variety are given by polynomials, the paper derives an explicit, uniform lower bound that depends only on the degrees involved: at least 1/(d·D^m), where d is the degree of the functions, D the degree of the equations cutting out the variety, and m the codimension. The bound is optimal, as a curve defined by iterated D-th powers with a single d-th power function attains equality. These results matter because in positive characteristic resolution of singularities and the usual tools of singularity theory are unavailable, so previously no such uniform integrability statement was known. The proof works through elementary sublevel-measure estimates for monic polynomials over non-Archimedean fields, bypassing the missing resolution machinery.","feed_headline":"Log-canonical threshold never vanishes in positive characteristic","feed_subtitle":"For algebraic functions the integrability exponent is at least 1/(d·D^m), and the bound is sharp.","key_machinery":"The argument rests on two ingredients. First, a small-ball (sublevel) estimate: for a monic polynomial of degree d over a non-Archimedean local field, the Haar measure of the set where |f(x)| ≤ δ is at most d·δ^{1/d}; the same bound holds for a Weierstrass polynomial in several variables once the Weierstrass preparation theorem is used to reduce a general analytic function to a monic polynomial in the last variable. Second, in the algebraic setting, the order of vanishing of a degree-d polynomial along a smooth variety defined by degree-D equations is controlled by the Hilbert–Samuel multiplicity, bounded via Bézout by d·D^m; a conversion lemma (Lemma 2.11) then turns this order bound into t","core_discovery":"At the heart of the paper is Theorem 1.4: if X is a smooth n-dimensional subvariety of affine space defined by m polynomials of degree at most D, and if φ_1, ..., φ_r are polynomials of degree at most d that do not all vanish on any irreducible component of X, then for every F-point x0 of X the analytic log-canonical threshold lct_F(⟨φ_1,...,φ_r⟩; x0) is at least 1/(d·D^m). The constant is sharp: Example 1.5 shows that on the curve x_{i+1} = x_i^D, the function φ = x_{m+1}^d has threshold exactly 1/(d·D^m). Along the way the paper establishes the more basic positivity statement (Theorem 1.2) for arbitrary analytic functions on analytic manifolds, with no algebraic or degree hypotheses.","pith_inferences":["The proof's key step, Lemma 2.11, assumes that a generic shear and rescaling makes an order-K analytic function distinguished in the last variable; for a function like x^K(1+x) this step visibly fails, so the effective bound may rest on a repairable but currently unproven claim—or the bound may hold by a different route.","The small-ball estimation technique may be flexible enough to yield sublevel bounds for functions that are not Weierstrass polynomials, such as sums of monomials or products, potentially giving analogous thresholds without the algebraic assumptions.","The optimality example suggests a general heuristic: the smallest threshold on a variety tends to be realized along a branch where the function has maximal order, governed by the product of degrees; this could guide conjectures for the related F-pure threshold, which the paper notes is a different invariant.","The uniform-in-point nature of the algebraic bound may help in establishing global integrability theorems for distributions on reductive groups over positive-characteristic local fields, where previously no such uniform arithmetical control existed."],"forward_implications":["For any smooth algebraic F-variety and any regular function φ that does not vanish identically on a component, |φ|_F^{-s} is locally integrable for every s < 1/(d·D^m), uniformly at all F-points.","The lower bound is optimal in general: the curve x_{i+1}=x_i^D with φ=x_{m+1}^d realizes lct = 1/(d·D^m).","The positivity theorem covers arbitrary analytic functions on arbitrary F-analytic manifolds, with no bound depending on degrees.","For an ideal J generated by several functions, lct_F(J; x0) is bounded below by 1/(d·D^m) whenever the generating polynomials satisfy the stated degree and vanishing conditions.","These bounds supply a positive exponent of integrability for pushforwards of smooth compactly supported measures by analytic maps in positive characteristic, generalizing characteristic-zero results."],"fun_headline_variants":["Log-canonical threshold never zero in positive characteristic","Explicit lower bound for log-canonical threshold over char p fields","Sharp threshold bound for algebraic functions in positive characteristic","Integrability exponent always positive over local fields of char p"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole argument leans on the claim that an analytic function of order K can, after a shear and a rescaling of some coordinates, be made distinguished—meaning its K-th pure power term has strictly the largest coefficient—so that Weierstrass preparation yields a monic degree-K polynomial; for some functions this step is not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Log-canonical threshold never zero in positive characteristic","Explicit lower bound for log-canonical threshold over char p fields","Sharp threshold bound for algebraic functions in positive characteristic","Integrability exponent always positive over local fields of char p"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000361,"raw_usage":{"total_tokens":1779,"prompt_tokens":727,"completion_tokens":1052,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":985}},"tokens_in":471,"tokens_out":1052,"duration_ms":9617,"temperature":1.0,"reasoning_tokens":985,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T00:28:06.531929+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the proof's coordinate-change step on f(x)=x^K(1+x) over any positive-characteristic local field: the coefficients of x^K and x^{K+1} both have absolute value 1, so the function is never distinguished of order K after any allowed shear and rescaling (trivial in one variable). This one-line example settles whether Lemma 2.11's route from order K to lct ≥ 1/K is valid; if the construction fails, the effective lower bound remains unproven, and finding an actual function with lct < 1/K would refute Theorem 1.4.","supporting_citations":[],"review_version":1}