{"id":"4f52d106-ff49-47c6-a3e5-86812f071268","arxiv_id":"2511.16257","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For real analytic homogeneous functions that are Newton R-nondegenerate and convenient, the paper claims a strict inequality (even n) or equality (odd n) between the absolute oscillation index and the real log canonical threshold, while admitting an unresolved conflict with the literature.","lead":"This math paper studies when a value called the oscillation index of an oscillatory integral is strictly larger in absolute value than the real log canonical threshold of a real analytic function. It gives conditions in the homogeneous Newton-nondegenerate case where the two coincide or fail to coincide, but the authors conclude that their result conflicts with known formulas and that an error exists somewhere.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 is falsified by f(x,y)=x^3+y^3: direct scaling gives beta=-2/3, so the claimed strict inequality beta<-gamma fails.","rationale":"The reader's REJECT verdict is correct and is strengthened by a concrete counterexample. The central claim of Theorem 3 is not merely unsupported; it is false. The homogeneous cubic f=x^3+y^3 meets every stated hypothesis, and a two-line scaling argument gives a nonzero tau^{-2/3} coefficient, forcing beta=-2/3. This independently settles the 'incompatibility with the standard formula' noted by the authors: the standard formula beta=-gamma is right for this example, and the error lies in the paper's parity/cancellation argument. The cancellation claim (3.5) uses pi^*dx as a top form on the blowup, but for n even the real blowup of a point is non-orientable, so the density used for the oscillatory integral is |pi^*dx|, not pi^*dx; chart-wise oddness does not imply global vanishing. Thus the strict inequality fails. The reader's weakest assumption concerned the possibility that the proof is sound and the literature formula wrong; our counterexample shows instead that the proof is unsound, so I mark agreement as partial. The verdict remains REJECT/UNCHANGED.","tokens_in":5775,"tokens_out":25109,"duration_ms":205272,"concrete_test":"Compute the leading asymptotic coefficient for f=x^3+y^3 with phi in C_c^infty(R^2), phi(0)!=0, using the substitution u=tau^{1/3}x, v=tau^{1/3}y: I(tau,phi)=tau^{-2/3} int e^{i(u^3+v^3)} phi(tau^{-1/3}u,tau^{-1/3}v) du dv -> tau^{-2/3} phi(0) (int_{-infty}^{infty} e^{it^3}dt)^2. Evaluate the nonzero Airy constant (int_{-infty}^{infty} e^{it^3}dt)^2 = (2 int_0^infty cos(t^3)dt)^2; this gives beta=-2/3 and refutes Theorem 3. This is a single analytical check, confirmable numerically by evaluating the Airy constant.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3 is not merely unproved; it is false. Take n=2 and f(x,y)=x^3+y^3. This is homogeneous of degree d=3>n, convenient, and Newton R-nondegenerate: for the only positive-dimensional compact face, x^3+y^3, the critical set is {(0,0)} subset of {xy=0}; vertex faces give {x=0} or {y=0}, also contained in {xy=0}. For any compactly supported smooth phi with phi(0)!=0, the scaling u=tau^{1/3}x, v=tau^{1/3}y gives I(tau,phi)=tau^{-2/3} integral e^{i(u^3+v^3)} phi(tau^{-1/3}u,tau^{-1/3}v) du dv -> tau^{-2/3} phi(0) C, where C=(int_{-infty}^{infty} e^{it^3}dt)^2=(2 int_0^infty cos(t^3)dt)^2 != 0. Hence beta(f) >= -2/3. Since gamma(f)=n/d=2/3, Theorem 2 gives beta(f) <= -2/3, so beta(f)=-2/3, contradicting the strict inequality beta(f)<-n/d. The proof error is localized to the parity/cancellation step in Section 3: for n even the real blowup of the origin is non-orientable, so the signed form pi^*dx=y_i^{n-1}dy_i dby is not a global density; the claimed oddness cancellation is invalid. The d-odd involution argument can symmetrize only the imaginary part, while the leading real term survives.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the relation between the oscillation index β(f) of an oscillatory integral and the real log canonical threshold γ(f)=rlct(f) for a real analytic phase f. It recalls Theorem 1 (γ(f)=rlct(f)) and Theorem 2 (β(f)≤−γ(f)), then states Theorem 3: if f is Newton R-nondegenerate, convenient, homogeneous, n is even, and either deg f>n or f^{-1}(0)={0} with d even, then γ(f)=n/d and β(f)<−n/d. The proof in Section 3 uses blowups, a partition of unity, and a parity/cancellation argument in the coordinates of (3.1). The abstract and introduction explicitly acknowledge that the strict inequality seems incompatible with standard formulas and that 'there must be some error somewhere', without locating it.","tokens_in":6221,"tokens_out":12837,"duration_ms":135920,"significance":"If Theorem 3 were correct, it would identify a new class of phases where Varchenko's bound is strict, contradicting standard Newton-polyhedron formulas; that would be a substantial result. The paper also contains useful review material, and Corollaries 2.1–2.2 on independence of cutoff and of the Taylor expansion of the amplitude are interesting. However, the central theorem is false: f=x^3+y^3 is a direct counterexample satisfying all hypotheses of Theorem 3. The parity proof rests on a signed-form/density error, and the paper's own admission of an unlocated contradiction, together with the conceded non-cancellation in Remark 3.2, means the main claim is not established and cannot be repaired by a small local correction.","major_comments":[{"comment":"Theorem 3 is false as stated. Let n=2 and f(x,y)=x^3+y^3. This is homogeneous of degree d=3>2, convenient, and Newton R-nondegenerate: for the face x^3+y^3, ∂f=0 gives (0,0)∈{xy=0}; for the vertex faces the critical sets are {x=0} or {y=0}, also contained in {xy=0}. For any φ∈C_c^∞ with φ(0)≠0, the scaling u=τ^{1/3}x, v=τ^{1/3}y gives I(τ,φ)=τ^{-2/3}∫ e^{i(u^3+v^3)} φ(τ^{-1/3}u,τ^{-1/3}v) dudv → τ^{-2/3} φ(0) C, where C=(∫_{-∞}^{∞} e^{it^3}dt)^2=Γ(1/3)^2/3≠0. Hence β(f)≥−2/3. Since (3.2) gives γ(f)=n/d=2/3 and Theorem 2 gives β(f)≤−2/3, we obtain β(f)=−2/3, contradicting the strict inequality in (6).","section":"Theorem 3 / §3"},{"comment":"The parity/cancellation step is invalid because π^*dx is a signed n-form, not a density. The oscillatory integral (1) is an integral of the density |π^*dx|. In the chart (3.1), for n even the factor y_i^{n-1} changes sign under y_i↦−y_i, and π is orientation-reversing on {y_i<0}. Equation (3.5) integrates the signed form y_i^{n-1}dy_i∧..., so oddness in y_i can make the integral zero, but the original integral uses |y_i|^{n-1}. The correct integrand is even in y_i, and no cancellation follows. This is precisely the mechanism in the counterexample f=x^3+y^3.","section":"§3, Eq. (3.5)"},{"comment":"For odd d the argument is also not valid. The text says to apply the previous argument to the real part and then to show vanishing of the leading imaginary terms using the involution ι with ι^*x_i=−x_i. For n even, ι^*dx=dx; after symmetrizing χ′, I(τ,χ′)=∫ cos(τf)χ′ dx. This real part is not zero. In the counterexample f=x^3+y^3, the real part of the leading term is φ(0)C with C>0. The sentence in the same paragraph that a symmetry of f does not imply vanishing does not repair the gap, because the surviving term is a cosine integral over the density, not an odd integral.","section":"§3, d odd case"}],"minor_comments":[{"comment":"The text states that 'there must be some error somewhere, although it does not seem easy to find it inside this paper.' This is an explicit acknowledgment that the contradiction with [Va76]/[CKN13] is not resolved. A refereed theorem should either identify the error in the cited literature or be presented conditionally.","section":"Abstract/Introduction"},{"comment":"Remark 3.1 suggests that under the hypotheses of Theorem 3 one could get β(f)=−(n+1)/d from [CKN13, Section 7.1]. This is inconsistent with Theorem 3 and false for f=x^3+y^3, where β(f)=−2/3. The paper should reconcile these assertions.","section":"Remark 3.1"},{"comment":"Remark 3.2 concedes that without a nonnegativity assumption 'it does not seem easy to show the non-cancellation among the integrals associated with different faces'. Since Theorem 3 does not assume nonnegativity, this is a gap in the odd-d part of the proof that is acknowledged in the text.","section":"Remark 3.2"}],"recommendation":"reject","confidential_remarks":"The counterexample f=x^3+y^3 is decisive: it satisfies all hypotheses of Theorem 3 and gives β(f)=−γ(f), so the main theorem is false. The proof error is localized to the use of the signed pullback form instead of the density in the blowup coordinates. The paper might be salvageable as an expository note on this density pitfall, but not as a proof of the stated theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem of this paper is not just unproved; it is false. Take n=2 and f(x,y)=x^3+y^3. This f is homogeneous of degree 3>2, convenient, and Newton R-nondegenerate. The scaling x=τ^{-1/3}u, y=τ^{-1/3}v gives I(τ,φ) ~ τ^{-2/3} φ(0) ∫ e^{i(u^3+v^3)} du dv, and that integral is a nonzero real constant. Hence β(f) = -2/3 = -γ(f), contradicting the strict inequality in Theorem 3.\n\nThe error is in Section 3. The authors pull back dx as an oriented form y_i^{n-1} dy_i dby and argue that for even n the integrand is odd in y_i, so the integral vanishes. But oscillatory integrals integrate densities, not forms. The correct pullback is |y_i|^{n-1}. The real blowup of the origin is non-orientable for even n, so the signed form has no global meaning. For x^3+y^3 this is exactly what goes wrong: the parity cancellation is an artifact of forgetting the absolute value.\n\nTo give credit where it is due, the paper is unusually honest. The abstract and introduction openly state that the result is incompatible with the standard formula and that an error must exist somewhere. The discussion of real log canonical thresholds and Remark 3.3 on the difference between Newton nondegeneracy over R and C are useful. Remark 3.1 also signals the inconsistency by noting that the claimed inequality would follow from the standard [CKN13] formula. But acknowledging a problem is not the same as resolving it; the contradiction is left hanging.\n\nAs it stands, the paper should be rejected. The central claim is false, and the proof's flaw is load-bearing. That said, a serious referee is worth sending: the claim is important, the error turns on a subtle point that many readers could miss, and the counterexample makes for a clean pedagogical illustration.","headline":"Theorem 3 is false — f(x,y)=x^3+y^3 gives β=-2/3, so the claimed strict inequality β<-γ fails; the proof drops the absolute value in the Jacobian.","tokens_in":6667,"tokens_out":14210,"would_cite":false,"duration_ms":126914,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14B05","32S25","42B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a class of even-dimensional real analytic phases, the oscillation index is strictly below the negative of the real log canonical threshold, contradicting a standard formula in the literature.","keywords":["oscillation index","oscillatory integral","real log canonical threshold","Newton nondegenerate","real analytic function","asymptotic expansion","homogeneous polynomial","blowup"],"falsifier":"Compute the asymptotic expansion of ∫ e^{iτ(x^2+y^2)} φ dx dy for a bump φ supported near 0 with φ(0) ≠ 0. Stationary phase gives a nonzero coefficient of τ^{-1}, so β = −1 = −γ, directly violating the strict inequality β < −1 that Theorem 3 predicts for this phase.","tokens_in":5693,"feed_emoji":"📉","tokens_out":11170,"duration_ms":100834,"temperature":0.7,"pith_summary":"This paper studies when the oscillation index β(f) of an oscillatory integral equals the negative of the real log canonical threshold, −γ(f). For Newton R-nondegenerate convenient homogeneous functions in an even number of variables, the authors prove that equality fails — β(f) is strictly smaller than −n/d — when the degree exceeds the dimension, or when the zero set is just the origin with even degree. This conflicts with a widely used formula from the literature, and the authors explicitly acknowledge that the discrepancy means some error exists somewhere, though they do not identify whether it lies in their proof or in the earlier formula. If the paper is right, the oscillation index carries finer information than the real log canonical threshold and depends on parity in a way previously missed.","feed_headline":"Even dimensions break the oscillation-index equality","feed_subtitle":"For homogeneous Newton-R-nondegenerate phases with even n, β(f) < −γ(f), in conflict with older formulas.","key_machinery":"The key machinery is a blowup of the origin combined with the Newton polytope of f. The oscillation index β(f) is the leading exponent in the asymptotic expansion of ∫ e^{iτf}φ dx, while the real log canonical threshold γ(f) is the minimal real jumping number, computable from a resolution of singularities; for homogeneous degree d it equals n/d. Newton R-nondegenerate means the principal part on each face of the Newton polytope has no nonzero real critical points away from coordinate hyperplanes, and 'convenient' means the polytope meets every coordinate axis. The decisive calculation is the inner integral over the exceptional coordinate y: for even n and d, the integrand y^{n-1} e^{iτy^d h(","core_discovery":"The central claim is Theorem 3: assume f is Newton R-nondegenerate, convenient, and homogeneous of degree d, with n even and either d > n or f^{-1}(0) = {0} (and d even). Then the real log canonical threshold is γ(f) = n/d, but the oscillation index satisfies the strict inequality β(f) < −n/d. The proof uses the blowup of the origin, where the integral decomposes over charts; when d and n are even, the integral along the exceptional coordinate is an integral of an odd function and vanishes, and a symmetry argument with x ↦ −x extends the conclusion to odd d in the zero-set-origin case. The paper also notes that in the odd-dimensional case with f^{-1}(0) = {0}, equality holds, so the even/odd","pith_inferences":["If the strict inequality is genuine, the asymptotic expansion in these cases has no term at the threshold exponent, a property that could be checked numerically for low-degree phases and would immediately distinguish the two competing formulas.","The quadratic phase x^2 + y^2 on R^2 (n=2, d=2, zero set the origin) is the sharpest test case: standard stationary phase gives β = −1 = −γ, which appears to contradict Theorem 3; re-evaluating this one example with the paper's blowup method could reveal whether the error lies in the older formula or in the paper's own argument.","The parity phenomenon connects to a general principle: odd/even symmetry of the phase under negation can annihilate leading-order contributions to oscillatory integrals, a mechanism that may apply to other symmetric phase families.","The authors' inability to locate the error suggests the contradiction is subtle; a systematic comparison between the blowup computation and Newton-polytope estimates on a single explicit example would settle which side is mistaken."],"forward_implications":["If Theorem 3 holds, the decay of oscillatory integrals in these homogeneous cases is faster than the exponent suggested by the real log canonical threshold alone.","The strict inequality means any formula asserting β(f) = −γ(f) for all Newton R-nondegenerate convenient functions must be restricted by dimension parity and by the structure of the zero set.","The paper's Remark 3.1 indicates that in the covered cases one can in fact derive β(f) = −(n+1)/d from the same Newton-polyhedron techniques, providing a concrete corrected value.","The odd-dimensional singleton-zero-set case preserves equality, so the phenomenon is a parity effect, not a general failure of the bound.","The sketched extension to nonnegative non-homogeneous functions suggests that parity of certain exponents on Newton faces controls whether equality holds there as well."],"fun_headline_variants":["Even dimensions split oscillation-index from threshold","In even dimensions, oscillation index falls below threshold","Odd equality, even strict gap: new oscillation-index result","Homogeneous real analytic case: even n gives strict inequality","Oscillation index vs. threshold: even dimensions show gap"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The authors' proof of Theorem 3 is sound, and the contradiction with the earlier formula is due to an error in that formula rather than in this paper; if the standard formula is correct, Theorem 3 is false.","fun_headline_variants_meta":{"raw":{"variants":["Even dimensions split oscillation-index from threshold","In even dimensions, oscillation index falls below threshold","Odd equality, even strict gap: new oscillation-index result","Homogeneous real analytic case: even n gives strict inequality","Oscillation index vs. threshold: even dimensions show gap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"total_tokens":1145,"prompt_tokens":742,"completion_tokens":403,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":341}},"tokens_in":486,"tokens_out":403,"duration_ms":5161,"temperature":1.0,"reasoning_tokens":341,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T21:09:36.956056+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the asymptotic expansion of ∫ e^{iτ(x^2+y^2)} φ dx dy for a bump φ supported near 0 with φ(0) ≠ 0. Stationary phase gives a nonzero coefficient of τ^{-1}, so β = −1 = −γ, directly violating the strict inequality β < −1 that Theorem 3 predicts for this phase.","supporting_citations":[],"review_version":1}