{"id":"2e9ab996-8813-4fa9-bd17-d8b917a741d9","arxiv_id":"2607.06874","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A centered Gaussian with smaller covariance assigns at least as much probability as one with larger covariance to any closed convex set with reference probability at least 1/2.","lead":"This paper proves that for two centered Gaussian vectors where one has a larger covariance matrix, the smaller-covariance vector assigns at least as much probability to any closed convex set that already has probability at least 1/2 under the larger-covariance vector. This extends Anderson's Theorem to one-sided and asymmetric tests, enabling conservative statistical inference when covariance estimates are deliberately inflated.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified","rationale":"The reader correctly identified the Ehrhard-Borell concavity step as the structural linchpin and the boundary approximation as a secondary concern. On careful examination, both are handled correctly. The Ehrhard-Borel inequality applies to convex sets under Gaussian measure, the chain rule for the Hessian is verified explicitly, and the sufficient condition (G_t(v) ≥ 1/2 ⟹ H_t(v) ≥ 0 ⟹ ∇²_v G_t(v) ⪯ 0) is logically sound. The boundary case is handled by a standard enlargement-and-limits argument. The singular cases reduce correctly to the full-rank case. The anisotropic heat equation identity and the dominated convergence arguments are verified with appropriate uniform bounds. The counterexample in Appendix C correctly demonstrates that the Loewner ordering cannot be replaced by marginal variance dominance plus Sudakov-Fernique conditions. The statistical application (Proposition 1) follows from standard weak convergence arguments with appropriate handling of atoms in chi-bar-square distributions. No adjustment to the ACCEPT verdict is warranted.","tokens_in":12964,"tokens_out":815,"duration_ms":226278,"concrete_test":"Independently verify the key chain-rule Hessian computation in Appendix A.4: starting from G_t(v) = Φ(H_t(v)) with H_t(v) = Φ^{-1}(G_t(v)), confirm that ∇²_v G_t(v) = φ(H_t(v))[−H_t(v)∇_v H_t(v)(∇_v H_t(v))^T + ∇²_v H_t(v)], and that ∇²_v H_t(v) ⪯ 0 (from Ehrhard-Borell concavity) combined with H_t(v) ≥ 0 (from G_t(v) ≥ 1/2) implies ∇²_v G_t(v) ⪯ 0. This is the single computation on which the sign of ∂G_t/∂t depends.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof's structural linchpin is the application of the Ehrhard-Borell inequality to establish that H_t(v) = Φ^{-1}(G_t(v)) is concave in v, which yields ∇²_v G_t(v) ⪯ 0 whenever G_t(v) ≥ 1/2 (Appendix A.1). This step is sound: Ehrhard-Borell applies because Σ_t^{-1/2}(K+v) is convex when K is convex, and the Gaussian measure γ_d is log-concave. The chain rule computation of the Hessian is verified explicitly in A.4, and the sufficient condition H_t(v) ≥ 0 ⟺ G_t(v) ≥ 1/2 is correct. The interpolation argument (defining J₁ and showing it must equal [0,1] by a continuity/contradiction argument) is standard and correctly handles the strict inequality case. The boundary case pr(Y∈K)=1/2 (A.5) uses a clean enlargement argument: K_δ = K + B_δ is closed convex, receives strictly more than 1/2 probability under the full-rank Y, and continuity from above gives the result. The singular cases (A.6, A.7) reduce to the full-rank case by projection onto the column space of Σ_Y, which is valid since both X and Y live in colspace(Σ_Y) a.s. The anisotropic heat equation identity is verified by direct computation in A.2. The continuity and differentiation-under-the-integral justifications in A.3 use uniform bounds exploiting mI_d ⪯ Σ_t ⪯ MI_d, which is correct for t∈[0,1]. I do not identify a load-bearing concern that would undermine the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"The manuscript proves a Gaussian comparison inequality (Theorem 1) stating that for centered Gaussian vectors X, Y with covariance matrices ordered as Σ_Y − Σ_X ⪰ 0, the smaller-covariance law assigns at least as much probability as the larger-covariance law to every closed convex set K with pr(Y ∈ K) ≥ 1/2. This provides a one-sided analogue of Anderson's Theorem (Anderson, 1955), which requires mirror symmetry of the set. The proof proceeds through several stages: the positive definite case with strict inequality (Appendix A.1), the boundary case pr(Y ∈ K) = 1/2 via set enlargement (A.5), singular Σ_X via L² convergence (A.6), and singular Σ_Y via projection (A.7). The key technical ingredients are a Gaussian interpolation satisfying the anisotropic heat equation and the Ehrhard–Borell inequality, which establishes concavity of Φ⁻¹(G_t(v)) in v and hence negative semidefiniteness of the Hessian ∇²_v G_t(v) whenever G_t(v) ≥ 1/2. A corollary extends the result to upper tail probabilities of lower semicontinuous quasiconvex functions above the median, and Proposition 1 provides a statistical application justifying conservative one-sided and order-restricted inference using conservative covariance estimators at significance levels α < 1/2. A counterexample in Appendix C disproves a conjecture of Cohen and Fogarty (2022) under weaker (Sudakov–Fernique) covariance conditions.","tokens_in":13571,"tokens_out":1183,"duration_ms":183695,"significance":"The result fills a genuine gap between Anderson's Theorem (which requires mirror symmetry and gives full stochastic dominance) and one-sided testing needs in statistical practice. The restriction to α ≤ 1/2 is natural for hypothesis testing and does not limit practical applicability. The proof is self-contained, building on the Ehrhard–Borell inequality and Anderson's Theorem, both external to the author. The explicit verification of the anisotropic heat equation identity (A.2), the careful continuity and differentiation-under-the-integral justifications (A.3), and the chain rule computation (A.4) are commendable. The counterexample in Appendix C is a valuable contribution that sharpens understanding of why the Loewner ordering is essential. The statistical application (Proposition 1) is well-motivated and correctly handles nuisance parameter estimation and plug-in tail probability estimation, including the discontinuous case arising from chi-bar-square laws.","major_comments":[],"minor_comments":[{"comment":"In Appendix A.1, the string 'Σ_t = Σ^{1/2}_t Σ^{1/2}_t' appears to state that Σ_t equals its own symmetric square root product, which is trivially true but does not define the square root. The intended statement is presumably that Σ_t admits a symmetric invertible square root, i.e., Σ_t = Σ^{1/2}_t Σ^{1/2}_t where Σ^{1/2}_t is the unique positive definite square root. A brief clarification would avoid momentary confusion.","section":null},{"comment":"In Corollary 1, the notation m^-_Y = inf{t : pr{f(Y) ≤ t} ≥ 1/2} defines a lower median, while Proposition 1 later uses m^+_Y = sup{t : S_ξ(t; Σ_Y) ≥ 1/2}, an upper median. The relationship between these two definitions (they coincide when the distribution is continuous) could be stated explicitly to aid readers connecting the corollary to the proposition.","section":null},{"comment":"In Proposition 1, the condition lim_{h→0+} S_ξ(m^+_Y + h; Σ_Y) = 1/2 ensures no downward jump from 1/2. The subsequent remark about chi-bar-square laws is helpful, but a brief explicit statement of what 'projection onto {0}' means in this context (i.e., the degenerate case where the chi-bar-square distribution is a point mass at 0) would improve accessibility for readers less familiar with order-restricted inference.","section":null},{"comment":"In §3, the statement 'The upper tail probabilities cross at a = 0.08' is followed by 'The median of g(Y) is m^-_Y = 0.23 > 0.08.' The logic is that the crossing occurs below the median, so Corollary 1's guarantee applies above 0.23. This is correct but the phrasing could more directly state that the crossing being below the median is precisely what Corollary 1 requires.","section":null},{"comment":"The reference 'Harshaw et al. (2026)' lists the journal as 'Journal of the American Statistical Association, page to appear.' If the paper has appeared by the time of publication, the reference should be updated with volume and page numbers.","section":null},{"comment":"In Appendix A.3, the bounds C_V, C_{V,2}, C_{V,3}, C_{V,4} are introduced without explicit definitions. While their existence follows from the surrounding inequalities, stating them as explicit constants (or at least noting they depend only on V, m, M, and d) would improve clarity.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a clean, self-contained proof of a natural and useful result. The stress-test note correctly identifies the Ehrhard–Borell concavity step as the structural linchpin; on reading the manuscript, this step is sound and well-executed. No load-bearing concerns arise. The minor comments are purely presentational. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper proves a one-sided analogue of Anderson's Theorem for Gaussian measures, and in doing so resolves conjectures from Ehm and Krüger (2018) and Cohen and Fogarty (2022). Theorem 1 says that if X and Y are centered Gaussians with Σ_Y ⪰ Σ_X in Loewner order, then for any closed convex set K with pr(Y ∈ K) ≥ 1/2, we have pr(X ∈ K) ≥ pr(Y ∈ K). This is a genuine new result, not a re-derivation, and it removes the mirror symmetry requirement that limited Anderson's original theorem to two-sided inference. The Ehm-Krüger conjecture gets proved under weaker conditions than originally stated (quasiconvexity instead of convex monotonicity). The counterexample in Appendix C is a nice addition — it shows the Loewner ordering can't be replaced by marginal variance dominance plus Sudakov-Fernique conditions, which is exactly the weaker condition Cohen and Fogarty had conjectured would suffice. The proof structure is well-organized: positive definite case with strict inequality first, then boundary, then singular cases by projection. The key ingredients are the Ehrhard-Borell inequality (to get concavity of Φ^{-1}(G_t(v)) in v, which controls the Hessian sign) and an anisotropic heat equation interpolation. Both are verified explicitly in the appendix. The continuity and differentiation-under-the-integral arguments in A.3 use uniform eigenvalue bounds and look correct. The stress-test flagged the Ehrhard-Borell application as the structural linchpin, and I agree it's sound — Σ_t^{-1/2}(K+v) is convex when K is, and the chain rule in A.4 checks out. The statistical application (Proposition 1) is well-motivated and follows from standard weak convergence arguments. It's not where the mathematical interest lives, but it justifies why the theorem matters: conservative covariance estimation is common in randomized experiments and design-based inference, and this extends that framework to one-sided and order-restricted tests at conventional significance levels. Soft spots are minor. The restriction to α < 1/2 is inherent to the result and the author is upfront about it. The boundary case handling via enlargement (A.5) is clean but standard. I didn't find any load-bearing concerns. This is a solid paper that deserves a serious referee. The core theorem is clean, the proof is self-contained, and it resolves open problems in the literature.","headline":"Clean Gaussian comparison inequality resolving two conjectures; proof is solid and self-contained","tokens_in":13710,"tokens_out":585,"would_cite":true,"duration_ms":85523,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Smaller-variance Gaussians dominate on convex sets above the median","keywords":[],"falsifier":"A counterexample would be: a pair of centered Gaussians with Σ_Y − Σ_X ⪰ 0 and a closed convex set K with pr(Y ∈ K) ≥ 1/2 but pr(X ∈ K) < pr(Y ∈ K). The paper itself provides a counterexample showing that replacing the Loewner order with weaker conditions (larger marginal variances plus Sudakov-Fernique increment conditions) is insufficient — there, the tail probabilities cross above the median, violating the conclusion.","tokens_in":13192,"feed_emoji":"📐","tokens_out":880,"duration_ms":190588,"temperature":0.7,"pith_summary":"The paper proves that if X and Y are centered Gaussian vectors with Y having the larger covariance matrix in the Loewner order (Σ_Y − Σ_X positive semidefinite), then for any closed convex set K that captures at least half the probability of Y, the smaller-variance vector X assigns at least as much probability to K as Y does. This is a one-sided analogue of Anderson's Theorem: Anderson's classical result requires K to be symmetric about the origin and then gives stochastic dominance for all thresholds; this paper drops the symmetry requirement and recovers the inequality above the median threshold. The proof works by interpolating between the two covariance matrices along a heat-equation path, showing that the Gaussian probability of each translate of K is nonincreasing along that path whenever it is at least 1/2. The key structural ingredient is the Ehrhard-Borell inequality, which gives concavity of the inverse-normal-transformed probability as a function of the translation parameter, forcing the relevant Hessian to be negative semidefinite above the 1/2 threshold. As a statistical corollary, conservative covariance estimation (using an estimator that overshoots the true asymptotic variance) yields valid one-sided and order-restricted inference at significance levels below 1/2, extending a tool previously limited to symmetric two-sided testing.","feed_headline":"Smaller-variance Gaussians win on convex sets above the median","feed_subtitle":"A one-sided Anderson's Theorem: drop the symmetry requirement, keep the inequality whenever the reference set captures at least half the概率.","key_machinery":"The proof interpolates between Σ_X and Σ_Y via Z_t = X + t^{1/2}W with Cov(W) = Σ_Y − Σ_X, so that Z_t satisfies an anisotropic heat equation ∂G_t/∂t = (1/2) tr(Δ ∇²_v G_t). The Ehrhard-Borell inequality implies Φ^{-1}(G_t(v)) is concave in v, which forces ∇²_v G_t(v) ⪯ 0 whenever G_t(v) ≥ 1/2. Since Δ ⪰ 0, the trace tr(Δ ∇²_v G_t) ≤ 0, so G_t is nonincreasing in t along the path wherever it stays above 1/2. A continuity argument shows it stays above 1/2 for all t ∈ [0,1] if it starts above 1/2 at t=1.","core_discovery":"The central discovery is that Anderson's symmetry requirement can be replaced by a median threshold: for centered Gaussians ordered by covariance, the smaller-covariance law dominates the larger-covariance law on every closed convex set whose reference probability is at least 1/2. The mechanism is that the Ehrhard-Borell inequality forces the inverse-normal transform of the translated set probability to be concave, which makes the time-derivative of the interpolated probability nonpositive whenever that probability is at least 1/2, so the probability cannot increase as one moves from the smaller to the larger covariance.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Covariance ordering controls convex sets above the median","Anderson's Theorem holds without symmetry past the median","Convex Gaussian comparison needs only a median threshold","Ehrhard-Borell yields one-sided Anderson for convex sets","Smaller covariance dominates convex sets at half measure"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire argument rests on the Ehrhard-Borell inequality providing concavity of the inverse-normal-transformed probability of translated convex sets. If that concavity fails or cannot be extended to the boundary case pr(Y ∈ K) = 1/2, the sign of the derivative along the interpolation path is uncontrolled and the comparison breaks down.","fun_headline_variants_meta":{"raw":{"variants":["Covariance ordering controls convex sets above the median","Anderson's Theorem holds without symmetry past the median","Convex Gaussian comparison needs only a median threshold","Ehrhard-Borell yields one-sided Anderson for convex sets","Smaller covariance dominates convex sets at half measure"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":517,"prompt_tokens":441,"completion_tokens":76,"prompt_tokens_details":null},"tokens_in":441,"tokens_out":76,"duration_ms":24265,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T23:45:26.836106+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A counterexample would be: a pair of centered Gaussians with Σ_Y − Σ_X ⪰ 0 and a closed convex set K with pr(Y ∈ K) ≥ 1/2 but pr(X ∈ K) < pr(Y ∈ K). The paper itself provides a counterexample showing that replacing the Loewner order with weaker conditions (larger marginal variances plus Sudakov-Fernique increment conditions) is insufficient — there, the tail probabilities cross above the median, violating the conclusion.","supporting_citations":[],"review_version":1}