{"id":"5c739124-5279-443c-ab9c-11e8f4bbf324","arxiv_id":"2507.05928","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For centered sub-Gaussian variables, the optimal ratio constants are sqrt(3/8) and sqrt(log 2), attained by the Gaussian and Rademacher laws respectively.","lead":"The paper finds the exact best constants that relate the sub-Gaussian norm to the sub-Gaussian parameter for centered random variables. These sharp constants matter because converting between the two most common sub-Gaussian descriptions appears throughout high-dimensional probability and statistics.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Upper bound rests on an unverified application of Winkler's theorem to the exponential moment constraint; Section 2.2 also contains incorrect derivative identities, so the proof needs verification before full acceptance.","rationale":"The reader's weakest assumption correctly identifies Winkler's theorem as the load-bearing external dependency. My independent check confirms this is the point where the proof is least secure: the paper cites the theorem without verifying hypotheses for a noncompact space with an unbounded exponential moment constraint, and even states the extremal set equality too broadly (H3 elements with a slack exponential constraint are not extreme; only the inclusion is needed). I also found concrete, demonstrable errors in the derivative computations of Proposition 2.9. These errors do not appear to affect the truth of Theorem 1.1: the corrected derivatives still imply h2≥0 and h≥0, and the reduction argument goes through once Winkler's hypotheses are confirmed. Because the central constants and the overall strategy are sound, I would not reject or mark the result unverified, but I would condition acceptance on a rigorous verification of the Winkler application and a correction of the derivative identities. The sharpness examples and the lower bound are correct and give independent support.","tokens_in":8745,"tokens_out":29987,"duration_ms":339370,"concrete_test":"Check the exact hypotheses of Winkler's Theorem 2.1(a) and Theorem 3.1 (or provide a self-contained proof) for the moment set H with unbounded functions x and e^{x^2} and the inequality ∫e^{x^2}dμ≤2; if the hypotheses fail, prove directly that every extreme point of H is in H1 ∪ H2 ∪ {μ∈H3 : ∫e^{x^2}dμ=2}. In parallel, recompute the derivatives in Proposition 2.9 with a CAS: verify h2(1)=h2'(1)=0, h2''(1)=8 log 2 -4, h2'''(u)=2 log 2 (6+2/u^2)-6, and h'''(u)=2 log 2 (6 -4/u +2/u^2); confirm h2,h≥0 and F'(u)≥0 for u≥1.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The main logical step is Lemma 2.6/2.7: an application of Winkler's theorem ([16], Theorems 2.1(a) and 3.1) to the moment set H = {μ: ∫dμ=1, ∫x dμ=0, ∫e^{x^2}dμ≤2}. The paper does not check the theorem's hypotheses: H lives on noncompact R, the functions x and e^{x^2} are unbounded, and the exponential constraint is an inequality. If the theorem requires compact support or bounded moment functions, the claimed finite-support representation of extreme points is unsupported and the reduction to H2 ∪ H3 (and then H2) fails. A second concrete defect is in Section 2.2: the printed derivative identities for h2 and h are false (e.g., h2''(1)=8 log 2 - 4 > 0, not 0, and the true h''' is 2 log 2 (6 - 4/u + 2/u^2), not the printed expression). The desired inequalities h2 ≥ 0 and h ≥ 0 appear to survive after correction, so the constants themselves are not in doubt, but the proof as written is not fully rigorous.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper identifies the sharp universal constants relating the sub-Gaussian norm \\|X\\|_{\\psi_2} and the variance-proxy parameter \\sigma_X for centered real-valued random variables. Theorem 1.1 claims that \\sqrt{3/8}\\,\\|X\\|_{\\psi_2} \\le \\sigma_X \\le \\sqrt{\\log 2}\\,\\|X\\|_{\\psi_2}, with sharpness attained by the standard normal distribution and the Rademacher distribution. The lower bound is proved by a Gaussian integral comparison; the upper bound is proved by reducing the maximization of the moment generating function over the set H = {centered laws with E e^{X^2} \\le 2} to binary laws, using compactness, an extreme-point reduction based on Winkler's moment-set theorems, and a calculus lemma for binary laws. The introduction also surveys earlier known constants and situates the new result in the literature.","tokens_in":8948,"tokens_out":24553,"duration_ms":273680,"significance":"If the proof is completed, these appear to be the first sharp constants in a pair of inequalities that are widely used in high-dimensional probability and concentration theory. The numerical improvement from the previous best upper constant (about 1.12) to \\sqrt{\\log 2} \\approx 0.83 is clean, and the two sharpness examples (standard Gaussian and Rademacher) are explicit and convincing. The lower-bound proof is short, correct, and self-contained. The paper also gives a useful overview of earlier bounds and references. However, the upper-bound proof as written relies on a moment-set theorem whose hypotheses are not checked, and it contains incorrect derivative identities in the binary calculus lemma; these points must be fixed before the result can be regarded as fully established.","major_comments":[{"comment":"The reduction to H2 \\cup H3 is the load-bearing step for the upper bound, but the paper does not state or verify the hypotheses of Winkler's theorems [16, Theorems 2.1(a) and 3.1] as applied to H in (2.7). The set H lives on noncompact R, the moment functions x and e^{x^2} are unbounded, and the exponential constraint is an inequality; these are exactly the features that require assumptions in moment-set theorems. In addition, Theorem 3.1 is invoked for a Borel-set-valued representation \\bar\\mu(B) = \\int \\nu(B)\\,\\pi(d\\nu), which is stronger than the representation of continuous affine functionals supplied by Choquet's theorem. Please either quote the precise theorems used and verify their hypotheses for H, or replace this step with a direct proof that every extreme point of H has support of size at most three (for example, using the three moment conditions and Lemma A.1), together with Choquet's theorem applied to the continuous affine functional \\mu \\mapsto M_\\mu(s).","section":"2.1.2 (Lemmas 2.6 and 2.7)"},{"comment":"The displayed derivative identities used to prove h2(u) \\ge 0 and h(u) \\ge 0 are incorrect. For h2(u) = 2c(u^3 - u - 2u\\ln u) - (1+u)(u-1)^2 with c = \\ln 2, one computes h2''(1) = 8c - 4 > 0, not 0, and h2'''(u) = 2c(6 + 2/u^2) - 6. For h(u) = 2c(u^3 + u^2 - u - 1 - 2u^2\\ln u - 2u\\ln u), the correct third derivative is h'''(u) = 2c(6 + 2/u^2 - 4/u), not 2c(6 - 2/u^2 - 4/u). The nonnegativity conclusions survive the correction: h2''(1) > 0 and h2''' > 0 on [1,\\infty), while h'''(u) \\ge 8c > 0 for u \\ge 1. Thus the constants are not in doubt, but the proof as printed is not rigorous and the equations in this subsection must be corrected.","section":"2.2 (proof of Proposition 2.9)"}],"minor_comments":[{"comment":"The Lagrange multiplier argument writes the multiplier \\lambda_2 for the inequality constraint g2 \\le 0 even when this constraint is inactive; the proof works with \\lambda_2 = 0 in that case, but this should be stated explicitly for completeness.","section":"2.1.3 (Lemma 2.8)"},{"comment":"The sentence 'Lemma 2.8 also shows that H3 is not a closed subset of the compact set H' appears before the lemma and is not part of the lemma's statement; it would be clearer placed after the proof.","section":"2.1.3 (Lemma 2.8)"},{"comment":"The symbol H3 is used both for the set of three-point measures in (2.2) and for the parameter set {(p,x) \\in G3 : g0=g1=0, g2\\le 0} inside the proof of Lemma 2.8; using a different symbol (e.g., \\tilde H_3) would avoid confusion.","section":"2.1.3 (Lemma 2.8)"},{"comment":"In the sentence 'This above inequality reveals...' the word 'above' is redundant; this is a minor typographical point.","section":"3"}],"recommendation":"major_revision","confidential_remarks":"I believe the constants are very likely correct and the lower-bound proof is solid. The main risk is the unverified application of Winkler's moment-set theorem; if that theorem does not cover the present noncompact, unbounded-moment setting, the upper-bound proof would need a different argument. The derivative errors in Section 2.2 are simple to fix and do not change the conclusion. The paper is well within the scope of a probability journal and should be encouraged after the requested revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi [Colleague],\n\nThe headline: this paper pins down the sharp constants in the two-sided inequality between the psi2 norm and the sub-Gaussian parameter, and the upper bound is a real improvement: sqrt(log 2) ≈ 0.83, versus the previous best published ~1.12. The lower bound sqrt(3/8) was implicit in earlier work, but the sharpness argument via the standard normal is new. That is a clean, useful result for anyone who converts between the Orlicz norm and the variance proxy.\n\nWhat the paper does well: the proof strategy is coherent. They prove compactness of the moment set H, then use extreme point arguments to reduce the maximization of the MGF to binary laws, and then handle the binary case by an explicit calculus lemma. The lower bound is a short Gaussian integral argument. The extremal examples (Gaussian and Rademacher) are computed correctly, and the citations to prior upper bounds look accurate.\n\nThe soft spots are real but patchable. The application of Winkler's theorem in Lemma 2.6 is not fully justified. The set H is noncompact, and the constraint function e^{x^2} is unbounded; the theorem's hypotheses are not checked. The authors do prove weak compactness of H just before, so the application is plausible, but a referee should ask them to spell out why Winkler's theorem applies, or to replace it with a direct convex-analysis argument. The other issue is in Section 2.2: several derivative identities are printed incorrectly. For instance, the claim that h2''(1)=0 is false; the correct value is 8 log 2 - 4 > 0. The printed formula for h'''(u) is also wrong; the correct expression is 2 log 2 (6 - 4/u + 2/u^2). These errors do not sink the proof — the corrected derivatives still give the needed inequalities, in fact more directly — but the text as written is not fully rigorous.\n\nBottom line: the main theorem is very likely correct, the constants are genuinely sharp, and the gaps are fixable. This deserves a serious referee, not a desk reject. I would cite it once it is cleaned up.\n\nBest,\n[Name]","headline":"Sharp constants for the psi2/sigma_X comparison, with mostly sound strategy and fixable lapses in the calculus; deserves a referee.","tokens_in":9512,"tokens_out":6416,"would_cite":true,"duration_ms":56457,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60E15","46E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper determines the exact constants in the inequalities relating the sub-Gaussian norm and the sub-Gaussian parameter: $\\sqrt{3/8}\\|X\\|_{\\psi_2} \\le \\sigma_X \\le \\sqrt{\\log 2}\\|X\\|_{\\psi_2}$, with both bounds sharp.","keywords":["sub-Gaussian norm","variance proxy","sharp constants","optimal variance proxy","Orlicz norm","Luxemburg norm","moment generating function","Rademacher distribution"],"falsifier":"Fix any real $s$ and numerically maximize $\\sum_{i=1}^3 p_i e^{s x_i}$ over three-point centered distributions with $\\sum_i p_i e^{x_i^2} \\le 2$; if any such maximum exceeds $e^{(\\log 2)s^2/2}$, the upper bound fails. Equivalently, a single three- or four-point distribution satisfying the constraint whose moment generating function exceeds the binary supremum at some $s$ would refute the theorem.","tokens_in":8531,"feed_emoji":"🎯","tokens_out":8992,"duration_ms":91879,"temperature":0.7,"pith_summary":"The paper determines the exact values of the universal constants connecting the two standard measures of sub-Gaussian behaviour: the Orlicz-style norm $\\|X\\|_{\\psi_2}$ and the variance proxy $\\sigma_X$. Its main theorem states that for every centered real-valued random variable $X$, $\\sqrt{3/8}\\,\\|X\\|_{\\psi_2} \\le \\sigma_X \\le \\sqrt{\\log 2}\\,\\|X\\|_{\\psi_2}$, and that both inequalities are sharp. The upper bound is the genuinely new contribution: the authors prove that, among centered variables normalized to have $\\|X\\|_{\\psi_2}\\le 1$, the moment generating function is maximized by a two-point (binary) distribution, namely the Rademacher law in the sharp case. The lower bound, attained by the standard Gaussian, is proved by a short Gaussian-integral argument. Together they pin the ratio $\\sigma_X/\\|X\\|_{\\psi_2}$ to the interval $[\\sqrt{3/8},\\sqrt{\\log 2}]$ with no room for improvement.","feed_headline":"Optimal constants found for sub-Gaussian norm and variance proxy","feed_subtitle":"For any centered variable, the ratio of the variance proxy to the psi-2 norm is between 0.612 and 0.833; both extremes are attained.","key_machinery":"The load-bearing object is the moment set $H$ of centered probability measures with $\\int e^{x^2}\\,d\\mu \\le 2$, together with the linear functional $\\mu \\mapsto \\int e^{sx}\\,d\\mu(x)$. The argument has four moves: (i) compactness and continuity on $H$ via Skorohod coupling and weak-compactness results, so the maximum is attained; (ii) a general theorem on extreme points of moment sets, applied to the constraints $\\mu(1)=1$, $\\mu(x)=0$, $\\mu(e^{x^2})\\le 2$, which forces any extremal maximizer to have at most three atoms; (iii) a Lagrange-multiplier/Rolle argument showing a three-atom maximizer cannot exist, leaving only two-point measures; and (iv) an explicit formula for the variance proxy of a binary random variable, after which the upper bound reduces to verifying that a certain function $F(u)$ stays above $2$ for $u\\ge 1$. The lower bound is carried by a separate short argument: writing $E e^{X^2/K^2}$ as an integral against the standard Gaussian density and applying the definition of $\\sigma_X$.","core_discovery":"On the paper's own terms, the central discovery is that the two classical inequalities relating $\\|X\\|_{\\psi_2}$ and $\\sigma_X$ are governed by sharp constants $\\sqrt{3/8}$ and $\\sqrt{\\log 2}$, and that the extremes are realized by the two canonical sub-Gaussian examples: the standard Gaussian distribution for the lower bound and the symmetric Bernoulli (Rademacher) distribution for the upper bound. The proof of the upper bound shows something stronger: the supremum of $E e^{sX}$ over the set $H = \\{\\mu : \\int x\\,d\\mu = 0,\\ \\int e^{x^2}\\,d\\mu \\le 2\\}$ is attained at a binary distribution for every real $s$. This reduces a potentially complex variational problem to a one-variable calculus inequality, which yields $\\sigma_\\mu \\le \\sqrt{\\log 2}$ for all $\\mu \\in H$.","pith_inferences":["The same reduction method, replacing $e^{x^2}$ with $e^{|x|^p}$ for $p>2$, would plausibly yield sharp constants between $\\|X\\|_{\\psi_p}$ and a suitable $p$-th-order variance proxy; the extremal support size would then depend on $p$, since the moment-set dimension increases.","Because the sharp upper bound is attained by a discrete law, the supremum over $H$ of $E e^{sX}$ is non-smooth in $s$ near the extremal direction; numerical schemes that approximate the sup by fine grids should approach the binary bound from below, with the Rademacher law as the limiting extremizer.","The two sharpness examples bracket the full ratio interval, but the paper does not say whether every value in $[\\sqrt{3/8},\\sqrt{\\log 2}]$ is realized; mixtures of a Gaussian and a Rademacher law, suitably scaled, would be natural candidates to test."],"forward_implications":["No universal inequality can improve the two constants: any centered sub-Gaussian $X$ satisfies $\\sigma_X \\le \\sqrt{\\log 2}\\,\\|X\\|_{\\psi_2}$ and $\\|X\\|_{\\psi_2} \\le \\sqrt{8/3}\\,\\sigma_X$, and both bounds are attained.","The upper-bound proof yields a parameter-free recipe: whenever $E e^{X^2/K^2}\\le 2$, one immediately gets $E e^{sX}\\le e^{(K^2\\log 2)s^2/2}$ for all real $s$, so the variance proxy is at most $\\sqrt{\\log 2}\\,K$.","The lower bound shows that a bound on the variance proxy alone, $\\sigma_X\\le \\sigma$, implies $\\|X\\|_{\\psi_2}\\le \\sqrt{8/3}\\,\\sigma$, transferring Gaussian-type MGF control back to Orlicz-norm control with a concrete constant.","The extremal laws are the standard Gaussian and Rademacher distributions, so these are the test cases for any refinement of related inequalities for centered sub-Gaussian variables."],"supporting_citations":[{"why":"Supplies the extreme-point and Choquet-representation theorems for moment sets; the reduction of the MGF maximization to at most three atoms stands on it.","marker":"[16]"},{"why":"Provides the explicit formula for the variance proxy of a binary random variable used to convert the binary bound into $\\sqrt{\\log 2}$.","marker":"[3]"},{"why":"Supplies the Skorohod coupling, Fatou lemma, and weak-compactness results used to prove the maximum over $H$ is attained.","marker":"[6]"},{"why":"Characterizes uniform integrability and tightness, used to verify that the feasible set $H$ is compact.","marker":"[7]"},{"why":"Provides the Lagrange multiplier rule applied to exclude three-point maximizers.","marker":"[9]"}],"fun_headline_variants":["Exact constants: Gaussian and Rademacher achieve both bounds","Sub-Gaussian norm and variance proxy: optimal constants pinned","Sharp inequalities: best constants for sub-Gaussian tail events","Optimal constants proven for sub-Gaussian norm and variance proxy","Gaussian and Rademacher hit exact bounds for sub-Gaussian norm"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the sharp upper bound rests on a general theorem about extreme points of moment sets applied to the constraint $\\int e^{x^2}\\,d\\mu \\le 2$; the paper does not verify all hypotheses of that theorem for this non-polynomial constraint, so if the theorem does not apply, the reduction to binary distributions collapses.","fun_headline_variants_meta":{"raw":{"variants":["Exact constants: Gaussian and Rademacher achieve both bounds","Sub-Gaussian norm and variance proxy: optimal constants pinned","Sharp inequalities: best constants for sub-Gaussian tail events","Optimal constants proven for sub-Gaussian norm and variance proxy","Gaussian and Rademacher hit exact bounds for sub-Gaussian norm"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00079,"raw_usage":{"total_tokens":3422,"prompt_tokens":827,"completion_tokens":2595,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":2509}},"tokens_in":443,"tokens_out":2595,"duration_ms":19395,"temperature":1.0,"reasoning_tokens":2509,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:16:10.412587+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix any real $s$ and numerically maximize $\\sum_{i=1}^3 p_i e^{s x_i}$ over three-point centered distributions with $\\sum_i p_i e^{x_i^2} \\le 2$; if any such maximum exceeds $e^{(\\log 2)s^2/2}$, the upper bound fails. Equivalently, a single three- or four-point distribution satisfying the constraint whose moment generating function exceeds the binary supremum at some $s$ would refute the theorem.","supporting_citations":[{"cited_title":"4, 581–587","cited_arxiv_id":null,"evidence_quote":"Supplies the extreme-point and Choquet-representation theorems for moment sets; the reduction of the MGF maximization to at most three atoms stands on it."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the explicit formula for the variance proxy of a binary random variable used to convert the binary bound into $\\sqrt{\\log 2}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Skorohod coupling, Fatou lemma, and weak-compactness results used to prove the maximum over $H$ is attained."},{"cited_title":"1, 382–389","cited_arxiv_id":null,"evidence_quote":"Characterizes uniform integrability and tightness, used to verify that the feasible set $H$ is compact."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Lagrange multiplier rule applied to exclude three-point maximizers."}],"review_version":1}