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Three sharp inequalities fuse entropy, Fisher, and moment measures

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2026-07-10 04:38 UTC pith:RGFJIS66

load-bearing objection Three new sharp inequalities built by multiplying two known inequalities and canceling a divergence term; sharpness inherited from cited preprints. the 2 major comments →

arxiv 2607.08599 v1 pith:RGFJIS66 submitted 2026-07-09 cs.IT math.IT

New sharp inequalities involving non-relative, relative and cross informational functionals with some remarkable minimizers of generalized Gaussian and Beta types

Razvan Gabriel Iagar , David Puertas-Centeno This is my paper
classification cs.IT math.IT
keywords inequalityenyicrossentropyinequalitiesrelativegaussianinformational
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves three new sharp inequalities that bind together informational functionals of three types—non-relative (depending on a single probability density), relative (depending on a pair), and cross (a hybrid). The central mechanism is a multiplicative cancellation trick: the authors take a recently established inequality relating Rényi entropy, Rényi divergence, and Rényi cross-entropy, and multiply it by a separate relative Stam-like or moment-entropy-like inequality. The Rényi divergence term appears in both and cancels, leaving a clean inequality that directly connects a non-relative functional, a relative functional, and a cross functional. This yields a Stam-type inequality linking Rényi entropy power, relative Fisher information, and Rényi cross-entropy; a moment-type inequality linking absolute deviation, relative cumulative deviation, and cross-deviation; and a Fisher-type inequality linking biparametric Fisher information, relative Fisher information, and a new generalized cross-Fisher information. All three are sharp, and their equality cases are characterized by stretched Gaussian densities, generalized trigonometric/hyperbolic functions, and generalized Beta distributions.

Core claim

The core discovery is that multiplying a three-term Rényi inequality (entropy + divergence ≤ cross-entropy) by a relative Stam or moment-entropy inequality causes the Rényi divergence factor to cancel, producing a new sharp inequality that directly links one-parameter non-relative functionals, biparametric relative functionals, and cross functionals. The equality cases are explicit for the first two inequalities—pairs of stretched Gaussian or generalized Beta densities—and are characterized by a second-order ODE for the third.

What carries the argument

The multiplicative cancellation of Rényi divergence between inequality (2.8) and the relative Stam/moment-entropy inequalities from the relative framework. Equality cases are then found by simultaneously solving the proportionality condition h ∝ f^{(β−1)/(β−α)} from (2.8) with the optimizer structure of the relative inequalities, which involves a change of variables y(x) driven by the stretched Gaussian g_{p,λ}.

Load-bearing premise

The sharpness of all three new inequalities is inherited from a relative Stam inequality and a relative moment-entropy inequality established in a cited companion preprint. If the sharpness analysis in that companion result has a gap, the sharpness claims here collapse—though the inequalities themselves, being products of two individually valid bounds, would still hold as non-sharp estimates.

What would settle it

A concrete pair of densities (f, h) satisfying all regularity conditions for which the product N_α[f]·φ_{p,λ_α}[f||h] falls strictly below K·exp{H_γ[f;h]}, or a proof that the relative Stam inequality (3.6) from the companion paper is not actually sharp under conditions (3.3)/(3.4), which would invalidate the equality-case analysis for all three theorems.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The cancellation trick can be iterated: any future relative inequality involving Rényi divergence can be combined with (2.8) to produce a new non-relative/relative/cross inequality at the corresponding functional level.
  • The generalized Beta and stretched Gaussian minimizers provide explicit extremal distributions that can serve as reference cases for testing numerical optimization algorithms in information geometry.
  • The cross-deviation and generalized cross-Fisher information functionals introduced here are new objects; their properties (convexity, monotonicity, data-processing behavior) are natural next targets.
  • The moment-inequality minimizers being generalized Beta distributions suggests a deeper structural analogy between the Stam and moment-entropy hierarchies at the cross-functional level.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The paper derives three new sharp informational inequalities that combine non-relative, relative, and cross functionals. The technique is to multiply two known inequalities (one from [29] mixing Rényi entropy, divergence, and cross-entropy; one from [28] providing relative Stam and moment-entropy bounds) and cancel the shared Rényi divergence term. Theorem 3.1 gives a Stam-like inequality involving the Rényi entropy power, relative Fisher information, and Rényi cross-entropy, with stretched Gaussian / generalized trigonometric optimizers. Theorem 4.1 gives a moment-type inequality with generalized Beta optimizers. Theorem 5.1 gives a Fisher-type inequality whose optimizers satisfy a second-order ODE that cannot be solved in closed form. The derivations are short and mechanically checkable.

Significance. The paper provides a clean and systematic way to generate cross-functional inequalities from existing relative-framework results. The optimizers for Theorems 3.1 and 4.1 are given explicitly in terms of known special functions (stretched Gaussians, generalized trigonometric functions, Beta distributions), which is a concrete strength. The technique of combining a three-term Rényi inequality with relative Stam/moment-entropy bounds to eliminate the divergence is natural and the resulting inequalities are genuinely new. However, the novelty is incremental: each theorem is a direct product of two cited inequalities, and the paper's contribution is primarily the identification of simultaneous saturation conditions and the resulting optimizer calculations.

major comments (2)
  1. Theorem 5.1, §5: The sharpness claim is not fully substantiated. The optimizers are characterized by the second-order ODE (5.9), which the authors state 'cannot be explicitly integrated.' The paper then asserts that 'any solution y(x) of (5.9) produces a pair of optimizers (f,h)' but does not prove that such a solution exists, nor that it yields valid probability densities (non-negative, integrable, normalizable). Without an existence argument—e.g., a fixed-point or variational existence result—the sharpness of Theorem 5.1 is asserted but not demonstrated. The authors should either provide an existence proof for solutions to (5.9) yielding valid densities, or downgrade the sharpness claim for Theorem 5.1 to conditional on such existence.
  2. Theorems 3.1, 4.1, and 5.1: The sharpness of all three results is transitive, inheriting from the relative Stam inequality (3.6) and relative moment-entropy inequality (4.5) of [28, Theorem 4.1], which is cited as an arXiv preprint. The multiplication-of-inequalities technique preserves sharpness only if both factor inequalities are simultaneously saturated by the same pair (f,h). For Theorems 3.1 and 4.1, the paper does verify this compatibility by deriving explicit optimizers (eqs. 3.10–3.13, 4.10–4.13). However, the paper should explicitly state that the sharpness of all results is contingent on [28] being correct, and ideally summarize the key steps of [28, Theorem 4.1] so the reader can verify the simultaneous-saturation argument without consulting the preprint.
minor comments (7)
  1. The paper depends critically on two unpublished preprints [28] and [29] by the same authors. The central inequality (2.8) from [29] and the relative Stam/moment-entropy inequalities from [28] are load-bearing. The authors should ensure these are accepted or forthcoming, or at minimum make the preprints available for review.
  2. §2, definition of Rényi cross-entropy: the limiting case writes 'lim_{α→1} H_α[f]' but should be 'lim_{α→1} H_α[f;g]' to be consistent with the notation used elsewhere.
  3. §3, equation (3.1): the condition '(α(2−λ)−1))(α−γ) = (α−1)²' has an extra closing parenthesis. Please verify the intended expression.
  4. §3, Remark 1: the statement that 'up to a scaling change, g^{1/α*}_{p,λ} = g_{p,λ} with λ = λ_{α−1}/(α−1)' is not immediately obvious. A brief derivation would help.
  5. §4, equation (4.4): the exponents on σ are typeset in a way that makes the expression hard to parse. Clarifying the grouping (e.g., with explicit braces) would aid readability.
  6. §5, equation (5.1): the notation ϕ^{(cr)}_{a,b,c}[f;h] is introduced here but the superscript '(cr)' is not used consistently in the subsequent text. Standardize the notation.
  7. The paper states (§3, last paragraph) that the optimal constant K can be 'explicitly computed in terms of the optimal constant of the biparametric Stam inequality' but does not give the formula. Providing the explicit expression (or at least the relationship) would be helpful.

Circularity Check

0 steps flagged

No significant circularity; derivation is a valid multiplication of independently-stated inequalities, with sharpness inherited transitively from cited preprints.

full rationale

The paper's three theorems are each derived by multiplying two previously established inequalities and canceling a shared divergence term. This is a standard and non-circular mathematical technique. The factor inequalities come from [28] (relative Stam/moment-entropy inequalities) and [29] (the three-term Rényi inequality (2.8)). While both are self-citations, the present paper does not redefine its outputs in terms of its inputs, nor does it fit a parameter and call the result a prediction. The optimal constants K are explicitly identified as those from [28], not repackaged as new results. The sharpness argument requires simultaneous saturation of both factor inequalities by the same pair (f,h); the paper addresses this by solving the resulting compatibility conditions (ODE (3.11) for Theorem 3.1, yielding explicit generalized trigonometric/hyperbolic optimizers; algebraic conditions (4.10)-(4.13) for Theorem 4.1, yielding Beta-type optimizers). For Theorem 5.1, the compatibility ODE (5.9) is not explicitly solved, which is a correctness/completeness gap (the sharpness is asserted without exhibiting the optimizer), but this is not a circularity issue. The derivation chain is self-contained against the stated assumptions; the self-citations provide the load-bearing inputs but those inputs are not defined in terms of the present paper's outputs. Score 2 reflects the heavy self-citation load without rising to the level of definitional or constructional circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 2 invented entities

The paper introduces no free parameters fitted to data (pure theory). The optimal constants K are inherited from prior work and not computed. Two new functionals (cross-deviation, cross-Fisher information) are introduced as the right-hand sides of the new inequalities, defined by analogy with cross-entropy. The main axioms are three inequalities from cited preprints [28, 29] by the same authors, which are assumed valid.

free parameters (3)
  • K (optimal constant in Theorem 3.1)
    The constant K is stated to depend on the parameters (p, λ, α, γ) and to equal the optimal constant of the biparametric Stam inequality from [21, 22]. It is not given a closed-form value in the paper.
  • K (optimal constant in Theorem 4.1)
    Stated to equal (K^(0)_{p,λ})^{1/ξ}, the optimal constant of the moment-entropy inequality from [20, 21]. Not computed explicitly.
  • K (optimal constant in Theorem 5.1)
    Stated to be the same constant as in Theorem 3.1. Not computed explicitly.
axioms (4)
  • domain assumption Inequality (2.8): R_α[f] + D_β[f||h] ≤ H_γ[f;h] under condition (α−β)(α−γ) = (α−1)²
    This is the starting inequality from [29], cited as a recently established result by the same authors. It is the algebraic identity that, combined with relative Stam/moment-entropy bounds, produces all three theorems. Its validity is assumed, not re-proven.
  • domain assumption Relative Stam inequality (3.6): e^{−D_{λ_α}[f||h]} φ_{p,λ_α}[f||h] ≥ α^{−1/(αλ)} (K^(1)_{p,λ})^{1/α}
    Cited from [28, Theorem 4.1], a 2025 arXiv preprint by the same authors. The sharpness of Theorems 3.1 and 5.1 depends entirely on the sharpness of this inequality.
  • domain assumption Relative moment-entropy inequality (4.5): e^{D_{λ_ξ}[f||h]} σ_{p*,ξ}[f||h] ≥ (K^(0)_{p,λ})^{1/ξ}
    Cited from [28, Theorem 4.1]. The sharpness of Theorem 4.1 depends on this result.
  • ad hoc to paper Existence and regularity of solutions to the ODE (5.9) for Theorem 5.1 optimizers
    The authors state (5.9) 'cannot be explicitly integrated' but claim sharpness conditional on solutions existing. The existence of suitable solutions is assumed without proof.
invented entities (2)
  • Cross-deviation σ_{p,γ}[f;h] no independent evidence
    purpose: A moment-type functional of two densities, defined in (4.1), serving as the right-hand side bound in Theorem 4.1.
    Introduced in [29] as a transported functional. No independent physical or information-theoretic interpretation is provided beyond its role in the inequality.
  • Generalized cross-Fisher information φ^(cr)_{a,b,c}[f;h] no independent evidence
    purpose: A Fisher-type functional of two densities, defined in (5.1), serving as the right-hand side bound in Theorem 5.1.
    Introduced for this inequality. No independent evidence or prior literature motivation is provided.

pith-pipeline@v1.1.0-glm · 16211 in / 3070 out tokens · 344706 ms · 2026-07-10T04:38:05.889217+00:00 · methodology

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read the original abstract

Several new and sharp informational inequalities are derived as a byproduct of Stam-like and moment-entropy-like inequalities in the relative framework and a recently established inequality mixing the R\'enyi entropy, the R\'enyi divergence and the R\'enyi cross entropy of suitable probability density functions. More precisely, we obtain a Stam-like inequality connecting the R\'enyi entropy power, the recently introduced scaling-invariant relative Fisher information and the R\'enyi cross entropy. Furthermore, we derive an inequality involving only Fisher-like informational measures and another inequality involving only moment-like functionals of non-relative, relative and cross types, respectively. All the inequalities are sharp. The minimizers of the Stam-like inequality are, in certain cases, pairs of Gaussian or stretched Gaussian probability densities; in contrast, each minimizer of the moment-like inequality is the probability density of the generalized Beta distribution.

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