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arxiv: 2605.18600 · v2 · pith:573XK2TMnew · submitted 2026-05-18 · 💻 cs.IT · math.CO· math.IT· math.PR

The Sharma-Mittal Entropy is Subadditive and Supermodular on the Majorization Lattice

Roberto Bruno , Ugo Vaccaro This is my paper

Pith reviewed 2026-05-22 09:29 UTC · model grok-4.3

classification 💻 cs.IT math.COmath.ITmath.PR
keywords Sharma-Mittal entropysubadditivitysupermodularitymajorizationprobability distributionsentropy measuresinformation theory
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The pith

Sharma-Mittal entropy is subadditive and supermodular on the majorization lattice of probability distributions.

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

The paper proves that Sharma-Mittal entropy, a two-parameter family that includes Shannon, Tsallis, and Rényi entropies as special cases, is subadditive and supermodular when probability distributions are partially ordered by majorization. This ordering turns the set of n-dimensional probability vectors into a lattice, and the entropy function respects the lattice operations in the stated way. A reader would care because subadditivity and supermodularity supply concrete inequalities that bound entropy values once one distribution majorizes another. The result therefore supplies a single proof that recovers and extends the known properties of the three classical entropies.

Core claim

We prove that Sharma-Mittal entropy is a subadditive and supermodular function on the lattice of all n-dimensional probability distributions, ordered according to the partial order relation defined by majorization among vectors. Our result unifies and extends analogous results presented in the literature for the Shannon entropy, the Tsallis entropy, and the Rényi entropy.

What carries the argument

The two-parameter Sharma-Mittal entropy function evaluated on the lattice formed by the majorization partial order on the probability simplex.

If this is right

  • The same subadditivity and supermodularity hold for Shannon entropy, Tsallis entropy, and Rényi entropy as immediate special cases.
  • Any inequality that follows from subadditivity or supermodularity on a lattice now applies directly to Sharma-Mittal entropy.
  • The result supplies a uniform method for deriving entropy bounds once one probability vector majorizes another.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The lattice properties may be useful for optimization problems that maximize or minimize entropy subject to majorization constraints.
  • Similar proofs could be attempted for continuous distributions or for quantum states ordered by majorization.
  • Direct computation for low dimensions and selected parameters offers an immediate numerical check of the claim.

Load-bearing premise

The majorization relation must form a lattice on the probability simplex and the Sharma-Mittal entropy must be given by its standard two-parameter definition.

What would settle it

Explicit numerical counterexamples for small n and valid parameter values where either subadditivity or supermodularity fails for a pair of distributions related by majorization.

read the original abstract

We prove that Sharma-Mittal entropy is a subadditive and supermodular function on the lattice of all $n$-dimensional probability distributions, ordered according to the partial order relation defined by majorization among vectors. Our result unifies and extends analogous results presented in the literature for the Shannon entropy, the Tsallis entropy, and the R\'enyi entropy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves that the Sharma-Mittal entropy (a two-parameter family recovering Shannon, Tsallis, and Rényi entropies as special cases) is subadditive and supermodular on the lattice of n-dimensional probability distributions equipped with the majorization partial order. The result is presented as a unification and extension of prior lattice-theoretic properties known for the special cases.

Significance. If the central claims hold, the work offers a unified lattice-theoretic treatment of several important entropy measures under majorization, which is a standard tool in information theory and majorization theory. This could streamline proofs for related inequalities and support applications in optimization or inequality analysis over probability simplices. The explicit use of lattice operations (suprema and infima induced by sorted partial sums) is a methodological strength.

major comments (2)
  1. [§3, Theorem 3.2] §3, Theorem 3.2 (subadditivity): the proof invokes the monotonicity of the Sharma-Mittal function under majorization for admissible (α,β); an explicit check that the two-parameter form preserves the required inequality direction for all β in the admissible range (including the Tsallis limit) would strengthen the argument, as the reduction to known cases is only sketched.
  2. [§4, Proposition 4.1] §4, Proposition 4.1 (supermodularity): the lattice join operation is defined via componentwise suprema on sorted vectors; the derivation assumes these suprema remain probability vectors, but the boundary case when the join saturates the simplex constraints is not separately verified for the entropy expression.
minor comments (2)
  1. [Eq. (2)] The notation for the Sharma-Mittal entropy (Eq. (2)) uses α and β without restating the conventional range restrictions (α ≠ 1, β > 0, etc.) in the statement of the main theorems; adding a short reminder would improve readability.
  2. [Figure 1] Figure 1 (illustration of lattice operations) would benefit from labeling the sorted partial-sum vectors explicitly to match the definitions in §2.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2 (subadditivity): the proof invokes the monotonicity of the Sharma-Mittal function under majorization for admissible (α,β); an explicit check that the two-parameter form preserves the required inequality direction for all β in the admissible range (including the Tsallis limit) would strengthen the argument, as the reduction to known cases is only sketched.

    Authors: We agree that an explicit verification strengthens the argument. In the revised manuscript we will add a short lemma immediately before Theorem 3.2 that directly confirms the monotonicity inequality for the full admissible range of (α, β), with a separate paragraph treating the Tsallis limit β → 0. This replaces the sketched reduction with a self-contained check. revision: yes

  2. Referee: [§4, Proposition 4.1] §4, Proposition 4.1 (supermodularity): the lattice join operation is defined via componentwise suprema on sorted vectors; the derivation assumes these suprema remain probability vectors, but the boundary case when the join saturates the simplex constraints is not separately verified for the entropy expression.

    Authors: We agree that the boundary case deserves explicit treatment. Although the majorization lattice join is known to preserve the probability simplex, we will insert a brief subcase analysis in the proof of Proposition 4.1 that verifies the supermodularity inequality when the join saturates the simplex boundaries, either by direct substitution or by continuity of the entropy. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes subadditivity and supermodularity of the Sharma-Mittal entropy directly via lattice arguments on the majorization partial order over the probability simplex. The entropy definition is the standard two-parameter form (recovering Shannon, Tsallis, and Rényi as special cases) with no parameters fitted to the target properties. The lattice structure follows from the existence of componentwise suprema and infima on sorted partial sums, a standard fact independent of the entropy result. The unification of prior results for special cases is presented as an extension rather than a load-bearing premise, and no derivation step reduces to self-definition, fitted inputs renamed as predictions, or self-citation chains that substitute for independent verification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of Sharma-Mittal entropy and the lattice structure induced by majorization; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Majorization defines a lattice partial order on the probability simplex
    Invoked when stating the ordering among vectors of probability distributions
  • domain assumption Sharma-Mittal entropy reduces to Shannon, Tsallis, and Rényi entropies for specific parameter values
    Used to claim unification with prior results

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