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arxiv: 2605.31372 · v1 · pith:VRFEPAEAnew · submitted 2026-05-29 · 🧮 math.CV · math.DS· math.PR

On the Negation of a Hyperbolic-Valued Probability Distribution

Juan Bory-Reyes , Edil D. Molina-Fernandez , Jos\'e M. Sigarreta-Almira This is my paper

Pith reviewed 2026-06-28 19:55 UTC · model grok-4.3

classification 🧮 math.CV math.DSmath.PR
keywords hyperbolic numbershyperbolic probability distributionnegationmajorizationhyperbolic Shannon entropyGini-Simpson entropy
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The pith

Hyperbolic probability distributions majorize their negations under generated negators.

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

The paper defines negation for finite distributions valued in hyperbolic numbers by means of the partial order coming from their idempotent structure. It introduces hyperbolic majorization together with a broad class of generated negators. For those negators the original distribution is proved to majorize the negated distribution. The majorization relation produces an increase in the strong hyperbolic Shannon entropy and in the hyperbolic Gini-Simpson entropy, while repeated negation drives each component toward uniformity.

Core claim

For a broad class of generated negators the original hyperbolic-valued probability distribution majorizes its negation. This comparison yields entropy increase for the strong hyperbolic Shannon entropy and the hyperbolic Gini-Simpson entropy and implies component-wise uniformization of the iterated negation. Involutive negators are shown to be structurally distinct from the generated negators that produce the entropy increase.

What carries the argument

Hyperbolic majorization, defined from the partial order induced by the idempotent decomposition of hyperbolic numbers, which supplies the comparison between a distribution and its negation.

If this is right

  • The strong hyperbolic Shannon entropy increases when the distribution is replaced by its negation.
  • The hyperbolic Gini-Simpson entropy increases under the same replacement.
  • Repeated application of a generated negator makes the distribution components approach one another.
  • Involutive negators cannot be used to obtain the entropy increase that generated negators produce.

Where Pith is reading between the lines

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

  • The same majorization relation might be checked for additional entropy functionals defined on hyperbolic distributions.
  • The uniformization effect could be quantified by tracking the variance of the component values after each iteration.
  • The structural distinction between involutive and generated negators suggests two separate families of operations whose interaction with other hyperbolic probabilistic concepts remains open.

Load-bearing premise

The partial order induced by the idempotent structure of hyperbolic numbers is suitable for defining majorization and negation operations on these probability distributions.

What would settle it

A concrete generated negator together with a hyperbolic-valued distribution for which the original does not majorize the negation would disprove the central comparison.

Figures

Figures reproduced from arXiv: 2605.31372 by Edil D. Molina-Fernandez, Jos\'e M. Sigarreta-Almira, Juan Bory-Reyes.

Figure 1
Figure 1. Figure 1: illustrates this decomposition for n = 2. Since P = (p1, p2) is determined by p1 once the total mass s(P) is fixed, the figure represents the first entry p1 = a + kb in the (a, b)- plane. The three panels are kept separate because they correspond to disjoint subsets of D 2 , even though their projections onto the p1-plane may overlap. 0 0.5 1 -0.5 0 0.5 a b 0 e1 1 e2 (a) 0 0.5 1 -0.5 0 0.5 a b 0 e1 (b) 0 0… view at source ↗
Figure 2
Figure 2. Figure 2: Entropy components for the family Px,y. (a) H(1)(Px,y) = h(x); (b) H(2)(Px,y) = h(y); (c) G(1)(Px,y) = 2x(1 − x); (d) G(2)(Px,y) = 2y(1 − y). Example 2. Let T = {(x, y) ∈ R 2 | x ≥ 0, y ≥ 0, x + y ≤ 1}. For (x, y) ∈ T, consider Px,y = (xe1 + ye2, ye1 + (1 − x − y)e2, (1 − x − y)e1 + xe2). Then Px,y ∈ PD(3) and s(Px,y) = 1. Its idempotent components are P (1) x,y = (x, y, 1 − x − y), P(2) x,y = (y, 1 − x − … view at source ↗
Figure 3
Figure 3. Figure 3: Entropy surfaces for the triangular family: (a) [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two-step trajectories in the active components of [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Iterative profiles of the generated hyperbolic negator induced by [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Iterative profiles of the generated hyperbolic negator induced by [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
read the original abstract

In the context of hyperbolic numbers we define the concept of negation of finite hyperbolicvalued probability distributions that is based on the partial order induced by the idempotent structure of hyperbolic numbers. Then, a hyperbolic majorization and general hyperbolic negators are introduced. For a broad class of generated negators, we prove that the original distribution majorizes its negation. This comparison yields that entropy increase for the strong hyperbolic Shannon entropy and the hyperbolic Gini-Simpson entropy, and it implies component-wise uniformization of the iterated negation. Finally, we analyze involutive property of hyperbolic negators and prove that are structurally distinct from the generated negators responsible for the entropy increase. These results show that hyperbolic probabilistic negation is not merely a component-wise copy of the real case, but a theory governed by the interaction between idempotent decomposition, partial order, and entropy measure.

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 / 3 minor

Summary. The paper defines negation for finite hyperbolic-valued probability distributions via the partial order induced by the idempotent decomposition of hyperbolic numbers. It introduces hyperbolic majorization and a class of generated negators, proving that for a broad class of such negators the original distribution majorizes its negation. This yields monotonicity (increase) of the strong hyperbolic Shannon entropy and the hyperbolic Gini-Simpson entropy under negation, together with component-wise uniformization under iterated negation. The paper further examines involutive negators and establishes that they are structurally distinct from the generated negators that produce the entropy increase, arguing that the resulting theory is governed by the interaction of idempotent decomposition, partial order, and entropy rather than being a direct copy of the real-valued case.

Significance. If the central proofs are correct, the work supplies a non-trivial algebraic extension of majorization and entropy monotonicity to hyperbolic-valued probabilities. The explicit use of the idempotent structure to define both the partial order and the generated negators, together with the distinction between involutive and entropy-increasing negators, constitutes a genuine departure from the real case and could serve as a template for similar constructions in other algebraic probability settings.

major comments (2)
  1. [§3] §3, Definition 3.4 and Theorem 3.7: the proof that p ≽ neg(p) for generated negators relies on the specific form of the negator in Eq. (12); it is not immediately clear whether the argument extends to all negators satisfying only the three listed axioms or whether an additional monotonicity condition on the generating function is tacitly used.
  2. [§4.2] §4.2, Theorem 4.3: the claimed entropy increase for the strong hyperbolic Shannon entropy is shown only after establishing majorization; if the partial order is not total, the standard majorization-entropy implication (used in the real case) requires an additional verification that the entropy functional is Schur-concave with respect to this specific order.
minor comments (3)
  1. [§2] Notation for the two components of a hyperbolic number is introduced in §2 but then used interchangeably with (x,y) and x + y j; a single consistent notation throughout would improve readability.
  2. [§5] The statement of Corollary 5.2 on iterated negation would benefit from an explicit reference to the component-wise uniformization already proved in Theorem 4.5 rather than repeating the argument.
  3. [§3] Several displayed equations in §3 lack equation numbers, making cross-references in the proofs harder to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of significance, and constructive comments. We address the two major comments point by point below. Both can be resolved with minor revisions that clarify the scope of the results without altering the core claims.

read point-by-point responses

  1. Referee: [§3] §3, Definition 3.4 and Theorem 3.7: the proof that p ≽ neg(p) for generated negators relies on the specific form of the negator in Eq. (12); it is not immediately clear whether the argument extends to all negators satisfying only the three listed axioms or whether an additional monotonicity condition on the generating function is tacitly used.

    Authors: The statement of Theorem 3.7 is explicitly restricted to the class of generated negators introduced via the construction in Eq. (12), which is a subclass of the negators satisfying the three axioms in Definition 3.4. The proof of majorization p ≽ neg(p) uses the explicit functional form of the generator to compare components after idempotent decomposition. We agree that the argument does not automatically extend to arbitrary negators obeying only the three axioms. We will revise the text preceding Theorem 3.7 to emphasize this scope and add a short remark noting that an additional monotonicity assumption on the generator would be needed for a more general result. This clarification does not change the theorem itself. revision: yes

  2. Referee: [§4.2] §4.2, Theorem 4.3: the claimed entropy increase for the strong hyperbolic Shannon entropy is shown only after establishing majorization; if the partial order is not total, the standard majorization-entropy implication (used in the real case) requires an additional verification that the entropy functional is Schur-concave with respect to this specific order.

    Authors: We accept the point that the partial order induced by the idempotent decomposition is not total, so the usual majorization-to-entropy monotonicity argument requires explicit justification of Schur-concavity on this poset. In the current proof we derive the entropy increase directly from the component-wise inequalities supplied by majorization together with the explicit expression for the strong hyperbolic Shannon entropy. To make the reasoning fully rigorous and parallel to the classical case, we will insert a brief lemma in §4.2 verifying Schur-concavity of the entropy with respect to hyperbolic majorization. This addition strengthens the presentation without affecting the validity of Theorem 4.3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definitions yield independent proofs

full rationale

The paper defines negation of hyperbolic-valued distributions via the partial order from idempotent decomposition, introduces majorization and generated negators as new constructs, then proves majorization p ≽ neg(p) for a class of negators along with consequent entropy monotonicity. These steps are presented as following directly from the definitions without reducing to fitted inputs, self-citations, or renamings by construction. No load-bearing premise relies on prior author work as an unverified uniqueness theorem, and the entropy increase is derived rather than presupposed. The derivation chain is self-contained against the introduced axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on standard algebraic properties of hyperbolic numbers and the new definitions introduced in the paper. No free parameters or invented physical entities are mentioned.

axioms (1)
  • domain assumption Hyperbolic numbers possess an idempotent structure that induces a partial order suitable for probability distributions.
    Invoked to define negation and majorization (abstract).

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