Interval-Lattice Fixed-Point Semantics for

Certified Implicit Hypergraph Neural Operators

Sawsan Rateb Almokabaa1,* Maissam Ahamad Jdid2

1 Master’s Student, Faculty of Science, Damascus University, Damascus, Syria

2 Faculty of Science, Damascus University, Damascus, Syria

Received: July 02, 2025 Revised: October 04, 2025 Accepted: December 08, 2025 ⋆ Corresponding author

ABSTRACT

Implicit hypergraph models represent higher-order propagation by an equilibrium equation, but standard wellposedness

arguments typically force a contraction and therefore exclude noncontractive yet order-preserving

dynamics. This paper introduces an interval-lattice semantics for implicit hypergraph neural operators. On the box

lattice L = {X ∈ Rn×d : ℓ ≤ Xi j ≤ u}, a normalized hypergraph propagation PH ≥ 0, an entrywise nonnegative

channel map A ≥ 0, and an isotone activation generate an order-preserving operator T : L →L. The Knaster–

Tarski theorem then yields a nonempty complete lattice of equilibria without requiring ∥A∥ < 1. Coupled iterations

from the bottom and top elements produce certified lower and upper enclosures for every equilibrium. Under

the optional metric condition q = Lσ ∥PH ∥2 ∥A∥2 < 1, the extremal equilibria coincide; geometric convergence, a

residual-to-solution certificate, and a structural perturbation bound follow. Permutation equivariance and monotone

dependence on input features are also proved. Two small arithmetic tables illustrate certificate scaling rather than

empirical performance. The paper concludes with seven open problems on noncontractive uniqueness, signed

hypergraphs, finite certificate complexity, topology-aware perturbation metrics, expressivity, differentiable extremal

selection, and asynchronous lattice iteration.

Keywords: Complete lattice Fixed-point semantics Hypergraph neural operator Monotone map Certified inference

Implicit graph learning Open problems

1. INTRODUCTION

Let H = (V,E ,w) be a weighted hypergraph with incidence

matrix B ∈ {0,1}n×m. Hypergraph neural networks

replace pairwise propagation by node–hyperedge–node transport

and thereby encode higher-order relations [1, 2, 3, 4].

Implicit models instead define the representation as a fixed

point X∗ = T (X∗), corresponding formally to infinite-depth

weight sharing [5, 6, 7]. These two ideas were joined in recent

implicit hypergraph architectures [8, 9]; their well-posedness

mechanisms are metric, commonly based on projection, spectral

control, or contraction.

Metric uniqueness is sufficient but not logically necessary for

existence. If T is isotone on a complete lattice (L,⪯), then

X ⪯Y =⇒T (X) ⪯ T (Y), Fix(T ) ̸= ∅, (1)

by the Knaster–Tarski principle [10, 11, 12]. This separates

two questions:

order layer: existence and extrema,

metric layer: uniqueness and rates.

(2)

The separation is useful when q ≥ 1 prevents a contraction

argument but positivity still preserves order.