Interval-Lattice Fixed-Point Semantics for Certified Implicit Hypergraph Neural Operators
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.
Volume & Issue
Vol. Volume 6 / Iss. Issue 1