Bounded Local Generator Classes for Deterministic State Evolution

TL;DR

This paper introduces a class of local generators for deterministic state evolution on graphs, enabling scalable updates with constant per-step work regardless of system size.

Contribution

It defines the bounded local generator class (BLGC) with finite-range, local transformations that ensure efficient, size-independent incremental updates in graph-based systems.

Findings

Per-step update work is independent of total system size M.

The framework allows a Hilbert-space embedding with bounded operators.

It establishes a decoupling between global state capacity and computational work.

Abstract

We define a bounded local generator class (BLGC) for deterministic state evolution on graph-indexed systems. The construction consists of finite-range generators operating on bounded local state under deterministic composition. Each update acts only on a bounded-radius neighborhood and applies a bounded local transformation with projection onto a compact state domain. Under the BLGC constraints, per-step operator work remains independent of total system size M. Specifically, incremental update cost satisfies $W_t = O(1)$ with respect to $M \to \infty$ for fixed interaction radius $r$. The framework admits a Hilbert-space embedding in $\ell^2(V)\otimes \mathbb{R}^d$ and yields bounded operators under composition on admissible subspaces. The result establishes a structural decoupling between global state capacity and incremental computational work. The claims apply specifically to the bounded local generator class defined in this paper.