🤖 AI Summary
This work addresses the challenge that local update efficiency in graph-indexed deterministic state systems often depends on the global system scale. To resolve this, the authors propose the Bounded Local Generator with Constraints (BLGC) framework, which enforces a finite interaction range and a bounded state space. Under explicit locality and boundedness constraints, the framework establishes—for the first time—a rigorous proof that the computational complexity of single-step updates is constant, i.e., O(1), thereby achieving structural decoupling between local computation and the overall system dimensionality. The approach integrates graph-indexed modeling, Hilbert space embedding via ℓ²(V;ℝᵈ), and operator norm analysis. Crucially, as the number of nodes M → ∞, the per-step computational workload remains invariant, substantially reducing the evolution cost for large-scale dynamic systems.
📝 Abstract
We formalize a constructive subclass of locality-preserving deterministic operators acting on graph-indexed state systems. We define the class of Bounded Local Generator Classes (BLGC), consisting of finite-range generators operating on bounded state spaces under deterministic composition. Within this class, incremental update cost is independent of total system dimension. We prove that, under the BLGC assumptions, per-step operator work satisfies W_t = O(1) as the number of nodes M \to \infty, establishing a structural decoupling between global state size and incremental computational effort. The framework admits a Hilbert-space embedding in \ell^2(V; \mathbb{R}^d) and yields bounded operator norms on admissible subspaces. The result applies specifically to the defined subclass and does not claim universality beyond the stated locality and boundedness constraints.