We develop a machine–independent complexity theory for programs realised as trajectories of “law–time semigroups” on computable metric spaces. Each program is implementedas a controlled discrete sampling of an effective semigroup with non–negative cost increments representing step counts, energy dissipation, or holographic boundary activity.Under four explicit effectivity and growth assumptions, we prove that the resulting total cost defines a Blum complexity measure in the sense of Blum’s classical axioms.Within this framework we identify a canonical cost built from persistence–graded gradient flows and a boundary “holographic observation quotient” (PFHS–HOQ). We prove that deterministic polynomial–time computability is representation independent: replacing the canonical cost by any programwise polynomially equivalent Blum measure preserves the class P. We also formalise an HOQ–based lower–bound scheme that, if instantiated in concrete models, converts geometric boundary constraints into complexity lower bounds.Examples illustrate the framework in finite–state and continuous settings.
Takahashi, K (Fri,) studied this question.