Algebraic Foundations of Free Hilbert Expansion Computing (HEC0)

This paper introduces Free Hilbert Expansion Computing (HEC₀) — a self-contained framework derived from the State-Parallel Computing (SPC) paradigm, describing the free-propagation limit where the governance cost (Ω₍gov₎ → 0) vanishes and semantics emerge from the physics itself. HEC₀ formalizes computation as the physical propagation of optical–electronic states in a graphene–photonics substrate. These states span polarization, amplitude, phase, and frequency subspaces that together form a composite Hilbert space governed by a realizable operator algebra (A_HEC). Within this algebra, the feasibility ideal (F) and feasibility mask (M) define the physically sustainable manifold on which trajectories naturally converge to ω-limit attractors representing computational outcomes. Unlike SPC, which enforces convergence through explicit governance, HEC₀ realizes the same denotational semantics intrinsically: dissipation replaces feedback, feasibility replaces control, and convergence replaces iteration. Computation is thus executed not by instruction or supervision, but by the substrate itself — a regime where governance dissolves and convergence becomes computation. Through this graphene–photonics physical semantics, HEC₀ establishes a foundational model in which evolution and computation are indistinguishable, revealing a substrate-native dynamics that redefines how information acquires structure in free Hilbert spaces.