Zero-Dimensional Origin and Structural Expansion

We present a measure–operator framework in which structural degrees of freedom emerge from a zero-dimensional origin under unitary evolution. Structural processes and constraints are represented as measurable subsets of a configuration space and lifted to projection operators on a Hilbert space. A structural dimension operator is defined as the sum of these projections, and its expectation value quantifies the effective number of structural subspaces occupied by the state. The formalism unifies set-theoretic measure identities, projection algebra, and quantum dynamics within a single minimal axiomatic structure.