Emergent Structural Dimensionality and Operator-Induced Complex Structure from a Zero-Dimensional Origin

We develop a minimal measure–operator framework in which structural dimensionality emerges from a zero-dimensional origin using real operator algebra. Structural driving and constraint are represented by measurable sets lifted to orthogonal projections on a real Hilbert space. The dimension operator (D) quantifies occupied structural sectors. Non-commutativity of the projections generates cyclic subspaces from the origin. On certain low-dimensional invariant subspaces a skew-symmetric generator induces a canonical complex structure (J) with (J² = -I). A positive-definite G/H ratio () provides irreversible growth along an entropy-directed parameter (s). When the effective generator is quadratic on higher-dimensional invariant subspaces, Hermite–Gaussian mode hierarchies and conjugate pairs appear. The framework clearly separates rigorous theorems from additional assumptions required for rotational and modal emergence.