BU33 Hilbert Space under B_U Object-Layer Operator Formalism under Boundary-Admissibility

Abstract Hilbert Space under B_U: Object-Layer Operator Formalism under Boundary-Admissibility repositions Hilbert space within the B_U framework as an object-layer operator stage governed by boundary admissibility. The paper argues that Hilbert space supplies a highly effective formal arena for state representation, inner-product binding, orthogonal decomposition, closed-subspace projection, Cauchy-limit closure, and stable operator readout. Its mathematical completeness gives projection-layer background terms a closed domain in which approximation chains, high-frequency fluctuations, and microscopic residuals can be organized, stabilized, and cleared. However, Hilbert space does not grant standing by itself. Its vectors, subspaces, projections, spectra, and operators obtain physical significance only when admitted under single-universe boundary consistency, the fixed boundary B_0, and settlement projection \Pi_{B_0}. The core shift is from space-determinism to admission-determinism: dimensional freedom belongs to object-layer construction, while physical admissibility belongs to boundary-level adjudication. The note further clarifies that multidimensionality carries resolution gradients, projection partitions admissible domains, orthogonality performs stratified clearing, and completeness guarantees limit closure. In this sense, Hilbert space organizes projection; B_U grants admissibility; \Pi_{B_0} performs settlement; and single-valued standing locks the final readout.