STW Foundation: Compact Boundary Closure, Character Lattices, and Core Quantities

This paper presents the foundational construction of the Space–Time–Wave (STW) boundary-closure framework, a speculative geometric model in which particle sectors are represented as projected states of a common compact boundary system. The construction separates four-dimensional spacetime geometry from the finite internal phase support and formulates the latter through an admissible compact-cell projector, its character lattice, and the associated closure holonomy. The minimal admissible cell has rank 48, while removal of the common carrier direction produces a 47-dimensional active closure lattice. Together with the reduced fold contribution and charged self-loading correction, this defines the effective closure depth nSTW and distortion scale DSTW. These quantities provide shared input for downstream neutrino, charged-lepton, quark, boson, and hadron calculations. A common variational architecture is introduced in which sector spectra arise from projected Hessians of the reduced boundary functional, while mixing and decay amplitudes are assigned to its higher variations. The paper also distinguishes functional quark masses, unrenormalized boundary endpoint anchors, and externally matched running masses. The framework is presented as a computable reduced normal form rather than a completed microscopic or ultraviolet theory. Derived statements, normalization-dependent identifications, sector assumptions, conditional predictions, and unresolved proof obligations are explicitly separated. The purpose of this foundation is to establish a consistent source of notation, dependencies, and mathematical structure for critical evaluation of the accompanying STW sector papers. The supplement papers contain reduced boundary action derivation and technical note.