From Zero-Dimensional Projection to the Real World: Heterodimensional Expansion and Alignment with the Standard Model plus Cosmology Framework Collaboration (Integrated Version)

We propose a minimal mathematical framework in which structure degrees of freedom emerge dynamically from an initially structureless zero-dimensional state in a Hilbert space H, driven by two non-commuting orthogonal projection operators P and Q. The framework is organized into five rigorous layers---functional analysis → operator algebra → representation theory → Lie algebra emergence → physical interpretation and holographic alignment---naturally yielding the Standard Model gauge group SU(3)×SU(2)×U(1), a hierarchical fermion mass spectrum, and quantitative alignment with cosmological observations (dark-energy density and the holographic area law). We clarify that the pivotal “i-axis” is a purely algebraic feature of the complex structure operator J in the operator algebra; it guarantees Lie-algebra closure and, under holographic duality, corresponds to an internal phase direction in symmetry space. The holographic principle (AdS/CFT correspondence) manifests here as a bulk-boundary measure correspondence in the heterodimensional space. All checkpoints are verified by strict proofs.