The Trunk–Branch Orthogonal Projection of the Topological Ladder: A Unified Homology Architecture for the Gauge Trichotomy, Particle Triple Degrees of Freedom, and the Cosmic Trichotomy of Scales

From the odd-order suspension ladder and the level-truncation axiom of the GDUM pre-geometric three-layer architecture, all "triple structures" within the system are not isolated coincidences, but originate from a single rigorous trunk: a second-order, doubly-branched orthogonal projection mechanism. The three gauge channels U(1)/SU(2)/SU(3) constitute the unique topological trunk of the cosmic trichotomy; the threefold internal degrees of freedom of particles (color, flavor, generation) and the threefold cosmic spatio-temporal scales (ordinary matter, dark matter, dark energy) are two mutually orthogonal second-order branches projected outward from this trunk. The three trichotomy spectra share the same set of topological constraints, yet differ completely in hierarchical attribution, projection space, and physical realization; no simple one-to-one correspondence exists among them.