Pre-Geometric Topological Lineage of Gauge Symmetry Groups

The programmatic layer contains only the ℤ₂ double-sheet involution primitive; there exist no manifolds, Lie groups, fiber bundles, dimensions, or any other geometric objects at this level. U(1), SU(2), and SU(3) are none of them pre-given foundational symmetry axioms of the program; they are all, at the computational layer, topologically generated from the program-derived S⁰ as seed: multi-stage suspension unblocking generates the topological base spheres, and bundle construction then generates the full gauge-group spaces, which are synchronously projected onto the phenomenal layer as the carriers of interaction gauge dynamics. Throughout this chapter we strictly distinguish: programmatic primitive → computational-layer topological generation (suspension base + bundle construction) → phenomenal-layer gauge dynamics, without any cross-level violation.