Gauge Structure from Admissible Non-Injective Projection

Starting from the axiomatic foundation of admissible non-injective transitions, we derive the structure of gauge theory without introducing it as a postulate. Gauge charges are identified as invariants of equivalence classes in the projection fibre: two configurations are gauge-equivalent when they project to the same observable state. The group of fibre-preserving transformations constitutes the gauge group. SU (2) is identified as the unique minimal non-abelian admissible group from the quaternionic structure of the admissible fibre; U (1) arises as the residual phase symmetry; SU (3) is a structurally motivated candidate extension whose derivation remains open. The gauge connection arises as the infinitesimal generator of fibre transport under varying projection, and Yang Mills dynamics emerge from minimising the resulting curvature. No dynamical substrate is required at any step; the derivations rest solely on Axioms A1-A4 of the the program Foundations and the spectral results of the O-series. Effective spacetime geometry, requiring the continuum limit of Q5a, is left for a companion paper.