Meta-Geometry: From Two Axioms to Thirty-Five Physical Constants

Starting from two minimalist geometric axioms—Structure (Minimal Completeness: existence is the minimal discrete structure bearing maximal symmetry) and Dynamics (Symmetry Breaking: evolution is the irreversible breaking of symmetry toward proper subgroups under refinement limit)—we rigorously derive, through exhaustive mathematical screening, the core theorem -: the two-dimensional sphere as the continuous geometric carrier, and the 24-order dihedral group as the discrete symmetry basis, whose coupling achieves the ontological unification of continuous and discrete geometry. On the mathematical side, this framework repairs the foundational fissures left by the three historical crises of mathematics, anchoring the geometric boundaries of the Continuum Hypothesis, the Axiom of Choice paradox, and Gödel incompleteness. On the physical side, continuing the 300-year geometrization of physics, we prove that the four fundamental interactions are geometric connection fields derived from adjacency edges of different angles on ; all thirty-five free parameters of the Standard Model—gauge group structure, fine-structure constant, fermion mass spectrum, CKM matrix elements, cosmological density parameters, quantum gravitational area quantum—are invariants of the representation theory of , uniquely derived from geometric first principles. The entire system has zero free parameters, zero external inputs, and zero empirical fitting. Two axioms, three meta-rules, one core theorem, thirty-five constants, zero free parameters.