Discrete Sampling Geometry: A Rigorous Axiomatic Reconstruction

This paper establishes a rigorous axiomatic system for discrete sampling geometry. The system is built upon a unique existence — the unit interval I = [0,1] equipped with an involution σ(x)=1−x — and three selective axioms: Symmetry Priority, Minimality Principle, and Parameter Uniqueness. We first prove the categoricity of this axiomatic system, demonstrating that all models are uniquely isomorphic. From this foundation, we rigorously construct the unique standard model, deriving a complete geometric and algebraic chain: the natural numbers N, Z, R → the circle S¹ → the sphere S² → the dihedral group D₁₂ → a discrete curvature spectrum {0,2,5} → the Lie algebra u(1)⊕su(2)⊕su(3) → space-time S³×R → the unified Einstein–Yang–Mills dynamics. Key results include: (i) The absolute length scale is uniquely fixed by the intrinsic length of the quotient space M = I/σ, yielding a sphere S² with circumference C=1 and constant curvature K=4π². (ii) The dihedral group D₁₂ emerges as the unique maximal symmetry group on the 12-point equi-partition of the great circle. (iii) Under the refinement limit, the operator algebra on the 12-dimensional regular representation subspace V₋ contracts to the compact Lie algebra u(1)⊕su(2)⊕su(3), with dimensions 1+3+8=12 uniquely determined by the irreducible representation structure of D₁₂. (iv) The physical space-time S³×R of dimension 3+1 emerges as the minimal manifold carrying the SU(2) gauge bundle. Each step in the construction chain is uniquely forced by the axioms. The theory admits no free parameters, no external choices, and no empirical fitting. The paper forms a closed categorical axiomatic system whose fundamental principle is: to exist is to be unique.