Discrete Sampling Geometry: A Rigorous Axiomatic Reconstruction

This paper constructs a self-grounding axiomatic system of primordial geometry. Unlike traditional axiomatic methods, the meta-axiom of this system is not an artificially presupposed initial assumption, but the necessary and unique consequence of the Principle of Complete Minimality (symmetry priority, structural minimality, parameter uniqueness) applying itself as the criterion through self-referential selection. Among all possible mathematical forms, only the unit interval I = [0,1] and its involution σ(x)=1−x can perfectly satisfy these three constraints. Based on this unique foundation, we first prove the categoricity of the axiomatic system—all models are uniquely isomorphic. We then rigorously construct the unique standard model, deriving a complete chain of geometric and algebraic constructions: natural numbers ℕ, ℤ, ℝ → circle S¹ → sphere S² → dihedral group D₁₂ → discrete curvature spectrum {0,2,5} → Lie algebra u(1)⊕su(2)⊕su(3) → spacetime S³×R → unified Einstein–Yang-Mills dynamics. Core results include: (i) The absolute length scale is uniquely determined by the intrinsic geometry 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 equally spaced points 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) (dimension 1+3+8=12), a decomposition uniquely determined by the irreducible representation structure of D₁₂. (iv) The minimal physical spacetime carrying this gauge structure is uniquely identified as S³×R (3+1 dimensions). Every step in the construction chain is uniquely forced by the axioms. The resulting geometric and algebraic structures reciprocally confirm the uniqueness of the meta-rule, forming a complete self-referential loop: "rule selects foundation—axioms generate structures—structures verify rules." The entire system has no external assumptions, no free parameters, and no empirical fitting. It is an internally consistent, self-contained theory of primordial geometry whose fundamental principle is: To exist is to be unique; the rule is the structure.