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. From this construction chain, we further derive the geometric values of 35 fundamental physical constants, with zero free parameters and zero experimental anchors. The τ lepton mass (m_τ = 1. 7767 GeV) is derived purely geometrically from the S³ volume normalization, the orbit invariant I_τ = 32. 616, and the O4-C4 fluctuation correction T_τ = 4. 000058, agreeing with the experimental value of 1. 777 GeV to within 0. 017%. All 35 constants, including m_τ, are derived from the axioms with zero experimental input. These 35 constants fall into three precision tiers. First, 18 exact mathematical theorems: the gauge group structure U (1) ×SU (2) ×SU (3), the gauge coupling ratios 2: 3: 4, the high-energy Weinberg angle sin²θW=0. 6, the QCD vacuum angle θQCD=0, the neutrino mass-squared differences Δm²₂₁=7. 53×10⁻⁵ eV² and Δm²₃₁=2. 45×10⁻³ eV², the cosmic density fractions Ωb=5%, Ωdm=27%, Ωde=68%, the equation-of-state parameter w=−1, the Bekenstein–Hawking entropy formula SBH=A/4lP², the discrete curvature spectrum 0, 2, 5, and the critical dimensions DF=12, Dₛuper=10, DM=11, Dbosonic=26. Second, 2 pure theoretical predictions: the cosmological constant Λ=4π² and the area quantum Aₘin=4. 3944 lP², which currently lack direct experimental counterparts. Third, 15 experimentally comparable constants: the inverse fine-structure constant α⁻¹=137. 0359991847, the Higgs vacuum expectation value v=246. 2200 GeV, the electron mass mₑ=0. 51099893 MeV, the tau lepton mass m_τ=1. 7767 GeV, the muon-to-electron mass ratio m_μ/mₑ=206. 76828803, the tau-to-electron mass ratio m_τ/mₑ=3476. 9144, the six quark masses (mᵤ=2. 1580 MeV, mc=1. 2696 GeV, mₜ=173. 0000 GeV, md=4. 6650 MeV, mₛ=93. 3813 MeV, mb=4. 1809 GeV), the CKM matrix elements (Vᵤs=0. 2240, Vcb=0. 0410, Vᵤb=0. 0037), and the CP-violating phase δCP=66. 42°. All 15 experimentally comparable constants agree with measured values to within 1. 7%, with the majority deviating by less than 0. 1%; the fine-structure constant and Higgs vacuum expectation value match experiment at the 10⁻⁴% level. 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). (v) A universal discrete residual factor ηL2 = 1 + C (G) /N^rank (G) · Tr (Δ⁻¹|Type I⊕Type II) emerges from the S³ quantum curvature fluctuation at O (1/N⁴), unifying the corrections to all six quark masses and the cosmological constant. All its components—gauge group dimensions, curvature labels, lattice number N, and gauge rank—are rigid geometric invariants fixed by the axioms, requiring no external input. (vi) The framework yields 12 definitive predictions: 6 existential (L0) —including the exact gauge group, exactly three fermion generations, absolute proton stability, a unique Higgs scalar, charge quantization, and the absence of supersymmetric particles; and 6 precision numerical (L1) —including m_τ = 1. 7767 GeV, m_τ/mₑ = 3476. 9144, α⁻¹ = 137. 0359991847, mₜ = 173. 0078 GeV, normal neutrino mass ordering with Σm_ν = 0. 157 eV, v = 246. 2200 GeV, and δCP = 66. 4218°. To date, all 12 predictions survive experimental scrutiny with zero falsifications. 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.
Cheng Xi (Tue,) studied this question.