An Axiomatic Model: Axiomatic Derivation and Interpretation of the (5, 2) Signature in Wyler's Fine-Structure Constant Formula

v5.0 update Six figures added (vector PDF): structure of the Banyaequation, cost-signature correspondence, factor tracing of the Wylerformula, derivation chain, two forward chains, and the CAS structuralinterpretation of 137 = 128 + 9. v4.0 update Four axioms (orthogonality, atomicity, minimal cost, causality preservation) and one proposition define a discrete computational system whose sole primitive is the Compare-And-Swap (CAS) operation. Theorem 1 proves that the irreversible/reversible cost classification of its seven internal degrees of freedom (4 domain axes + 3 CAS stages) maps uniquely onto the (5,2) signature partition; SO(7) and SO(4,3) are explicitly excluded. This forces SO₀(5,2) ==> D₅ ==> the Wyler volume ratio, yielding 1/α = 137.036 082 (6×10⁻⁷ vs CODATA). All four Wyler-formula factors are accounted for by the axiomatic structure: 9 is axiom-derived (I), π⁵/(2⁴·5!), 8π⁴, and (·)1/4 are axiom-determined (II). A different combination of the same structural constants gives sin²θ_W = 7/(2+9π) = 0.23122 (4×10⁻⁶ vs MS-bar at M_Z). The baryon-to-photon ratio η_B = 6.14×10⁻¹⁰ (0.5σ vs Planck) is a forward-testable independent production. Sakharov's three baryogenesis conditions are structurally embedded in the CAS axioms. The CAS running mechanism reproduces the QED 1-loop β-function coefficient (denominator 3 = CAS stages) and the QCD β₀ numerator (7 = complete descriptive DOF). The CAS 3-stage structure maps to U(1), SU(2), SU(3) with 12 gauge bosons = C(4,2)×2. The internal degree count 7 = 4+3 coincides with the Hamming [7,4,3] code. Five unique falsifiable predictions are produced: higher-loop β-coefficients must contain CAS structural numbers; nearest integer to 1/α(M_Z) = 128 = 2⁷; J_CKM = 3.10×10⁻⁵; no 4th gauge interaction (Z'); Sakharov conditions embedded in CAS. All 38 results are classified into four epistemic tiers: axiom-derived (13), axiom-determined (9), axiom-described (15), out of scope (1). The complete system [10] has accumulated 1,096 derivation items (155 experimentally matched, 941 awaiting verification) and 130 unique predictions. This paper uses 4 of the 15 axioms in the complete Banya axiomatic system: Zenodo: https://doi.org/10.5281/zenodo.19562484 Figshare: https://doi.org/10.6084/m9.figshare.31987026 English (47 pp) + Korean (38 pp). The complete Banya axiomatic system (15 axioms): https://zenodo.org/records/19222015