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

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 sin²θW = 7/ (2+9π) = 0. 23122 (4×10⁻⁶ vs MS-bar at MZ). The baryon-to-photon ratio ηB = 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 (MZ) = 128 = 2⁷; JCKM = 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