An Axiomatic Model: A Computational Motivation for the (5,2) Signature in Wyler's Fine-Structure Constant Formula

The principal contribution of this paper is *not* a first-principles derivation of the fine-structure constant α, but rather a *computational motivation* for the metric signature (5,2) underlying Wyler's 1969 geometric formula D_5 = SO_0(5,2)/[SO(5)×SO(2)]. We introduce an axiomatic computational framework that models every physical change as a Compare-And-Swap (CAS) operation — four explicit axioms plus one lemma (together with two working definitions, four framework-internal meta-rules, and two external assumptions; thirteen items in total, disclosed in §3.1) — and prove (Theorem 1, qualified as a Structural Proposition) that the irreversible/reversible classification of cost corresponds structurally to the positive/negative signs of a quadratic form. Among the eight possible sign partitions of the seven macroscopic degrees of freedom (four domain axes plus three CAS stages), (5,2) is the unique admissible partition; other simple Lie groups of the same dimension 21, namely SO(7) and SO(4,3), are explicitly excluded on categorical grounds. The framework-internal status of the Wyler formula factors is honestly classified: 9 is self-derived from the framework cost accounting; π^5/(2^4·5!) is directly identical to the Hua volume of the type IV_5 Lie ball (Hua 1963, Ch. IV §1.4 Theorem 1.1.2) — and since Theorem 1 forces SO_0(5,2) → D_5, this factor carries a framework-internal motivation; the denominator 8π^4 and the 1/4 exponent are classified as borrowings from the Wyler–Robertson reformulation (the 1/4 exponent is independently rediscovered via the framework's cost-ratio interpretation, but an axiom-level derivation is left for future work). Result: 1/α = 137.036082, with a relative deviation of 6×10^-7 from the CODATA value. A different combination of structural constants in the same axiomatic system yields the Weinberg angle sin²θ_W = 7/(2+9π) = 0.23122, agreeing with the MS-bar value at the M_Z scale (the agreement of the renormalization scheme is stated as a conjecture). All five steps of the derivation are framework-internal forward chains, forced by the framework's two-reading principle (cost reading vs norm reading) plus standard arithmetic. The internal degree count 7 = 4+3 (four domain axes + three CAS stages) coincides structurally with the classical Hamming code [7,4,3] decomposition (four message bits + three parity bits); the quantum Steane code [[7,1,3]] is a CSS construction built on this Hamming code, and is noted as a quantum lift rather than a direct correspondence. The baryon-to-photon ratio η_B = 6.14×10^-10 is computed from the two framework-derived quantities alone (α and sin²θ_W) and agrees with the Planck 2018 measurement (6.12 ± 0.04)×10^-10 to within 0.4σ. The framework-internal mechanisms supporting this estimate (the 1/π RLU identification ratio, the prefactor 2 from meta-level orthogonality) are explicitly designated as a working hypothesis (RLU coupled-access regime), falsifiable by future CMB precision measurements. The genuinely new contributions of this paper are: (i) the correspondence between cost classification and metric signature (Structural Proposition 1); (ii) the simultaneous framework-internal production of α and sin²θ_W from the same axiomatic system; and (iii) the structural match between the seven internal degrees of freedom and the Hamming code. We position the framework as a third, cost-theoretic branch of the axiomatic-reconstruction tradition, lying alongside the full-reconstruction school of Hardy 2001 / CDP 2011 and the partial-reconstruction school of Spekkens 2007 / Masanes–Müller 2011 / Pawlowski 2009. This v2.0 record contains both the English version (alpha137_en.pdf, primary, 44 pages) and the Korean version (alpha137_kr.pdf, 35 pages).