Three Independent Derivations of the Universal Evolution Constant in UD Theory Converging to the Fine Structure Constant

We derive the universal evolution constant E of UD theory from first principles along three independent paths. The first traces the sequential cosmic constraints that must be simultaneously satisfied: symmetry breaking from the critical state $U = D = 1/2$, vacuum rejection e^π, singularity rejection π, the input-output ratio $3/2$ from the ontological classification of irreducible entities, and the quantum projection 1/√2. The second derives the matter condensation threshold from the S → M conversion: when the spatial-form fields Uᵤ + Ud convert to the material-form fields Dd + Dᵤ, the Hessian analysis directly yields the threshold strength carrying the same five-factor structure; the inverse M → S conversion, i.e., matter annihilation, is shown to obey the identical threshold by the U ↔ D duality. The third derives the electromagnetic coupling directly from the electron–photon vertex: the electron, a Dd–Dᵤ bound state, couples to the photon, a Ud–Dᵤ collective excitation, via the fundamental UD interaction. All three paths converge to E = 3/(4π e^π √2), giving E⁻¹ ≈ 137.082, which differs from the CODATA 2022 value αₑₓₚ⁻¹ = 137.035999084 by $0.034%$. This deviation is quantitatively accounted for by low-energy QED vacuum polarization corrections. The numerical identity across three independent physical roles establishes E as α. An observational prediction for the S → M conversion threshold in cosmic voids is given.