Logarithmic Closure Flow and the Origin of the Fine-Structure Constant in Time–Scalar Field Theory

The fine-structure constant α ≈ 1/137 governs the strength of electromagnetic interaction and appears throughout quantum electrodynamics, atomic spectroscopy, and particle physics. Despite its central importance, the numerical value of α is treated as an empirical input within the Standard Model.  This paper proposes a structural origin for α within the framework of Time–Scalar Field Theory (TSFT). Instead of introducing α as a fundamental coupling constant, we interpret it as a closure residue emerging from repeated scalar-time projection. Physical observables are required to remain stable under cyclic dimensional projection, which imposes a discrete spectral closure condition on the scalar-time evolution operator.  We show that stable closure requires the eigenvalues of the projection operator to lie on the unit circle, producing a quantized phase condition. When scale dependence is introduced through logarithmic scalar-time flow, the resulting closure phase generates an effective dimensionless coupling.  The fine-structure constant then appears as the minimal nontrivial closure residue of this operator. Within this framework, quantum electrodynamics is recovered as an effective theory operating on top of a deeper scalar-time closure structure. The numerical proximity of the resulting coupling to α^−1 ≈ 137 emerges as a natural consequence of minimal stable closure rather than an unexplained empirical parameter.