Toward a Derivation of the Fine-Structure Constant from Energetic First Principles: 𝛼 α as the self-consistency condition of the tetrads' Active function

This paper develops the E1P alpha-bridge programme: a structural route toward the fine-structure constant from Energetic First Principles. It brings together two prior results in the E1P framework: the electromagnetic closure result, which fixes the propagation scaffold and identifies vacuum impedance, and the self-similar cycling result, which fixes the golden-ratio scale grammar. The paper interprets the fine-structure constant as the compatibility condition between discrete source, discrete action, and vacuum ground-state balance. Under a compact tetrad-phase framework, charge and action fall into discrete spectra associated with the same compact phase structure. A nonlinear admissibility map is then introduced for the loading variable. In inverse-loading space this map becomes exactly affine, allowing a precise bridge to the restoration law from the self-similar cycling theorem. A central result is that the golden-ratio restoration grammar fixes the flow eigenvalue of the admissibility map. Under the magnitude-form bridge principle, the additive restoration law determines the universal linear gain and yields an explicit one-parameter flow for the loading variable. The remaining parameter is localized as metric data: the nonlinear saturation ceiling, equivalently the curvature parameter, or equivalently an AC aperture. The paper identifies the AC three-stream structure as the natural source of this metric normalization. In this form, the low-energy fine-structure constant is determined by the golden-ratio flow together with the AC aperture selected by the closure, spectra, and remainder separation. The paper presents the theorem structure, the compact-phase quantization framework, the Möbius–Riccati admissibility ansatz, the linearizing-coordinate lemma, the magnitude-form bridge principle, the projective-indeterminacy proposition, and the AC three-stream metric axiom that reduces the remaining numerical frontier to the derivation of a single geometric aperture.