Lemma: The Fine-Structure Constant as a Projection Impedance in Log-Harmonic Field Theory

This preliminary theoretical lemma presents a first Log-Harmonic Field Theory (LHFT) normal-form interpretation of the fine-structure constant alpha. Within Log-Harmonic Field Theory, alpha is not introduced as a detached numerical constant, but as the observer-readable permeability of the selected electromagnetic diagonal channel U (1) diag. Its inverse α⁻¹ is interpreted as the corresponding projection impedance after finite layer closure, Schur reduction, and observer compression. The manuscript develops the structural chain: F = 1 → cF = 5 → N_* = 2 cF² = 50 → M₂ (50), M₄ (50) → ρ₅₀ → K_αᵒbs (50) → α₅₀. The central LHFT readout is: α₅₀⁻¹ = 4π³ + M₂ (50) /16 − (7/16) ρ₅₀ − (1/16) ρ₅₀² + (2/3) ρ₅₀³ = 137. 0359991962043724444756547350674965. . . Equivalently: α₅₀ = 0. 00729735256330876619 The Alpha bridge is closed at the level of an explicit finite 1+7 Schur normal form, while the full microscopic derivation from the one-layer LHFT action S₁L remains open. The manuscript therefore uses the status phrase: “normal-form closed, microscopic derivation open. ” The document distinguishes between standard recovery physics, LHFT-specific ontological interpretation, finite normal-form calculation, measurement-space readouts, reproducibility notes, and open proof obligations. It does not claim to replace QED, atomic recoil metrology, CODATA adjustment procedures, or the empirical determination of alpha. Instead, it proposes a structural interpretation of what is being read as alpha in different measurement spaces.