The Fine-Structure Constant as a Finite Projection-Readout in Log-Harmonic Field Theory

The proof up to Hₛtr is already built into the mathematical core files 26. 04. 15LHFT₀1foundation. html and 26. 04. 15LHFT₀2ₘathematicalcore. html, because S₁L leads to the calibrated background, then to S^ (2), then to Kₛtruct, and finally to the structural Hilbert space Hₛtr with its spherical mode structure. reference: https: //zenodo. org/records/19637224 This preliminary theoretical lemma presents a Log-Harmonic Field Theory (LHFT) normal-form interpretation of the fine-structure constant α. In LHFT, α is not introduced as a detached numerical constant, nor as a smallest energy quantum or fundamental structural unit. Instead, α is interpreted as the dimensionless accessibility coefficient of the selected electromagnetic diagonal channel U (1) diagO within an admissible observer projection ΠO^Ψ. Its inverse α⁻¹ is read as the corresponding dimensionless projection impedance after finite projection-layer readout, Schur reduction, and observer compression. The construction introduces no adjustable free degrees of freedom in the α readout. It uses only long-established mathematical and physical methods: dimensionless coupling analysis, Hilbert-space decomposition, projection operators, block-operator reduction, Schur complements, normal-form analysis, quadratic Hessian expansion, finite-moment calculations, symmetry-channel selection, and standard electromagnetic recovery notation. This statement is therefore a normal-form interpretation using established tools. The manuscript develops the structural readout chain: F = 1 → cF = 5 → N_* = 2cF² = 50 → M₂ (50), M₄ (50) → ρ₅₀ → K_αᵒbs (50) → α₅₀ The central LHFT readout is: M₂ (N) = (N² - 1) / 12 M₂ (50) = 208. 25 width M₄ (N) = ( (N² - 1) (3N² - 7) ) / 240 M₄ (50) = 78020. 8625 stiffness ρ₅₀ = (23/110) * sqrt (M₂ (50) / M₄ (50) ) visible-hidden mixing degree of the closed 50-layer electromagnetic projection channel α₅₀⁻¹ = 4π³ geometric carrier of the visible projection channel + M₂ (50) /16 quadratic width of the closed 50-layer block − (7/16) ρ₅₀ leading Schur backaction of the hidden 7-dimensional complement − (1/16) ρ₅₀² quadratic self-compression of the visible channel + (2/3) ρ₅₀³ net cubic observer-compression term = α₅₀⁻¹ = 4π³ + M₂ (50) /16 − (7/16) ρ₅₀− (1/16) ρ₅₀² + (2/3) ρ₅₀³ = 137. 035999196204372444475654735067. . . Equivalently: α₅₀ = 0. 007297352563308766186. . . Calculation The Alpha bridge is closed at the level of an explicit finite 1+7 Schur normal form. In the deeper LHFT ontology, however, this closure must not be misunderstood as a zero-point structure or a smallest physical layer. Since u = ln (r/r₀) maps r → 0 to u → −∞, LHFT permits arbitrary structural zoom. Thus the finite normal-form statement may be written as: D_αNF = 0 ⇔ α = α₅₀ The current one-layer core is: S₁LDf, Ψ = ∫ ds du dΩ 1/2 G (Df) (∂ᵤ Df) ² − U (Df) + Lᵣel (Df, Ψ) + Lᵣes^ (fract) (Df, Ψ). Here G (Df) is the structural field-space metric, U (Df) is the log-harmonic structural potential, Lᵣel (Df, Ψ) describes the regular coupling between the structural contrast field Df and the structural wave field Ψ, and Lᵣes^ (fract) (Df, Ψ) contains the residual fractal and fine-structure sector. 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 α. Instead, it proposes a structural interpretation of what is read as α across different observer and measurement spaces.