Osmotic Selectivity as a Deterministic Function of the Fine Structure Constant

This paper derives the osmotic selectivity coefficient σ as a deterministic function of the fine structure constant α through a seven-step chain: α → Bohr radius a₀ → O–H bond length → water molecule diameter → aquaporin pore width → σ → Banach fixed point ω₀ = 0. Each link is either an exact definition, a quantum-mechanical eigenvalue, a geometric identity, or a constrained optimum confirmed by experiment. One conventional parameter (van der Waals radius of hydrogen) and one empirical input (measured pore width) enter the chain; neither affects the qualitative result: water passes, ions don't, q < 1, the fixed point exists and is unique. The fine structure constant formula α⁻¹ + S·α = 4π³ + π² + π (where S = Σ(2n−1)!!/(4n)!) is solved to 100 decimal places. The full series (23 terms to convergence) predicts α⁻¹ = 137.035999176335 from no measured inputs — only π, factorials, double factorials, and the integer 4. This value lies within 0.03 standard deviations of the most precise independent measurement (Fan et al. 2023, cesium recoil). A structural parallel is identified between the formula's self-referential series (antiparallel strands converging on a geometric axis) and DNA (antiparallel strands, four bases, exact complementarity). The product of the two strands reproduces the selectivity operator: INPUT × OUTPUT = S. The formula's internal selectivity S∞ ≈ 0.0417 falls within the biological σ range. The formula does not merely predict α — it encodes σ within its own structure.