Oxygen as Fixed Point: Wind, Blood, and the Constraint Surface of Existence

Molecular oxygen is maintained at a fixed-point concentration at every scale where chemistry operates: 98% arterial saturation in blood, 20.946% in the troposphere, and approximately 1% of all baryonic mass in the universe. This paper presents the oxygen fixed point as a cross-scale instantiation of the Banach contraction mapping, showing that the fine structure constant α enters the atmospheric chain through the Chapman cycle (σ ∝ α⁻⁴), that the Brewer-Dobson circulation is structurally isomorphic to cardiovascular circulation, that the Great Oxidation Event represents a discrete fixed-point transition, and that iron is the instrument of the oxygen fixed point at every scale from haemoglobin to the geodynamo to stellar nucleosynthesis. Fire and stellar fusion are identified as the same contraction mapping at chemical and nuclear scales, with photosynthesis as T⁻¹.