From Measurement to Reality I(V) = γ²(V) and the Geometry of Binary Distinction

The Fisher information of a binary quantum measurement with visibility V equals the squared Lorentz factor: I(V) = 1/(1 − V²) = γ²(V). This mathematical identity, confirmed through six independent derivations, underlies a conformal equivalence between the Bures and Beltrami-Klein metrics on the qubit Bloch ball, unique to N = 2 via a Weyl tensor obstruction for N ≥ 3. Consequences are traced across quantum metrology, thermodynamics, population genetics, cooperative biology, and pure mathematics. Results are organized by epistemic confidence, with proven theorems separated from conjectural extensions. Falsifiable predictions — including a Thomas-Wigner rotation angle for sequential qubit measurements testable on existing hardware — distinguish this framework from existing alternatives. Three honest tensions constrain its scope. This is the expanded companion preprint to the mathematical proof published as DOI: 10.5281/zenodo.19363449.