The Bridge Operator as Gauge-Gravity Unifier: One Fisher-Study Metric, Two Readouts in Two-Sided Closure Theory

For any smooth family of orthogonal projectors P on a Hilbert space, parametrised by coordinates xA on a manifold, the bridge operator BA: = Q (∂A P) P (with Q = I − P) defines a real quadratic form gAB: = Re tr (BA† BB). This is the quantum Fisher information metric on the projector Grassmannian. In Two-Sided Closure Theory (TSCT): when P is the Jones projector on a (3+1) D null Rindler wedge, gAB is the gravitational screen metric, and the identity P P̈ P = −2 B†B is the Raychaudhuri focusing law. When P is the generation projector Pᵥ₀ on CP², the same gAB is the Yukawa coupling geometry, and the same identity is the RG second-variation seed. Spacetime geometry and Yukawa geometry are therefore pullbacks of the same Fisher-Study metric along different parametrising maps. This is a metric-level gauge-gravity unification without compactification, extra dimensions, or an embedding larger gauge group.