Fixing the Measure: Deriving |Ψ|² From Symmetry in Deterministic Geometry

This paper derives the Born rule from first principles by identifying the unique measure over complex Hilbert space that is invariant under two physically motivated symmetries: complex-scaling homogeneity, and unitary covariance. Assuming only that quantum amplitudes \(\Psi\) inhabit a finite-dimensional Hilbert space \({\mathbb{C}}^{n}\), we show that the only measure consistent with deterministic, volume-preserving dynamics and these symmetries is proportional to \(|\Psi|^{2}\). This result explains the empirical success of the Born rule as a geometric necessity, not a probabilistic axiom. When applied to systems with disjoint outcome regions, as in volume-based formulations of branching dynamics, this measure yields outcome frequencies that match quantum predictions exactly. The derivation introduces no probabilistic or postulated elements beyond geometry and symmetry. Outcome weights arise solely from the invariant structure of finite-dimensional amplitude space under deterministic, volume-preserving flow.