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

We present a symmetry-first account of Born weights for finite-dimensional quantum systems that does not take probabilistic postulates as primitive. The core observation is geometric: pure states are rays in \({\mathbb{C}}^{n}\), hence the operational state space is complex projective space \(CP^{n - 1}\), equipped with its natural \(SU{(n)}\) action. We show that this symmetry fixes, up to normalisation, a unique \(SU{(n)}\)-invariant Borel probability measure on \(CP^{n - 1}\), namely the Fubini–Study measure \(\mu_{FS}\)[1] (Ashtekar & Schilling 1999). Symmetry alone, however, does not determine how weights attach to specific outcome projectors. To close this gap without appealing to stochastic axioms, we introduce minimal operational consistency requirements for a probability assignment \(p{({P \mid \psi})}\) over projectors: normalisation on orthonormal resolutions, additivity on orthogonal projectors, noncontextuality, and unitary covariance. For \(n \geq 3\) these assumptions imply the quadratic form \({p{({P \mid \psi})}} = {\langle\psi|P|\psi\rangle}\) by a Gleason-class theorem[2], and the qubit case is handled by a standard strengthening such as extension to POVMs (Busch 2003). Finally, combining these Born weights with deterministic volume-typicality results from our companion work (Paper A) yields observed outcome frequencies as long-run volume ratios under measure-preserving dynamics. The result is a compact, finite-dimensional foundation in which Born statistics arise from symmetry plus operational consistency, with empirical content residing in the typicality and invariance assumptions.