Born Rule and Indivisibility from the Moment Polytope of the Probability Simplex

This paper derives three theorems and one corollary from a single geometric construction: the moment-map projection of unitary dynamics on the projective state space CP^ (N−1) onto the probability simplex Δ^ (N−1). Theorem A establishes that the projected stochastic process is generically indivisible in Barandes' sense, with the qubit case recovering Barandes' canonical |Uₖj|² example exactly. Theorem B derives the Born rule in two components: the uniform Dirichlet distribution on the simplex arises as the PU (N) -Haar pushforward of the Fubini–Study measure on CP^ (N−1), and for a specific state the Born probability of outcome k is the k-th moment-map coordinate |ψₖ|² / ‖ψ‖². Theorem C shows that the class of moment-map-projected unitary processes coincides with Barandes' QM-correspondent class. As a corollary, the per-branch quantum-potential structure that Lohmiller and Slotine (2026) formulate as a spatial claim about ΔM √ρⱼ / √ρⱼ has its natural home on CP^ (N−1), where the modulus-direction Fubini–Study Laplacian admits the closed form ΔFSₘod √pₖ / √pₖ = 1/pₖ − (2N−1) globally on the simplex interior. The spatial form is the Kähler–Lagrangian section translation; outside this regime it acquires the standard Madelung–Bohm quantum potential. Vattay (2026) recently identified the spatial form's failure mechanism explicitly via a published comment on Lohmiller–Slotine; this paper provides the complementary geometric resolution. The Foldy–Wouthuysen transform from relativistic quantum mechanics appears as a one-parameter geodesic in PU (4) under the bi-invariant Fubini–Study metric.