Born Rule from Projector Measures in SU(N)-Covariant Time–Scalar Spectral Geometry

We derive the Born probability rule as a theorem within the Time–Scalar Field Theory (TSFT) spectral-geometry program. Assuming only that measurement outcomes correspond to orthogonal spectral projectors on the TSFT Hilbert space and that probability assignments are additive on exclusive outcomes and noncontextual with respect to commuting decompositions, we obtain a unique projector measure. By Gleason-type arguments (with a qubit patch via ancilla embedding or POVM extension), the probability of an outcome projector P must take the trace form μ(P) = Tr(ρP) for some density operator ρ. For pure states this reduces to μ(P) = ⟨ψ|P|ψ⟩, yielding the Born rule without invoking external quantum axioms.