Schrödinger Dynamics as the Unique Minimal Lift of Newtonian Mechanics in Charlton Geometry with a Lossy-Quotient Macrostate

We derive the canonical three-dimensional Schrödinger equation as the unique minimal local lift of Newtonian (Hamilton–Jacobi) particle dynamics on the ω-adapted spatial slice supplied by Charlton Geometry (CG). The geometric input is the CG substrate @ = (E4, δ, ω) with δ(ω, ω) = 1 and the Householder-reflected Lorentzian metric η := δ−2 ω⊗ω, which canonically defines a 3+1 split and spatial projector γ = δ−ω⊗ω. The observational input is the CG “noninvertibility route”: physically accessible macrostates form an observational quotient q : e S ↠ S by the lost-state group G = Z2 °ø U(1) with no preferred lift, so probabilities are forced as Ginvariant pushforwards and the Born density is fixed as |ψ|2. Given these inputs, and imposing a compact choice package on the lift representative ψ (U(1) covariance, norm preservation, correct ℏ → 0 Hamilton–Jacobi limit, and minimal local linear differential order: first in time, second in space), the only compatible evolution is iℏ ∂tψ =[− (ℏ^2 * ΔΣω) / 2m + V ] ψ. We also record (i) how the scalar sector generalizes canonically to a vector-potential sector by replacing ∇ with the covariant derivative of the CG-induced U(1) connection, where the electromagnetic potential arises from ω via A := η♭(ω) in the CG “reflected exterior calculus”, and (ii) falsifiers and open obligations, in particular the geometric origin of ℏ and the scale at which non-minimal corrections appear.