Born's Rule from Geometry: A Non-Invertibility Route to Quantum Probabilities

We derive Born’s rule as a geometric theorem inside Charlton Geometry (CG), removing probability as a primitive axiom. The core mechanism is non-invertibility: the Charlton Operator η = δ − 2 ω ⊗ ω forgets polarity, since η(ω) = η(−ω), and therefore admits no global inverse back to Euclidean parent data. This lost state is compounded by intrinsic phase: a U(1) sector whose periodicity identifies θ ∼ θ +2π, further collapsing distinguishable microstate into a single Lorentz-observable macrostate. We formalize both effects as an observational quotient q : e S ↠ S with fiber group Z2 °ø U(1), and show that any observer confined to S must represent the unresolved fiber by an invariant measure. The Born distribution then appears as the unique gauge- and rotation-invariant pushforward of Haar measure from the non-invertible fiber to measurement outcomes, yielding P(i) = ∥Πiψ∥2 without postulating ψ, Πi, or the squared modulus rule. Crucially, the multiplicative composition of these probabilities is recovered not by postulate, but as a theorem of holonomy functoriality within the lift-bundle connection. In this view, quantum probabilities are not ontic propensities but a geometric accounting ofirrecoverable information discarded by reflection and phase identification.