From Deterministic Counterspace to Stochastic Shadow: Deriving the Born Rule as a Projection Invariant in TCGS-SEQUENTION

The TCGS-SEQUENTION framework posits that observable 3D reality is a projection of a deterministic, non-linear 4D counterspace. A companion paper demonstrated that source-level nonlinearity is compatible with shadow-level no-signaling through the quotient map structure. This paper addresses the deeper challenge: why does a deterministic source generate probabilistic outcomes at the shadow level, and specifically, why the Born rule $P = |\psi|^2$? We show that the Born measure is the \emph{unique} probability assignment on the shadow state space $\Sspace$ that satisfies three geometric requirements: (i) fiber-independence (well-definedness on equivalence classes), (ii) foliation-invariance (independence of parameterization choices), and (iii) non-contextuality (Gleason's constraint). The projection geometry does not merely \emph{permit} the Born rule---it \emph{forces} it as the only consistent bridge between deterministic source dynamics and operational shadow statistics. Probability in TCGS is neither ontic randomness nor subjective ignorance, but a \emph{projection invariant}: the unique measure that renders the quotient map mathematically coherent.