Routing as a Singular Reparameterization: A Closed-Form Pushforward Prior and Exact Bayesian Complexity in a Minimal Proxy

A two-branch routed regressor realizes exactly the function class of ordinary quadratic regression. Because routing does not enlarge the observable function class here, the routed model differs from quadratic regression only through the prior induced by routed coordinates. We compute this induced prior in closed form and show that fiber integration along the routing map creates a logarithmically singular density on the linear submodel. The induced posterior over quadratic coefficients then coincides exactly with the quadratic-regression posterior under the pushed-forward prior, isolating the role of prior geometry in this proxy model. For compactly supported priors on linear truth ($q_2=0$), the support-relative global real log canonical threshold (RLCT) is $(3/2,2)$ when the support reaches the collapsed stratum and $(3/2,1)$ when it avoids it; under the full-support Gaussian-coordinate prior, the global RLCT is $(3/2,2)$. Against smooth coefficient-space baselines with the same known noise variance on linear truth, this singular-versus-smooth prior comparison yields a rigorous $\log\log n$ Bayes-factor advantage. The same multiplicity-$2$ product singularity transfers locally to sigmoid gates on linear truth and extends locally to $d$-dimensional inputs.