Regularity, Phase Transitions, and Uniform Inference for Proximal Counterfactual Quantile Processes

This paper develops semiparametric theory for counterfactual distribution, quantile, and lower-tail risk processes under unmeasured confounding using proximal negative-control proxies. Rather than treating each threshold as a separate proximal mean problem with outcome $\mathbf 1\{Y\le y\}$, we study the continuum of inverse problems indexed by $y$. For each treatment arm $a$, the counterfactual CDF $F_a(y)=P\{Y(a)\le y\}$ is represented by the primal bridge equation $T_a h_{a,y}=g_{a,y}$ and the linear functional $\ell(h)=E\{h(W,X)\}$. The dual bridge $q_a$ solves $T_a^*q_a=1$, equivalently $E[\mathbf 1(A=a)q_a(Z,X)-1\mid W,X]=0$. We show that this dual equation, together with the minimal residual-moment condition required for the influence function to lie in $L_2(P_0)$, is the exact regularity boundary in a threshold-saturated observed-data proximal bridge model: $F_a(y)$ is pathwise differentiable if and only if a regular square-integrable dual bridge exists. The canonical gradient is $$h_{a,y}(W,X)-F_a(y)+\mathbf 1(A=a)q_a(Z,X)\{\mathbf 1(Y\le y)-h_{a,y}(W,X)\}.$$ A singular-system characterization gives a Picard-type phase transition: root-$n$ regular estimation is possible exactly when $\sum_j\ell_{a,j}^2/s_{a,j}^2<\infty$ and the residual moment is finite. Outside this region, finite-dimensional efficiency bounds diverge under residual-noise nondegeneracy, and Gaussian inverse benchmarks yield slower minimax rates. We further establish efficient CDF-process inference, cross-fitted uniform doubly robust expansions, finite-rank weak-proxy rate conditions, density-free simultaneous quantile bands by inversion of CDF bands, and lower-tail CVaR inference via a shortfall representation. The estimators rely on closed-form linear algebra, convex Tikhonov regularization, and isotonic projection for shape enforcement.