Sample-Only Agnostic Learning of Bounded Monotone Single-Index Models

We give a proper polynomial-time constant-factor agnostic learner for all bounded nondecreasing single-index links under unknown centered positive C2 densities p=exp(-V) with fixed global bounds mI <= Hessian(V) <= LI. The activation and normal are both unknown, labels are arbitrary, and the learner uses ordinary independent samples alone. It returns one normal and a monotone step activation with squared loss at most C_(m,L) OPT + epsilon, using samples and real-arithmetic operations polynomial jointly in dimension and inverse accuracy, with logarithmic confidence dependence. The proof replaces Gaussian rank-one moment alignment by a cone of conditional centered covariates and a constant-dimensional spectral direction list. A close-projection conditional replica-kernel estimate relates prediction variance to vector-field strength. Established threshold-energy initialization is combined with a sample-only non-Gaussian affine-halfspace initializer, and a finite branching tree preserves a useful refinement path. The theorem includes discontinuous bounded links. It retains fixed lower and upper curvature bounds and does not assert general log-concave, unbounded-link, prescribed-Lipschitz-output, uniform condition-number, or finite-bit guarantees.