Does sufficiently low curvature make neural networks easy to learn under the manifold hypothesis? Recent work nearly identified a sharp geometric boundary. It proved exponential hardness on manifolds of reach $O(n^\alpha)$ for every $\alpha<1/2$, while reach $\omega(\sqrt n)$ forces polynomial intrinsic volume and permits interpolation. The critical regime $\Theta(\sqrt n)$ was left open. We close this boundary on the hard side. For every large ambient dimension $n$, we construct a connected closed $C^\infty$ curve in $[0,1]^n$ with reach $\Theta(\sqrt n)$ and a full-support, polynomial-time samplable distribution whose density is within constant factors of arclength. Learning linear-width one-hidden-layer ReLU networks on this distribution from noise-free labels requires $\tau^2\exp(\Omega(n))$ full statistical queries of tolerance $\tau$. Under polynomial-modulus Learning with Rounding, no polynomial-time learner exists even with arbitrary output hypotheses. The construction replaces coordinate repetition by coding. We traverse a reflected Gray code of $\mathbb F_2^m$, embed its messages with an explicit systematic small-bias code, and smoothly round the resulting polygon. Character orthogonality gives a restricted isometry for every sparse signed measure needed by a chord and tangent. Federer's tangent formula then amplifies constant abstract reach to $\Omega(\sqrt n)$. The systematic coordinates preserve a linear Boolean prefix on all but an exponentially small fraction of local pieces, which yields an almost pairwise-independent family of continuous parity networks. This shows that the $\sqrt n$ threshold is a genuine critical point rather than a limitation of the previous construction.
Keywords: manifold hypothesis, hardness of learning, reach, data manifolds, neural networks, computational lower bounds