A Homotopy-Adequacy Test for Neural Output Heads

Neural networks are often asked to regress objects whose natural state space is not Euclidean: phases live on $S^1$, directors on $\mathbb{RP}^2$, and rotations on $SO(3)$. We give an exact design test for such output heads. For a head $(U,\rho)$ with degeneracy set $Z=\mathbb{R}^n\setminus U$, a target class $[f]\in\pi_k(M)$ is representable exactly when $[f]\in\operatorname{im}\rho_*$. A transversality corollary turns this into a fast rejection filter, $\operatorname{codim} Z\leq k+1$, but not a sufficiency test. The induced-map calculation rejects the visually natural $3$D radial head for $\mathbb{RP}^2$: $\mathbb{R}^3\setminus\{0\}\simeq S^2$ has trivial $\pi_1$, whereas the $5$D Veronese, or nematic $Q$-tensor, head is adequate. A double-cover control holds $U$, $Z$, and codimension fixed while changing $\rho_*$, making the distinction observable: all five incompatible runs show a macroscopic probe step ($0.25$–$2.07$ rad) versus $10^{-4}$ rad for compatible ones. The criterion concerns dense, sign-aware fields rather than every localization task. In a controlled noiseless realization of a non-trivial $\pi_1(\mathbb{RP}^2)$ loop, a five-seed experiment drives the Veronese top-eigenvalue gap from $1.47\pm0.04$ at the boundary to an interior minimum of $0.013\pm0.008$, while a dense boundary probe exposes a $1.05\pm0.33$ rad jump for the inadequate radial head versus $(7.5\pm0.7)\times10^{-4}$ rad for the Veronese head. The gap is a label-free proxy for distance to the readout's degeneracy locus.