Nerve Faithfulness Certifies Data Geometry in Learned Dictionaries

Reconstruction error and sparsity do not determine whether a learned dictionary preserves the geometry of the activations it encodes. We introduce nerve faithfulness, a certified comparison between data geometry and code co-activation geometry. For a non-negative code $M$, $K(i,i^{\prime })={\max }_j\min (M_{ij},M_{i^{\prime }j})$ is exactly the Dowker $1$-skeleton; after normalization it induces a metric $d_{\text{Dow}}$, and ${\delta }_{\mathrm{nerve}}=\frac{1}{2}\Vert d_{\text{Dow}}-d_X{\Vert }_{\infty }$ certifies an interleaving through the identity on samples. The certificate is hypothesis-free and costs $O(m^{2}n)$ dense or $O(m{k}^2)$ for $k$-sparse codes. It remains nearly invariant across a $27\times$ sweep of tuning width for a fixed antipodal polysemantic defect ($0.510$–$0.539$), while coherence varies by a factor of $5.4$. As a training objective, a correlation form combined with coherence raises rank faithfulness on rotated MNIST from $0.301$ to $0.826$, lowers the certificate from $0.516$ to $0.329$, and improves reconstruction. On released GPT-2 dictionaries with FVU $0.136$–$0.155$, code–data rank correlations of only $0.070$–$0.084$ show that good reconstruction can coexist with severe geometric mismatch. Nerve faithfulness adds a distinct, certifiable axis for evaluating and training learned dictionaries.