Missing training categories leave structural imprints in learned representations, yet the mechanisms governing their detectability remain poorly understood. We formalize the aggregated posterior of a variational autoencoder (VAE) trained on K separable classes as an approximate K-component Gaussian mixture, whose negative log-density induces an energy landscape functionally equivalent to Modern Hopfield energy. Within this framework, removing a class corresponds to removing an attractor, producing predictable geometric deformations. We derive two key results in the analytical mixture setting: (i) a directed aspiration effect, whereby surviving modes shift toward the missing attractor, and (ii) a numerical corollary describing the emergence of a topological scar, a persistent void structure in the induced energy landscape. A central result is a pre-registered negative finding: a geometric proxy based on surviving-class resolvability fails to predict the topological detection optimum (1 PASS, 1 MARGINAL, 1 FAIL across ablation conditions). This demonstrates that the scar geometry is not reducible to cluster compactness, but instead reflects the structure of the missing volume itself. Empirically, we observe a consistent monotonic aspiration signal across five seeds (ρ = −0.25, p < 10⁻⁴), absent in autoencoder controls, supporting the role of variational regularization. Topological persistence increases with gap size in the expected order, though statistical significance remains borderline (ρ = 0.35, p ≈ 0.08). Overall, these results support the view that missing attractors induce a genuine geometric object — the topological scar — whose detection requires a void-centered rather than survivor-centered analysis. This work provides a theoretical companion to our prior empirical study (DOI: 10.5281/zenodo.19309874), and explicitly separates analytical derivations (Gaussian mixture setting) from empirical observations (trained VAEs), making the transfer assumptions testable.
Régis RIGAUD (Sat,) studied this question.