Model Collapse is a Singular Complexity Trajectory

Recursive training on synthetic data can drive generative models toward collapse, yet most explanations treat collapse as a statistical phenomenon: tail loss, variance shrinkage, or distributional drift. We develop a singular-geometric view of collapse. In a labeled Gaussian mixture abstraction of recursive self-training, we show that mode extinction induces a monotone trajectory in the real log-canonical threshold: as recursive resampling eliminates active modes, the overcomplete mixture moves through increasingly singular strata, with strictly decreasing singular complexity and growing Fisher-information nullity. We further show that this loss of identifiability begins before exact extinction: as a component's weight becomes small, Fisher curvature in its associated parameter directions shrinks proportionally, so exact singularity is the endpoint of a continuous near-nullity trajectory. Thus collapse corresponds not only to reduced distributional diversity, but also to a progressive loss of locally identifiable parameter directions. This theory separates the singular geometry of collapse from the multinomial/Wright--Fisher absorption dynamics. We also prove that, in the mixture abstraction, injecting enough real data prevents component extinction and keeps the singular-complexity trajectory from descending over a fixed horizon with high probability. We then use the theory to motivate loss-landscape diagnostics for neural recursive training, including Hessian spectra, Gauss--Newton effective rank, decoder-block curvature, feature-rank concentration, and local learning coefficient estimates. Across Gaussian mixtures, VaDE, standard VAEs, and a compact DDPM, we find that synthetic-only recursion exhibits the predicted geometric degeneration, that VaDE geometry diagnostics predict future component loss, and that real-data injection stabilizes both distributional and geometric collapse metrics.