Monitoring Neural Training with Topology: A Footprint-Predictable Collapse Index

While neural collapse (NC) predicts that a $K$-class-balanced classifier should organize terminal representations as a $(K-1)$-dimensional simplex equiangular tight frame (ETF), modular addition consistently enters a different regime: networks compress to a two-dimensional cyclic geometry in which both classifier weights and token embeddings lie on circles. We refine the explanation of this phenomenon in three directions. First, we formalize a layerwise non-uniform training mechanism: downstream classifier weights are driven by dense cross-entropy gradients into a rank-2 equiangular configuration before upstream embeddings fully reorganize, and once this classifier plane forms, backpropagated feature gradients constrain embedding motion to the same plane while weight decay suppresses orthogonal components. Second, after this subspace locking, the induced in-plane dynamics admit an entropy-regularized transport interpretation on $S^1$; combined with modular-addition labels, this reduces embedding formation to phase alignment, whose minimizers are single-frequency characters of $\mathbb{Z}/P\mathbb{Z}$ and hence equal-angle points on a circle. Third, we quantify why this solution prevails over NC: a simplex ETF gains only an $O(1)$ advantage in cross-entropy, whereas the cyclic rank-2 solution enjoys a $Θ(K)$ advantage under Schatten or weight-decay surrogates, yielding a critical threshold $λ_{\mathrm{crit}} = Θ(1/K)$. Our results explain both why classifier weights move first and why embeddings subsequently align with them, showing that grokking on modular arithmetic is governed not by maximal separation alone but by a task-structured trade-off between separation, symmetry, and complexity.

Recursive learning -- where models are trained on data generated by previous versions of themselves -- is increasingly common in large language models, autonomous agents, and self-supervised systems. However, standard performance metrics (loss, perplexity, accuracy) often fail to detect internal degradation before it becomes irreversible. Here we identify a phenomenon we call silent collapse: under broad recursive conditions, model internal distributions -- predictive entropy, representational diversity, and tail coverage -- progressively contract even as conventional metrics appear stable or improving. We discover that silent collapse is not abrupt. Its onset is reliably preceded by three trajectory-level precursors: (1) contraction of anchor entropy, (2) freezing of representation drift, and (3) erosion of tail coverage. These signals manifest multiple generations before any degradation in standard validation metrics, enabling early warning. Based on these precursors, we propose the MTR (Monitor--Trust--Regulator) framework, a lightweight metacognitive loop that monitors trajectory statistics, estimates a slow-timescale trust variable, and adaptively modulates the effective learning intensity. MTR provides early warning and actively prevents silent collapse without requiring access to pristine real data -- a critical advantage when original data is unavailable, contaminated, or private.

Frontier large language model training consumes massive accelerator fleets and long wall-clock computation, making stability failures costly when they occur. After a numerical or a hyperparameter fault has already destabilized the training dynamics, it may continue for thousands of steps while loss and gradient norms still appear normal. We study mechanism-driven detection of training instability by deriving internal monitors from the functional role of each critical module and from the earliest computational sites where failures are expected to produce measurable signatures. For low-precision flash attention, we monitor the spectral entropy of a QK bilinear decomposition, whose first-order term becomes abnormal before the loss fully collapses. For MoE routers, we derive indicators from their role in expert selection. Our fault-injection experiments on low-precision attention, large learning-rate, and combined faults show that these signals provide distinct signatures for different failures, triggering thousands of steps before loss divergence.