Features have life history. And we should care

Language models trained on natural text learn to represent numbers using periodic features with dominant periods at $T=2, 5, 10$. In this paper, we identify a two-tiered hierarchy of these features: while Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways all learn features that have period-$T$ spikes in the Fourier domain, only some learn geometrically separable features that can be used to linearly classify a number mod-$T$. To explain this incongruity, we prove that Fourier domain sparsity is necessary but not sufficient for mod-$T$ geometric separability. Empirically, we investigate when model training yields geometrically separable features, finding that the data, architecture, optimizer, and tokenizer all play key roles. In particular, we identify two different routes through which models can acquire geometrically separable features: they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence and cross-number interaction, or from multi-token (but not single-token) addition problems. Overall, our results highlight the phenomenon of convergent evolution in feature learning: A diverse range of models learn similar features from different training signals.

Successive self-training on a language model's own outputs is widely characterized as a process of flattening: diversity drops, distributions narrow, and the text becomes "more like itself." We provide evidence that this characterization is incomplete. Across eleven generations of self-training on five models (GPT-2 124M, Pythia-410M, Pythia-1.4B, OPT-1.3B, Pythia-2.8B), language is not flattened uniformly -- it is restructured. Surface markers (discourse connectives, hedges, em-dashes) rise, while mid- and deep-syntactic structures (questions, parentheticals, passives, subjunctives) collapse. We formalize this asymmetric collapse as the Structural Depth Hypothesis (SDH): the per-generation decay rate of a linguistic feature is predicted primarily by its structural depth -- the number of nested syntactic dependencies it requires -- and only secondarily by its generation-zero output frequency. Pooling 17-feature panels from five models spanning three architecture families (N=85), the pooled Spearman correlation is rho=0.540 (p < 10^{-6}; cluster-bootstrap 95% CI [0.434, 0.634]), while frequency is a substantially weaker predictor (rho=0.225). A matched human-text fine-tuning control yields rho=0.039 (p=0.88), confirming the gradient is self-training-specific. We further document a Superficial Complexity Paradox: aggregate complexity proxies (dep-tree depth, TTR, word length) all rise as the underlying clause structure dies, with direct implications for training-data curation and LLM-text detection.

Neural networks acquire structured representations at specific moments during training, yet identifying these transitions typically relies on retrospective, label-dependent metrics. We introduce a bifurcation theory of representation dynamics to detect these moments in real time. Analyzing a passive GMM probe attached to the evolving encoder, we show the onset of structure corresponds to a supercritical pitchfork bifurcation driven by the loss Hessian. The system exhibits a theoretically predictable zero-crossing ($β_c$) that, compared to the network's current state ($β$), yields a dynamic ratio $β(t)/β_c(t)$: a universal, label-free phase coordinate for representation dynamics, computable entirely from hidden states. We empirically validate four distinct transition regimes predicted by this coordinate across diverse settings: SAEs on language models (Pythia), SSL (CIFAR), and grokking (modular arithmetic). Crucially, under finite dissipation, macroscopic symmetry-breaking can lag the initial zero-crossing by orders of magnitude, which providing a rigorous dynamical account of the delayed escape observed in grokking. Microscopically, the bifurcation creates a shared unstable subspace, forcing collective symmetry breaking. We term this the "feature lottery" in SAE training: a feature's terminal interpretability becomes predictable remarkably early. By only 5% of training, early atom purity robustly predicts final convergence purity, with top-decile early atoms achieving over 12x the baseline purity at convergence. Beyond explaining concept emergence, $β/β_c$ provides a practical early-warning indicator for training health, detecting the onset of usable structure, the crystallization of feature identity, and representational collapse epochs before downstream metrics react.