Emergent Models: Intelligence from Tiny Substrates

Emergent intelligence have played a major role in the modern AI development. While existing studies primarily rely on empirical observations to characterize this phenomenon, a rigorous theoretical framework remains underexplored. This study attempts to develop a mathematical approach to formalize emergent intelligence from the perspective of limit theory. Specifically, we introduce a performance function E(N, P, K), dependent on data size N, model size P and training steps K, to quantify intelligence behavior. We posit that intelligence emerges as a transition from finite to effectively infinite knowledge, and thus recast emergent intelligence as existence of the limit $\lim_{N,P,K \to \infty} \mathcal{E}(N,P,K)$, with emergent abilities corresponding to the limiting behavior. This limit theory helps reveal that emergent intelligence originates from the existence of a parameter-limit architecture (referred to as the limit architecture), and that emergent intelligence rationally corresponds to the learning behavior of this limit system. By introducing tools from nonlinear Lipschitz operator theory, we prove that the necessary and sufficient conditions for existence of the limit architecture. Furthermore, we derive the scaling law of foundation models by leveraging tools of Lipschitz operator and covering number. Theoretical results show that: 1) emergent intelligence is governed by three key factors-training steps, data size and the model architecture, where the properties of basic blocks play a crucial role in constructing foundation models; 2) the critical condition Lip(T)=1 for emergent intelligence provides theoretical support for existing findings. 3) emergent intelligence is determined by an infinite-dimensional system, yet can be effectively realized in practice through a finite-dimensional architecture. Our empirical results corroborate these theoretical findings.

Across machine learning, biology, and physics, independently evolving systems often converge toward strikingly similar high-level structures despite radically different microscopic details. Grokking circuits converge across random seeds, evolutionary lineages rediscover similar metabolic solutions, and renormalization flows approach common fixed points. We propose the Hierarchical Emergence Framework (HEF) as a candidate universality framework for such convergence phenomena. HEF models emergence as a phase transition in a mechanism landscape constrained by thermodynamic and information-theoretic laws. The framework introduces a critical energy threshold Ec separating an exploration regime with competing mechanisms from a convergence regime governed by a unique minimum-cost mechanism. Under structural assumptions, we prove physical feasibility, derive strict metric contraction, and establish convergence toward a unique fixed-point representation independent of initial conditions. We further connect this convergence structure to causal emergence through Effective Information and mechanism competition entropy. To test the framework, we study delayed generalization ("grokking") in modular arithmetic transformers across 111 experiments. We identify a reproducible empirical fingerprint of the Ec transition: the weight norm peaks systematically before grokking in 92% of runs. Normalized accuracy curves collapse onto a tanh kink (R^2=0.93) consistent with a Landau-Ginzburg universality class, and all grokked models converge to 0.9745+/-0.014 regardless of initialization, weight decay, or training fraction (ANOVA p>0.13). HEF is not presented as a universal theory of emergence, but as a falsifiable mathematical scaffold for studying convergence phenomena across complex systems.

Emergent misalignment (EM) is a phenomenon in which models generalize with narrow fine-tuning, leading to broad (yet uneven) misalignment across evaluation questions. We study EM and its variability directly through the components of fine-tuning: training dynamics, model priors, and data. (1) We first explored how in-domain training loss relates to out-of-domain alignment scores across datasets and model families. Then, we tried to induce potential alternative local minima through different learning schedules for one narrow fine-tuning, but did not find strong runs with better broad alignment scores conditioned on similar or lower training loss. (2) We found that although the mean and standard deviations of the misaligned model scores are usually statistically different from those of the pre-trained model, there are some potential signals on overall positive correlation. The evaluation prompt-only activations from both the pre-trained and the original instruct models (prior to narrow fine-tuning) could predict fine-grained alignment scores after narrow fine-tuning. (3) Finally, we compared activation deltas before and after narrow fine-tuning and found moderate-to-high subspace overlap and similarity between the resulting activation shifts for training and evaluation prompts. Subspace overlaps between training and evaluation prompt activations correlate with their shifts' similarities when measuring with the last prompt-token activations. The train-evaluation data prompt overlap is controlled against overlap computed from random vectors and evaluation prompts activations.