Artificial intelligence systems are commonly evaluated through task performance and behavioral imitation, but such evaluations leave open whether an artificial agent can acquire, stabilize, and use new lexical meanings from grounded experience. This paper introduces Lexical Consensus, an experimental framework for studying grounded word learning over a structured perceptual substrate. Using frozen DINOv2 visual embeddings, Carroll-style nonce words, and interpretable lexical learners plus linear baselines, we test whether agents can acquire artificial labels for visual concepts, generalize them bidirectionally, and stabilize them across controlled settings. The main result is a robust perceptual-coherence gradient: native categories are easiest to learn, coherent overextensions remain learnable, mid-range disjunctive concepts degrade, and far-disjunctive concepts approach chance. A pre-registered CIFAR-100 dissociation experiment confirms that this gradient is governed by perceptual distance rather than semantic relatedness: perceptual distance predicts acquisition accuracy (partial R^2 = 0.245, p < 1e-7), while semantic distance adds no significant explanatory power (partial R^2 = 0.002, p = 0.660). Bidirectional evaluation shows that naming and retrieval are distinct: exemplar-based mechanisms outperform centroid prototypes in label-to-image retrieval, exposing a memory-fidelity dimension separate from naming accuracy. Falsification controls, homogeneous candidate-pool evaluations, and null results on representational restructuring indicate that frozen perceptual geometry both enables lexical grounding and limits what can be acquired without representational adaptation.
We propose a unified mathematical framework for a geometric understanding of concept learning and neuron interpretation in sparse autoencoders (SAEs). While SAEs improve interpretability of neural networks by learning sparse feature representations, a principled definition of ''concept'' and ''learning'' remains unclear. We formalize concepts as sets of data points and cast concept learning as a set-alignment problem between human-defined and model-induced concepts. This formulation distinguishes three increasingly strong notions of learning -- detection, separation, and approximation -- and yields geometric conditions, error bounds, and capacity constraints for when concepts can be represented by individual neurons or multi-neuron units. It also provides a set-theoretic account for common SAE phenomena, including feature splitting, feature absorption, feature families, and hierarchical concepts. Finally, we connect concept learning and neuron interpretation through formal concept analysis, showing that the two directions need not agree and that their many-to-many structure can be organized by concept lattices. Experiments on synthetic data with ReLU and Top-$K$ SAEs illustrate the theory and reveal the effects of SAE size and sparsity on concept learning.