Formalizes LLM representation manifolds by defining features as metric spaces, proving cosine similarity encodes geodesic feature distance, then validating homeomorphism and isometry empirically on colors, dates, and years.

Schaeffer et al. show LLM "emergent abilities" often reflect discontinuous metrics masking smooth capability growth. But grokking, induction heads, and Curie-point-style transitions prove genuine phase transitions exist.

From a statistical learning perspective, modern neural networks can indeed be understood as a large-scale maximum likelihood estimation (MLE) process. Specifically, a neural network is a parameterized function, and the most common way to train a neural network is to perform maximum likelihood estimation on the data.

The core purpose of UAT (Universal Approximation Theorem) is that neural networks can approximate any continuous function, and continuous functions can be approximated by polynomials (Weierstrass theorem).

Vectors are arrows, dual vectors are rulers—their pairing yields invariant scalars. Tensors are multi-slot linear machines with fillable slots, resolving the vector-in/vector-out vs. scalar-out paradox. The metric tensor bridges the two, keeping lengths and energies coordinate-invariant.