Three Measurable Failure Modes of Large Language Models

Human language is inherently ambiguous - not a deterministic code but an ensemble of overlapping meanings whose disambiguation depends on context that is often incomplete or absent. A system that processes natural language must therefore be probabilistic, not by architectural choice but by mathematical necessity. This paper argues that the resulting uncertainty has structure: what the field calls hallucinations is not one phenomenon but three structurally distinct failure modes of this probabilistic nature, each with a different causal origin, a different measurable signature, and a different class of solutions. Mode 1 (autoregressive reinforcement) is the self-consistent wrong trajectory produced when an error contaminates the model's own conditioning context. Mode 2 (confabulation) is fluent generation produced from parameter directions that received no training signal - the null space of the weight matrix. Mode 3 (irreducible uncertainty) is the correct response of a calibrated probabilistic system to a genuinely ambiguous query. Each mode has a computable quantitative metric: correction sensitivity (CS), dimensional excess (DE), and output entropy (Hₒᵤₜ). The three measurements rest on a single coding-theoretic construction, the syndrome table S = N(J̄ · V)^, whose full derivation is in the companion paper "A Syndrome Algebra for Differentiable Parametric Systems". A controlled experimental series on a synthetic LSTM ($D=256$, $L=10$, six fixed seeds) confirms the framework end to end. The three metrics separate cleanly: the CS gap between known and unknown domains narrows monotonically from 0.273 ± 0.095 at $k=1$ to 0.067 ± 0.037 at $k=10$. The Pearson correlation r(DE, CSunknown) = 0.9896 across k predicts out-of-domain failure from weight matrix alone. Causal localisation of an injected perturbation reaches $100%$ accuracy over $180$ trials with a pre/post residual ratio of approximately 2× 10⁸. Oracle correction is exact (cosine $1.000000$ over $36,000$ trials). A direct comparison of multicellular specialists against monolithic generalists shows the Singleton-bound multicellular advantage grows from 0.158 ± 0.049 at $N=5$ to 0.310 ± 0.054 at $N=10$ in CS gap, empirically justifying the modular hierarchy. Additional notes: This preprint is accompanied by the mathematical paper A Syndrome Algebra for Differentiable Parametric Systems (see related identifiers). Code and data are available at the linked GitHub repository. Model weights are not included due to size; they are regenerated deterministically from the provided scripts and canonical seeds.