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 $(\mathsf{CS})$, dimensional excess $(\mathsf{DE})$, and output entropy $(\mathsf{H}_{\mathrm{out}})$. The three measurements rest on a single coding-theoretic construction, the syndrome table $S = \mathcal{N}(\bar{J} \cdot V)^\top$, 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 $\mathsf{CS}$ gap between known and unknown domains narrows monotonically from $0.273 \pm 0.095$ at $k=1$ to $0.067 \pm 0.037$ at $k=10$. The Pearson correlation $r(\mathsf{DE}, \mathsf{CS}_{\mathrm{unknown}}) = 0.9896$ 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 $2\times 10^8$. 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 $0.158 \pm 0.049$ at $N=5$ to $0.310 \pm 0.054$ at $N=10$ in $\mathsf{CS}$ 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.