Language Models Compare Quantities Using Number-specific and Unit-specific Heuristics

Do language models preserve the ordinal meaning of intensity words when those words must produce numeric actions? I study a researcher-constructed scale of 10 English degree modifiers, from slightly to drastically, informed by the Quirk et al. degree-modifier taxonomy, in a controlled resource-allocation environment where Claude Haiku receives a natural-language instruction, produces a numeric allocation, and a deterministic backend converts that allocation into a measurable outcome. The only variable that changes between runs is the intensity word or the starting system state, isolating their effects on the model's numeric output. Across 6,620 runs at T=0.0 and T=0.7, three patterns emerge. First, the model compresses 10 intensity words into 5 distinct median outputs: four lower-tier words all map to the same value, while stronger words break into higher regimes (Spearman rho = 0.845, p < 0.001). Second, when the current system state is supplied as context, separate Kruskal-Wallis tests show that grouping by starting allocation captures far more rank-based variance than grouping by word (epsilon-squared baseline = 0.782 vs. epsilon-squared word = 0.079), and lexical differentiation collapses to zero as the system approaches capacity. Third, near feasibility limits the model exhibits three behavioral modes: weak words hedge with small adjustments, strong words abstain entirely, and the word drastically pushes to the local ceiling. These patterns persist across temperature, with stochastic sampling broadening distributions but not restoring ordinal distinctions between words. In this model and domain, the model's numeric interpretation of vague intensity words is compressed, state-dependent, and discontinuous near operational boundaries.

Large language model optimization is an active research area, spanning quantization of model weights, early-exit methods for skipping layers, and speculative decoding. Each track uses its own quality measures, typically an idiosyncratic benchmark score. Few approach the measurement precision required by other scientific disciplines. We propose a rigorous methodology for measuring output quality, suitable for cross-system and cross-technique comparison. We score outputs with an LLM as a judge, but calibrate the judge formally: we compare its scores on two ordinary runs of a model given the same prompts, verifying that it shows no systematic preference between statistically equivalent outputs and measuring its per-sample noise. Each design also includes a 'null' condition, provably identical in distribution to the unmodified model, whose measured difference must be zero. With this one instrument we measure several acceleration techniques on the same prompts, so their quality costs can be compared. Perceived quality proves highly dependent on the domain of discourse. A 4-bit model was indistinguishable from its 16-bit original down to our design's +/-0.3-point resolution, in English prose and Chinese alike. At 3-bit precision the same prompts lost 0.5 points in English prose, 0.9 in Chinese, and 1.1 on multi-step math; early exit that cost 0.7 points on prose cost 2.5 on math, cutting correctly solved problems from 19 of 27 to 6. The pattern held for models from Alibaba and from Meta, but not its magnitude: the same quantizer cost Meta's model 1.8 points where it cost Alibaba's 0.7. A model's certainty about a token predicts how likely it is to differ from the full model's choice, but not how much that difference affects judged quality, so acceptance rules relying on certainty cannot distinguish errors that matter from errors that don't.

Standard evaluation of large language models assumes stable model rankings across inference conditions. We challenge this assumption by varying the token generation budget, i.e., the maximum tokens a model may produce, across seven levels (64--4,096), evaluating four models on three reasoning benchmarks (56,476 inferences). We report four findings: (i) 3--19% of items exhibit non-monotone behavior (accuracy decreasing with more budget), even after controlling for truncation, and this phenomenon is model-specific (cross-model overlap: 6--14%). (ii) Model rankings reverse across budgets on all benchmarks ($p {<} 0.01$, McNemar). (iii) Oracle analysis reveals model complementarity up to $+27.8$pp, most pronounced at constrained budgets. (iv) A budget-aware router captures 14.1% of the oracle gap cross-domain; budget features help within-domain ($+1.6$ to $+5.7$pp) but are domain-specific and hurt transfer ($-1.2$pp). These results argue for budget-conditioned evaluation protocols.