Sequence Position Explains an Apparent Complexity Gradient in Mixture-of-Experts Routing Entropy

Sequence Position Explains an Apparent Complexity Gradient in Mixture-of-Experts Routing Entropy. Preprint, version 1.1 (August 2026; version 1.0 June 2026). Not peer reviewed. Routing entropy, the Shannon entropy of the distribution a mixture-of-experts (MoE) router places over its experts at a token, is often averaged over a prompt and read as a measure of how hard the model is working. We show that this average inherits a positional effect that can masquerade as a complexity effect. On a 168-prompt suite graded into twelve levels of intended cognitive complexity, all-token mean routing entropy rises with level in two MoE models (DeepSeek V3.1, Spearman ρ=+0.80; Qwen3.5-397B, ρ=+0.62; n=168 each) and tracks prompt token count more tightly still (ρ=+0.88 and +0.78). Per-token routing entropy rises with position inside the prefill in every prompt of a per-token diagnostic, so longer prompts, which the grading also makes longer, pool more high-entropy positions and lift the mean. A readout at the final prefill token, a single position whose causal context is the whole prompt, removes the gradient (ρ=+0.02, p=0.82, and ρ=-0.06, p=0.42) and reverses the extremes: the short level-1 prompts show slightly higher final-token entropy than the long level-12 prompts in both models (p<0.001). The all-token pattern replicates on DeepSeek R1. In this setting the aggregate gradient is a function of token count and token position, so a between-prompt difference in averaged routing entropy is interpretable as a difference in what the prompts demand only when position or length is controlled; a position-controlled readout is the minimal control. Changes in this version. Single-thesis revision (August 2026): one claim carried abstract to conclusion, position diagnostic and final-token readout foregrounded, proposed-experiment lists removed, terms defined at first use. All version 1.0 numeric values retained. This deposit contains the paper (LaTeX source and built PDF), a claim-by-claim source index (SOURCES.md), and the supporting data directory, unchanged from version 1.0.