A Two-Stage Structural Control Model of LLM Reasoning

We show that a compression-based structural metric $\phi(x) = \log(1 + K(x)/K_0)$, computable via \texttt{gzip} in microseconds, governs LLM reasoning success through a gate mechanism rather than computational throughput. Across five models (0.5B--7B parameters, two architecture families), GSM8K, and a 90-task suite, we find: (1)~$\phi(x)$ predicts accuracy with all GSM8K correlations and three of four 90-task results reaching significance; (2)~this survives after controlling for text length (partial $\rho = -0.235$, $p < 10^{-6}$); (3)~quintile accuracy drops from 68--80\% to 28--40\% across model scales. Crucially, a controlled runtime analysis with manual decode-loop timing reveals the conditional independence $\phi_{\text{pre}} \perp T_{\text{decode}} \mid N_{\text{tokens}}$ ($p > 0.25$, three models), falsifying the direct Structure--Time hypothesis. Problem structure controls \textit{whether} reasoning succeeds (the gate), not \textit{how long} it takes (the pipeline). We propose $\phi(x)$ as a zero-cost structural difficulty proxy for reasoning tasks.