Solver-Hard Is Not Model-Hard: A Hardness-Controlled Diagnostic for LLM Constraint Reasoning

Abstract

LLM constraint reasoners are often evaluated near the random-SAT phase transition, confounding density and solver hardness. We test instance-level transfer while near-matching clause density. At aligned size bins, with near-matched density and matched maximum clause width, we compare proof-hard expander-Tseitin and proof-easy ladder-Tseitin formulas, pigeonhole anchors, and density-mismatched controls. Theory separates their resolution hardness; a solver-specific Glucose mean-conflict proxy differs by up to $51\times$, and five other solvers preserve the direction. Across three included models (243 instances each; a fourth is excluded for abstention), the near-matched-density accuracy gaps range from $-32$ to $+20$ points, with a pooled gap of $+1.7$ points ($p=0.74$) and a wrong-signed correctness-versus-conflict association ($r=+0.15$). A proof-preserving relabeling lowers accuracy in all five clusters for one model (mean $-93$ points) but not another, exposing model-surface sensitivity. In a preregistered extension, provider-reported completion-token spend does not consistently increase with the proxy after accounting for formula length and censoring. At 16k, the reasoning model spends more on proof-easy matched formulas and exhausts its budget on the solver-easiest UNSAT family; the 32k C1 gap is absent. These scoped dissociations concern verdict accuracy and observed token spend, not certificate solving, exact proof length, or allocation efficiency.