Interface-Controlled Evaluation and Size Transfer of Language-Model Equilibrium Solvers on Procedural Zero-Sum Matrix Games

We study language models as tool-free equilibrium solvers that read a serialized payoff matrix and write a strategy pair, and ask whether they can learn to solve unseen zero-sum games and whether named-game evaluations of such models measure the game or the prompt. Every answer is scored by an exploitability certificate that needs no equilibrium label. First, in an interface-controlled audit of Qwen3.5-9B on ten named base games with 12,600 released observations, the correct-label advantage is 24.6 points under raw prompting but 1.6 points under a matched chat template, and the interaction stays positive when base-game identities are resampled. Second, training on $2\times 2$–$3\times 3$ games transfers to unseen sizes under a fixed short-answer protocol: at $5\times 5$, the best-of-four success rate is 61% for LP-label SFT, 37% for a residual-reward GRPO recipe that uses no equilibrium labels, 16% for a chat-template base that answers validly in 93% of samples, and 15% for a format-only SFT control; SFT reaches 33% at pass@1. Third, the dominated-action padding control used to probe size effects needs calibration: a payoff-blind policy passes 45 of 50 padded $12\times 12$ games under range-normalized scoring but 6 of 50 at the embedded game's precision. We release the records, the calibrated control, and the scoring code.