When Low-Rank Payoff Compression Changes Equilibrium Structure

Low-rank compression is a common way to shrink the payoff matrix of a large zero-sum game, and the best rank-$k$ spectral approximation is the default compressor. Spectral accuracy measures how faithfully payoffs are reproduced, so it leaves open whether the equilibrium set survives. We show that it can fail at any accuracy: the equilibrium sets of a game and of its best rank-$k$ approximation can stay a constant distance apart while the spectral error tends to zero. The same failure appears in the $337\times 337$ sequence form of Leduc poker and in three structurally different matrix-game families, at spectral errors down to $2.9\times 10^{-3}$. The mechanism is selection. When a perturbation of scale $ϵ$ breaks a tie among equilibria, a compression residual $\eta$ is amplified by ${ϵ}^{-1}$ on the equilibrium face once global support containment holds, and an explicit family attains the displacement $\eta /(2(4ϵ+\eta ))$. Two guarantees separate the two questions. Set fidelity is local and conditional: face exactness, a uniform separation of outside actions, and a local gap-to-distance bound jointly identify the equilibrium set near a selected face. Output fidelity is pointwise: best responses on the original matrix certify the exploitability of the returned strategy whatever face generated it, and the certificate lifts from actions to strategy populations. The experiments reproduce the failure at finite spectral error on four testbeds, measure the cost of discovering the protected face without an oracle, and run the certificate inside double-oracle loops on three OpenSpiel games. The certified meta-solver returns the exact meta-solution's support in all 396 rounds, with numerically evaluated exploitability below $10^{-13}$ on the perturbed current meta-game, while a blind meta-game truncation distorts the support and leaves exploitability at the payoff scale.