From 201 Bins to 1 Bit: What a Binary Advantage Interface Retains in Vision-Language-Action Models

Vision-language-action (VLA) models are promising for general robot control, yet advantage-conditioned post-training can use a fine-grained categorical value critic even when the actor receives only a binary condition. We study what information this interface retains. Equality of the induced observation-action-label law is sufficient for equality of the supervised actor objective. For a mean-based residual, equal critic means suffice but are not necessary; we also bound objective changes under value and empirical-quantile perturbations. A covariance identity characterizes when ideal positive conditioning improves a chosen action utility. Direct binary prediction has a Bernoulli posterior as its Bayes target. In an idealized finite-alphabet model, mean-estimation sample complexity is independent of bin count, unlike full categorical recovery. Guided by this analysis, we train a scalar bootstrap score, cache recorded residuals in a Persistent Advantage Buffer, and fine-tune a prompt-conditioned flow-matching actor with classifier-free guidance. Physical evaluations on four bimanual manipulation tasks illustrate the pipeline against a shared behavioral-cloning initialization. The results separate sufficiency at the actor interface from the challenge of learning useful labels.