ALAM: Algebraically Consistent Latent Transitions for Vision-Language-Action Models

Vision-language-action (VLA) models remain constrained by the scarcity of action-labeled robot data, whereas action-free videos provide abundant evidence of how the physical world changes. Latent action models offer a promising way to extract such priors from videos, but reconstruction-trained latent codes are not necessarily suitable for policy generation: they may predict future observations while lacking the structure needed to be reused or generated coherently with robot actions. We introduce \textbf{ALAM} (\textbf{A}lgebraic \textbf{L}atent \textbf{A}ction \textbf{M}odel), an Algebraically Consistent Latent Action Model that turns temporal relations in action-free video into structural supervision. Given frame triplets, ALAM learns latent transitions that are grounded by reconstruction while being regularized by composition and reversal consistency, encouraging a locally additive transition space. For downstream VLA learning, we freeze the pretrained encoder and use its latent transition sequences as auxiliary generative targets, co-generated with robot actions under a joint flow-matching objective. This couples structured latent transitions with flow-based policy generation, allowing the policy to exploit ALAM's locally consistent transition geometry without requiring latent-to-action decoding. Representation probes show that ALAM reduces additivity and reversibility errors by 25--85$\times$ over unstructured latent-action baselines and improves long-horizon cumulative reconstruction. When transferred to VLA policies, ALAM raises the average success rate from 47.9\% to 85.0\% on MetaWorld MT50 and from 94.1\% to 98.1\% on LIBERO, with consistent gains on real-world manipulation tasks. Ablations further confirm that the strongest improvements arise from the synergy between algebraically structured latent transitions and joint flow matching.