Robots increasingly run large vision-language-action policies on remote servers, making the camera uplink the bottleneck. Existing video codecs spend bits on content the receiver could predict: the robot's own motion. We introduce the kinematic hypothesis context, a zero-rate conditioning signal that both ends build from joint states, the robot model and the camera calibration. Ego-motion homographies, per-link robot reprojections, an end-effector plane and a background memory warp the previous reconstruction into hypotheses that condition a learned video codec. We prove that history-only and global-pose conditioning with a limited receptive field cannot access this information under large motion, and that coding gains become task gains at the policy's success cliff. On LIBERO, the context saves 23.9% bit rate over a capacity-matched history-conditioned codec and 40-56% over x264, x265 and SVT-AV1, and needs 23.2% fewer bits than DCVC-RT at equal policy action deviation; trained to lower rates, it matches DCVC-RT in PSNR. In closed loop at equal rate, OpenVLA-OFT success nearly doubles from 22.8% to 43.1% at 6 kbps (72.6% with lower-rate training, 23.6 points above DCVC-RT), pi_0.5 success rises from 55.5% to 77.4%, and the gain grows under packet loss. The context transfers across three codec hosts and to real robot data.