The Rigid–Articulated Boundary: When Neural-ODE Extrapolation Works in 3D Gaussian Splatting, and Why It Fails

Dynamic 3D Gaussian Splatting reconstructs scene motion within an observed time window; whether such representations can be made to predict future scene states — to extrapolate beyond the training horizon — is far less understood. We study this directly. We formulate scene dynamics as a Neural Ordinary Differential Equation (Neural ODE) over the positions of a canonical 3D Gaussian set: a learned velocity field is integrated forward in time and the deformed Gaussians are rendered directly, with no auxiliary decoder. We evaluate temporal extrapolation on five D-NeRF scenes spanning rigid, rigidly-anchored, and articulated motion, comparing against a memorization baseline that shares the same canonical geometry. The result is a sharp and consistent boundary. On rigid or rigidly-anchored structure the velocity field extrapolates well, beating memorization by up to +9 dB PSNR at long horizons with the advantage widening as the horizon grows. On articulated human bodies the same model fails: the figure blurs or disappears, and on two of three articulated scenes a trivial frozen-pose baseline matches or exceeds it. The boundary tracks structure, not motion speed: a slowly rising figure fails as surely as fast jumping motion. We further document that PSNR substantially misleads here: a near-empty prediction scores well on a mostly-static frame, and on articulated scenes PSNR, SSIM, and LPIPS disagree on which method is better, so single-metric evaluation of extrapolation is unreliable. Finally, neither a lightweight k-nearest-neighbour spatial-coupling variant nor evolving appearance within the ODE closes the gap, narrowing the cause to a loss of articulated coherence. We frame this as a controlled probe of one precondition a geometryaware predictive representation of the physical world would require — an explicit state whose future value is well-defined beyond the observed window — not as an action-conditioned or planning-capable system. We do not claim a new predictor; we delineate where ODE-based Gaussian extrapolation is and is not currently viable, and why. All experiments run on a single consumer GPU; code and configurations will be released.