Forecasting physical systems over long horizons from irregularly sampled observations demands models that are stable, computationally efficient, and free of fixed-timestep assumptions. We address this with a continuous-time Koopman autoencoder whose latent dynamics obey $dz/dt = \mathbf{K}_{\mathrm{cont}} z$, yielding closed-form inference via $z(\tau) = \exp(\mathbf{K}_{\mathrm{cont}} \tau) z(0)$ at any horizon $\tau$ in a single step. This decouples forecast cost from forecast length at inference time and supports data assimilation as gradient-based optimization with cost independent of the assimilation window. However, scaling continuous-time Koopman dynamics to high-dimensional chaotic systems causes severe latent instability, including spectral collapse and trajectory divergence over long horizons. In contrast, discrete Koopman methods train an operator $\mathbf{A}$ such that $z_{t+\Delta t} = \mathbf{A} z_t$; recovering the continuous generator could be theoretically done through matrix logarithm but requires conditions not guaranteed by training, and approximation errors grow with the $\Delta t$ imposed by the training data. These methods also require fixed, regular timesteps. We identify an empirically effective set of structural constraints -- rollout training, forward-backward consistency, latent regularization, and physics-conditioned LoRA -- sufficient for stable long-horizon latent dynamics. On challenging fluid benchmarks, our method outperforms strong diffusion and operator-learning baselines on long-horizon forecasting while achieving a 110$\times$ inference speedup.