Machines that Predict Trajectories from Templates

What the math gives to ML

The paper provides a constructive trajectory-continuation mechanism: represent an observed past window as a linear combination of stored trajectory templates, then apply the same coefficients to their future windows. For linear systems, exact prediction is possible when the past window is long enough to identify all output-visible modes of the new system and those modes are contained in the library spectrum. This can become a differentiable memory or state-space-model prediction layer that replaces nearest-neighbor retrieval with globally solved modal interpolation. The key falsifiable test is whether forecast error exhibits a transition when the context length reaches the effective number of visible modes.