Geometry from Histories: Categorical Reconstruction of Learned State Spaces

A state space reconstructed from histories must preserve the operations that give those histories their consequences. We formulate this requirement through a category of exact realizations of response laws and supported continuation. The complete response profile is terminal, and reduced exact realizations are therefore canonically isomorphic. A discounted operational metric gives a compact completion; finite-test approximation errors yield explicit Gromov–Hausdorff bounds with a separate term for unresolved continuation. Uniqueness also forces compatible local realizations to satisfy the cocycle condition. Temporal direction enters as an obstruction to representations that erase response-relevant order. Learned continuous-time signatures and attention provide a constructive realization whose compatibility can be examined under changes of recording and coordinates. ReLU components supply piecewise-affine local structure, with interface contributions retained by an exact distributional formula for finite contrasts. For systems with absorption, localization must intertwine killed continuation operators: this preserves termination laws, expected lifetime and transported responses, with quantitative bounds under approximate intertwining. Valuation and differential localization extend the construction when their additional operator hypotheses hold. The resulting framework reconstructs geometry through observable distinctions and specifies which dynamics and boundary effects must survive a change of representation.