Intervention-Relative Predictive Memory

This preprint develops a framework for intervention-relative predictive memory: the minimum information that must be retained about the past in order to predict every future experiment allowed by a declared intervention set.For finite-dimensional linear processes, predictive memory is characterized exactly by a reachable-observable quotient and the rank of the corresponding past-future Hankel response matrix. The paper derives finite certification procedures, monotonicity under richer control, a decomposition of memory growth into reachability expansion and visibility activation, symmetry-protected compression, and no-compression results under sufficiently rich intervention algebras.The framework is connected to process-tensor bond rank, hidden-memory realization, and physical memory costs. At a prepare-and-measure temporal cut, linear, classical, and quantum memory are distinguished through ordinary rank, nonnegative rank, and positive-semidefinite rank. A robust quantum-dimension witness is derived for an n-message decoding task.A proposed single-photon spatial-mode experiment provides a concrete test. Phase-coded histories have identical present path-intensity distributions but can be perfectly distinguished by permitted future phase and Fourier operations, giving an ideal response matrix equal to the identity. For 17 modes, sufficiently high decoding success directly certifies a 17-dimensional quantum memory.The paper distinguishes established realization, automata, process-tensor, quantum-control, and factorization results from the proposed synthesis and experimental interpretation. It concludes with two natural next directions: experimental implementation and a multitime completely-positive / PSD-rank theory of intervention-relative quantum memory.