For a controlled process, arrange population probabilities into a behavior table: columns index past histories and rows index future tests. Its full column span is the linear predictive behavior space.
Assume the full behavior space has rank at most \(r\). Choose an invertible \(r\)-by-\(r\) population core \(H\). Let \(G_s\) be the corresponding table with a declared action-observation symbol \(s\) inserted. With a compatible row/column convention,
The empty-test row supplies the mass functional and a covered history supplies the initial predictive state. The factorization then transports every future test through the observable coordinates.
The mathematical point
The dimension is determined by independent predictive behaviors. A global behavior-rank bound and an invertible core establish that the chosen coordinates cover the full behavior space.
Two distinct obstructions. An invertible sampled block does not establish the full-process rank bound. A fitted linear realization does not automatically preserve a probability domain. The next plates address the latter obligation.
Scope
The reconstruction uses population tables under the stated coverage hypotheses. No finite-sample rank test here authenticates those hypotheses or an unknown hidden-state model.
Formal route
BehaviorSpace.lean; BehaviorMinimality.lean; ObservableCore.lean; ObservableRealization.lean.