A Consequential Closure Diagnostic in Registration Theory: Future-Equivalence as a Supporting Lemma and the Boundary Between Incomplete Representation and Failed Closure

This focused technical note formulates a Consequential Closure Diagnostic for Registration Theory (RT). The result is not presented as a new future-equivalence construction; instead, it uses familiar quotient and predictive-state ideas as supporting machinery for an RT-specific completeness and closure test.

The note proves that, given causal closure, the No-Unaccounted-Consequence Principle, and representation coverage of all internally realized consequential carriers, a model that merges two reachable states cannot later produce different admissible physical registration consequences. If two states are represented identically but later yield different physical consequences.  At least one claim must fail: the representation is consequence-complete, the stipulated dynamics is causally closed, or both.

A short Future-Consequence Quotient Lemma is included to formalize consequence-equivalence classes, with explicit acknowledgment of prior art including Myhill–Nerode minimization and computational-mechanics causal-state constructions. The central contribution is therefore the RT closure diagnostic rather than the quotient construction itself.

The note also clarifies that delayed consequences may pass through arbitrarily many transformations, hosts, retained structures, combinations, branches, and feedback paths without requiring preservation of a fixed carrier or unchanged state variable. What must remain accounted for is the physically consequential distinction or its transformed successor.

This result is intended as a versioned technical preprint within the Registration Theory program. It does not yet derive cross-host interaction, physical propagation, spacetime locality, a universal carrier, or the No-Unaccounted-Consequence Principle itself. Those remain subsequent physical derivation problems.