Closed-Event Consequence Completeness in Registration Theory

This research note develops a formal sufficient criterion for closed-event consequence completeness in Registration Theory.

The central problem is to distinguish physical consequence completeness from stronger requirements such as global state injectivity, full predecessor reconstructibility, or information preservation. A many-to-one transition need not represent physical loss if the distinctions that merge have become consequence-equivalent under an independently specified future regime.

The paper introduces an operational future-consequence signature and proves a Closed-Event Consequence Completeness Theorem. Under explicit assumptions of causal closure, independently defined consequence criteria, exhaustive forward consequence correspondence, persistence fidelity, carrier efficacy, valid reconciliation, and representation coverage, the represented state is shown to be consequence-sufficient:

R(x)=R(y)⟹ΣT(x)=ΣT(y).

Equivalently,

kereqR⊆kereqΣT.

A related minimal-completion result shows that when a representation is consequence-incomplete, the smallest required refinement distinguishes only the different consequence classes occurring within each represented-state fiber. Full historical provenance need not be retained.

The framework is applied to the four many-to-one CI mergers in Toy Model 2. Under the model’s own deterministic autonomous future semantics, the paired successors have identical complete future orbits. They are therefore consequence-equivalent relative to the endogenous Toy Model 2 consequence signature, although this does not by itself establish a broader physical reconciliation principle for Registration Theory.

The note also establishes the boundary between consequence completeness and quantitative conservation. Once an explicit closed-event realization supplies complete consequence classes and event types, additive conserved valuations can be sought from the nullspace of the event matrix

Mαc=nαc+−nαc−,Mv=0.

A nonzero nullspace would establish an additive invariant of that specified event algebra, but would not by itself identify the invariant with energy, momentum, probability, or another known physical quantity.

The paper therefore provides the abstract mathematical completeness theorem and identifies the remaining physical realization obligations required before a Registration Theory conservation law can be claimed.