Abstract
This paper asks whether the cumulative value of a transition-derived observable can be represented exactly by the current state, without maintaining a dedicated event-log cache. We first define a minimal transition core consisting only of a state space, a move set, a legality predicate, and a next-state map. For a commutative target group \(M\), a current-state map \(\Phi\) and a transition label \(\sigma\) form an exact register when
\({\Phi{({{\mathsf{N}\mathsf{e}\mathsf{x}\mathsf{t}}{(s,m)}})}} = {{\Phi{(s)}} + {\sigma{(s,m)}}}\)
for every legal transition. The resulting path sum telescopes to an endpoint difference; this elementary identity provides the framework for the explicit construction developed below.
We then specialize to \(M = {\mathbb{Z}}_{2}\) in a finite state-transition realization, D1b, built on an Ammann–Kramer–Neri (AKN)-type icosahedral rhombohedral substrate. The substrate supplies the six-direction rhombohedral cell and face alphabet used by the realization, whereas the exact-one-face insertion rule, the LOW/HIGH relative-side transition label, and the frozen finite catalogue are specific to the D1b transition system studied here; they are not derived from AKN matching or inflation rules. The general exact-register result requires neither this substrate nor its rotation group \(A_{5}\); D1b is used as a finite, source-auditable, nontrivial test case. From current cell-presence and face-incidence data we construct an explicit potential \(\Phi\), and prove that its one-step change equals an independently defined LOW/HIGH relative-side transition parity. Under a registered source certificate, a reachable HIGH–HIGH branch exhibits
\({\Phi_{0}\longrightarrow{\Phi_{0} + 1}\longrightarrow\Phi_{0}},\)
which establishes certificate-conditional reachable non-vacuity. We further show that the register is recomputable from the current D1b residue and that a coherent dedicated cache contains no information independent of that residue for this observable.
The result is an exact current-state compilation theorem for one explicit history-derived parity. It does not imply reconstruction of the full event history or preservation of future behavior; additional scope boundaries are summarized in Appendix A.