Current-State Interfaces and the Future Quotient in a Finite Icosahedral (AKN-Type) Attachment Geometry: Untouchable Differences and Persistent History Parity
Year: 2026
Type: preprint
Abstract: On a frozen 691-cell patch of AKN-type rhombohedral geometry with a single-face attachment grammar, a companion paper constructed finite current-state codes that decide the complete legal future, without determining whether the normalised code \(R_{K}^{(2)}\) among them is minimal. Here we ask which differences between current states act on the legal future, and compare the partition of states by the normalised code \(R_{K}^{(2)}\) with the future quotient, the partition by equality of extension languages. Assuming that every face belongs to at most two catalogue cells (K1, an audited finite property of the frozen catalogue), we prove in Lean that \(R_{K}^{(2)}\) is an exact complete invariant of the one-step legal interface: two catalogue states have the same normalised code exactly when they have the same available cells and the same legal moves. Together with an exhaustive enumeration, this yields all identifications within eight insertions of the seed. The future quotient is strictly coarser even among reachable states that still have legal moves: a preregistered computation on complete descendant DAGs certifies 5,384,241 future-equivalent pairs separated by \(R_{K}^{(2)}\). Under the same assumption we prove that if a cell is available in a reachable state \(s\) but not in a state \(t\), and some legal continuation of \(s\) touches it, then an explicit word distinguishes the futures of \(s\) and \(t\); hence the availability differences absorbed by future equivalence lie in cells that the available side can never touch, and an exhaustive audit of all certified nonterminal pairs found no exception. Finally, an outcome-blind, preregistered test shows that a history-derived parity, evaluable from the current state, does not factor through the future quotient, even between states with identical normalised code and size; by an existing integration law, such a difference persists along every common legal continuation. The future quotient itself is classical; the contribution is its explicit realisation in a finite attachment geometry, with formally verified general theorems and computationally certified reachable witnesses. No physical claim is made.
Source: Qeios
DOI: 10.32388/8c43ek
Author: Masaru Numagaki
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7225629007