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plain-language theorem explainer
Successor and predecessor on the 8-element phase cycle are mutual inverses. Researchers modeling discrete recognition traces under the 8-tick hypothesis cite this to confirm reversibility of phase steps. The proof is a one-line simplification that unfolds the successor and predecessor definitions directly.
Claim. For any phase $p$ in the finite 8-cycle, the predecessor of the successor of $p$ equals $p$.
background
The module states the 8-tick hypothesis that observed folding and recognition traces exhibit 8-phase periodicity, with phases 0-3 for LOCK structure formation and 4-7 for BALANCE equilibration. Phase is the finite set Fin 8. Successor advances the phase by one step cyclically; predecessor retreats by one step cyclically. This rests on the 8-tick phase space from the Church-Turing Physics Structure module, which likewise sets Phase to Fin 8.
proof idea
The proof is a one-line wrapper that applies the simplifier to the successor and predecessor definitions.
why it matters
This secures the cyclic invertibility demanded by the 8-tick discretization hypothesis. It supports the forcing chain step toward eight-tick octave periodicity in the Recognition Science framework. The result remains open to falsification by traces that exhibit non-8 periodicity or fail 8-phase segmentation.
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