Twin‑Fixed Kaiso provides a minimal syntactic framework for hierarchical dynamics generated by pairs of quantities (“twins”). Each level of the hierarchy consists of a twin ^ (n) = (₁^ (n), ₂^ (n) ) and its twist\ ^ (n) =| ₁^ (n) - ₂^ (n) |\, which serves as the fundamental invariant that distinguishes regular and singular levels. A transition map ₙ propagates the hierarchy while preserving syntactic closure. Regular levels are characterized by the invariance of the twist, whereas singularities occur precisely when the twin collapses and the twist vanishes. The framework shows that hierarchical propagation is possible only within twist‑preserving strata, and that any violation of closure leads to structural collapse. Examples and computations are provided to illustrate the behavior of twins, twists, and transitions across multiple levels.
Masaru Morimoto (Sat,) studied this question.
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