Tsuihenkaku Kaiso

We introduce a new hierarchical framework, called the Tsuihenkaku Kaiso, in which two angular directions are fixed as a pair at every level of the hierarchy. Unlike conventional torsion, which is defined by a single directional difference, the twist in this framework is represented by a pair of angular directions, \[T⁽ⁿ⁾ = (τ⁽ⁿ⁾_1,\ τ⁽ⁿ⁾_2),\] and remains structurally invariant across all levels. This syntactic constraint ensures that the degrees of freedom do not increase, even when the hierarchy extends to infinity. We prove three central results: (1) hierarchical closure, (2) full measurability of twist except at singular points, and (3) uniqueness of singular collapse. These results eliminate the usual combinatorial explosion and chaotic behavior that typically arise in high-dimensional dynamical systems. Furthermore, the paired-angle structure naturally connects to double rotations in higher geometry (SU(2) × SU(2)), hierarchical chain complexes in topology, and double-phase structures in quantum mechanics. The Tsuihenkaku Kaiso thus provides a new mathematical foundation for understanding the boundary between measurable and non-measurable phenomena, revealing that infinite hierarchical systems can remain fully closed and analyzable under an appropriate syntactic constraint.