TITLE:
Prime-Quaternion Addresses for the Free-Group Skeleton of Banach-Tarski Decompositions
AUTHORS:
Payam Danesh, Raoul Bianchetti
KEYWORDS:
Banach-Tarski Theorem, Free Group, Rational Rotation, Quaternion, Paradoxical Decomposition, Non-Measurable Set, Cayley Graph
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.10,
October
16,
2026
ABSTRACT: We present an arithmetic address code for the free-group action used in the Banach-Tarski construction. Two integer quaternions of norm five induce rational rotations about perpendicular axes. Świerczkowski’s criterion proves that these rotations freely generate a group of rank two. For each reduced word, the norm of its unnormalized quaternion product is five raised to the word length, so its 5-adic valuation records Cayley-graph depth. We derive the scalar factor introduced by cancellation in unreduced words and transfer the code to the free part of the sphere after choosing an orbit transversal. The standard four-piece partition of the free group then gives two reconstructions of that free part. The four spherical pieces cannot all be measurable with respect to rotation-invariant surface-area measure. Accurate computation confirms the rotation identities and distinguishes all 4373 reduced words of length at most seven. The contribution is an explicit organization of these algebraic and orbit labels, with their dependence on reduction and on the chosen transversal made precise.