M08 TransAlgebra: The Number System Founded on Rank-3 Operations

This monograph develops TransAlgebra, a tower of algebraic structures founded on rank‑3 (exponential/branchation) operations, extending the classical Cayley–Dickson construction beyond rank‑2 multiplicative algebras. The central construction replaces the scalar notion of a number with the Configuration Number ring, defined as the group algebra of finite branch‑sequence groups over the complex numbers. The resulting algebras Tn have dimensions 1,2,4,8,… matching the Cayley–Dickson tower, but remain commutative and associative at every level, with generators corresponding to branch units rather than square‑root units. A fractional extension is introduced in which the fundamental rank‑3 defining relation replaces i2=−1 with its exponential analogue. Within this framework, rank‑3 multiplication is shown to act as a Fourier transform on coefficient functions, and classical rank‑2 multiplication becomes a convolution over branch sequences. The monograph formalises a rank‑reduction principle, whereby higher‑rank operations are linearised in TransAlgebra space, and clarifies how stochastic ensembles arising in infinite iteration correspond to probability measures on the TransAlgebra tower. The purpose of this deposit is to document the constructions, algebraic properties, and conceptual priority of TransAlgebra as a number system adapted to rank‑3 operations, independent of physical interpretation or further application.