Positional numeral systems over polyadic rings

Abstract

We construct positional numeral systems that work natively over nonderived polyadic $\left(m,n\right)$-rings whose addition takes $m$ arguments and multiplication takes $n$. In such rings, the length of an admissible additive word and a multiplicative tower are not arbitrary (as in the binary case), but "quantized". Our main contributions are the following. Existence: every commutative $\left(m,n\right)$-ring admits a base-$p$ place-value expansion that respects the word length constraint in terms of numbers of operation compositions ${\ell }_{mult}={\ell }_{add}(m-1)+1$. Lower bound: the minimum number of digits is greater than or equal to the arity of addition $m$. Representability gap: for $m,n\ge 3$ only a proper subset of ring elements possess finite expansions, characterized by congruence-class arity shape invariants $I^{(m)}$ and $J^{(n)}$. Mixed-base "polyadic clocks": allowing a different base at each position enlarges the design space quadratically in the digit count. Catalogues: explicit tables for the integer rings ${\mathbb{Z}}_{4,3}$ and ${\mathbb{Z}}_{6,5}$ illustrate how ordinary integers lift to distinct polyadic variables. These results lay the groundwork for faster arity-aware arithmetic, exotic coding schemes, and hardware that exploits operations beyond the binary pair.

13 pages, amslatex