Positional Numeral Systems over Polyadic Rings

We construct positional numeral systems that work natively over nonderived polyadic m,n-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 m,n-ring admits a base-p place–value expansion that respects the word length constraint in terms of numbers of operation compositions ℓmult=ℓadd(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≥3 only a proper subset of ring elements possesses 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. Catalogs: Explicit tables for the integer rings Z4,3 and Z6,5 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.