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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.12930 |
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| _version_ | 1866917451646631936 |
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| author | Duplij, Steven |
| author_facet | Duplij, Steven |
| contents | 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 $m,n\geq3$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positional numeral systems over polyadic rings Duplij, Steven Number Theory High Energy Physics - Theory Mathematical Physics Rings and Algebras Quantum Physics 11A07, 11A67, 20N15 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 $m,n\geq3$ 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. |
| title | Positional numeral systems over polyadic rings |
| topic | Number Theory High Energy Physics - Theory Mathematical Physics Rings and Algebras Quantum Physics 11A07, 11A67, 20N15 |
| url | https://arxiv.org/abs/2506.12930 |