Signed p-adic Residual Encodings of Finite-Domain All-Different Systems with a Sudoku Case Study

Abstract:We study signed, weighted affine $p$-adic residual objectives as native encodings of finite-domain constraints. For primes that separate the finite alphabet, sufficiently weighted positive unary rows pin each coefficient to its allowed set, while negative rows reward unequal endpoints or clause satisfaction. A coordinatewise domination theorem places every global minimiser in the finite domain; there the loss is, up to an additive constant, the all-different conflict count or the negative number of satisfied CNF clauses. Standard Sudoku provides an $81$-coefficient case study without a one-hot lift. A client-side implementation exposes the generated dataframes, arithmetic, diagnostics, and searches.

31 pages, 7 figures. Accepted for publication in p-Adic Numbers, Ultrametric Analysis and Applications

Machine Learning (cs.LG)

arXiv:2609.16063 [cs.LG]

(or arXiv:2609.16063v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.16063 Focus to learn more arXiv-issued DOI via DataCite

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