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

> Source: <https://arxiv.org/abs/2609.16063>
> Published: 2026-09-16 04:00:00+00:00

# Computer Science > Machine Learning

  [Submitted on 13 Sep 2026]

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

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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.
    

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