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

arXiv: https://arxiv.org/abs/2609.16063 · HTML: https://mlsanity.com/paper/2609.16063/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 18 references that are in the map.
- First posted on arXiv 13 Sep 2026.

## Builds on (most cited first, 2 of 2)

- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Learning with the p-adics](https://mlsanity.com/paper/2512.22692/) (2025)

## Abstract (arXiv)

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.