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

A paper submitted to arXiv on 13 Sep 2026 introduces signed, weighted affine p-adic residual objectives as native encodings of finite-domain constraints, with a coordinatewise domination theorem guaranteeing every global minimiser lies in the finite domain. The authors report that in that domain the loss equals, up to an additive constant, the all-different conflict count or the negative number of satisfied CNF clauses, and they present standard Sudoku as an 81-coefficient case study requiring no one-hot lift. A client-side implementation exposes the generated dataframes, arithmetic, diagnostics, and searches.

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 View PDF https://arxiv.org/pdf/2609.16063 HTML experimental https://arxiv.org/html/2609.16063v1 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. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender IArxiv Recommender What is IArxiv? https://iarxiv.org/about arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .