{"slug": "signed-p-adic-residual-encodings-of-finite-domain-all-different-systems-with-a", "title": "Signed p-adic Residual Encodings of Finite-Domain All-Different Systems with a Sudoku Case Study", "summary": "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.", "body_md": "# Computer Science > Machine Learning\n\n  [Submitted on 13 Sep 2026]\n\n# Title:Signed p-adic Residual Encodings of Finite-Domain All-Different Systems with a Sudoku Case Study\n\n[View PDF](https://arxiv.org/pdf/2609.16063)\n\n[HTML (experimental)](https://arxiv.org/html/2609.16063v1)\n\nAbstract: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.\n    \n\n### References & Citations\n\nLoading...\n\n# Bibliographic and Citation Tools\n\nBibliographic Explorer \n\n*(*[What is the Explorer?](https://info.arxiv.org/labs/showcase.html#arxiv-bibliographic-explorer))\nConnected Papers \n\n*(*[What is Connected Papers?](https://www.connectedpapers.com/about))\nLitmaps \n\n*(*[What is Litmaps?](https://www.litmaps.co/))\nscite Smart Citations \n\n*(*[What are Smart Citations?](https://www.scite.ai/))\n# Code, Data and Media Associated with this Article\n\nalphaXiv \n\n*(*[What is alphaXiv?](https://alphaxiv.org/))\nCatalyzeX Code Finder for Papers \n\n*(*[What is CatalyzeX?](https://www.catalyzex.com))\nDagsHub \n\n*(*[What is DagsHub?](https://dagshub.com/))\nGotit.pub \n\n*(*[What is GotitPub?](http://gotit.pub/faq))\nHugging Face \n\n*(*[What is Huggingface?](https://huggingface.co/huggingface))\nScienceCast \n\n*(*[What is ScienceCast?](https://sciencecast.org/welcome))\n# Demos\n\n# Recommenders and Search Tools\n\nInfluence Flower \n\n*(*[What are Influence Flowers?](https://influencemap.cmlab.dev/))\nCORE Recommender \n\n*(*[What is CORE?](https://core.ac.uk/services/recommender))\nIArxiv Recommender\n\n*(*[What is IArxiv?](https://iarxiv.org/about))\n# arXivLabs: experimental projects with community collaborators\n\narXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.\n\nBoth 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.\n\nHave an idea for a project that will add value for arXiv's community? [**Learn more about arXivLabs**](https://info.arxiv.org/labs/index.html).", "url": "https://wpnews.pro/news/signed-p-adic-residual-encodings-of-finite-domain-all-different-systems-with-a", "canonical_source": "https://arxiv.org/abs/2609.16063", "published_at": "2026-09-16 04:00:00+00:00", "updated_at": "2026-09-16 04:06:54.071325+00:00", "lang": "en", "topics": ["machine-learning", "ai-research"], "entities": ["arXiv", "Sudoku"], "alternates": {"html": "https://wpnews.pro/news/signed-p-adic-residual-encodings-of-finite-domain-all-different-systems-with-a", "markdown": "https://wpnews.pro/news/signed-p-adic-residual-encodings-of-finite-domain-all-different-systems-with-a.md", "text": "https://wpnews.pro/news/signed-p-adic-residual-encodings-of-finite-domain-all-different-systems-with-a.txt", "jsonld": "https://wpnews.pro/news/signed-p-adic-residual-encodings-of-finite-domain-all-different-systems-with-a.jsonld"}}