{"paper":{"title":"Topological Foundations of Multi-Field Instabilities in Continua: Part 1: Foundations Part 2:Analytical Formulation for 1-D Spin Chains Part 3: Numerical Upscaling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MP","nlin.PS"],"primary_cat":"math-ph","authors_text":"Amir Saker, Francois Nicot, Klaus Regenauer-Lieb","submitted_at":"2026-07-21T11:28:25Z","abstract_excerpt":"This three-part series establishes a parameter-free, topological classification of multi-field instability in granular continua, extending Maxwell's rigidity count to dynamic, non-equilibrium processes.\n  Part 1 (Foundations): a discrete Volumetric-Mechanical-Configurational (VMC) contact formulation maps contact-scale topology to macroscopic multiphysics coupling. A Parity Theorem, $\\det(\\mathsf{L})=(-1)^N\\det(\\mathsf{L})$, forces a structural null-mode for every odd channel count $N$, creating \"Gateway\" layers of broken time-reversal symmetry; once the basis-invariant Gateway number $\\mathca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.18995","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.18995/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}