{"paper":{"title":"Boundary Calculus, Rigidity Islands, and Deformation Theory in Algebraic Phase Structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Joe Gildea","submitted_at":"2026-01-26T08:29:13Z","abstract_excerpt":"We develop a general boundary calculus for algebraic phases and use it to formulate an intrinsic structural framework for deformation and obstruction phenomena. Structural boundaries are shown to be finitely detectable and canonically stratified by failure type and depth. For each boundary we construct a canonical boundary exact sequence and identify a maximal rigid subphase, called a rigidity island, that persists beyond global boundary failure.\n  Rigidity islands are organised by intrinsic invariants and serve as canonical base points for deformation theory. Deformation behaviour within the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.18257","kind":"arxiv","version":3},"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/2601.18257/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"}