{"paper":{"title":"Defect Spaces and Gram Operators for Tensor-Valued Incidence Maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kengo Miyamoto","submitted_at":"2026-05-27T14:31:03Z","abstract_excerpt":"We study vector-valued incidence maps obtained from ordinary graph incidence maps by linear observation of the free vertex space. Let $\\mathbb{F}$ be a field, $D = (X, E, s, t)$ a finite directed multigraph, $U$ an $\\mathbb{F}$-vector space, and $\\phi : X \\to U$ a vertex labeling with $\\mathbb{F}$-linear extension $\\hat{\\phi} : \\mathbb{F}^X \\to U$. The vector-valued incidence map $\\partial_\\phi : \\mathbb{F}^E \\to U$, $\\partial_\\phi(\\mathbf{1}_e) = \\phi(t(e)) - \\phi(s(e))$, factors as $\\partial_\\phi = \\hat{\\phi} \\circ B_D$, where $B_D$ is the classical incidence map of $D$. We prove the formula"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28535","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/2605.28535/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"}