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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.28535","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-27T14:31:03Z","cross_cats_sorted":[],"title_canon_sha256":"88a30b3e211398ee5995f24b038c92c8bc997aee78972d7c010e1ef062a39257","abstract_canon_sha256":"d9e48d1f7f43f934c31149e2cd225b9672d241ec00f65a88c630cbb9d8b812ba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T02:04:55.718446Z","signature_b64":"Qj062/+RWb0mH7WTZkkE0jSspiJGPkurGtJvvZu6CeTjpYlxVhykpIj72urB2kE7yfZkyIrgLRTxXilrzgm2Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9fa7bf6a6bf77fa74da3e35f2253b3db4ded2f95bbd71c96e6092974268fde68","last_reissued_at":"2026-05-28T02:04:55.717903Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T02:04:55.717903Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $D$. 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