{
  "annotation_id": "177a9c1f-1f6d-4263-856f-93af0063386d",
  "date_created": "2026-02-17T05:53:05.100000Z",
  "date_modified": "2026-02-17T05:53:05.100000Z",
  "file_hash": "41439f3d500973c335d9f62e5bf4f1c91dfde3347efcae7ce6d6505e168541e5",
  "private": false,
  "record": {
    "abstract": "We demonstrate that the Cayley-Hamilton theorem is a derived consequence of a more fundamental dimensional constraint: the syzygy formed by the tensor product of two Levi-Civita symbols, which vanishes identically in m-dimensional space. By shifting perspective from the tensor A to the isotropic operators that induce A\u0027s invariants through contraction, we reveal that the Cayley-Hamilton identity emerges when this vanishing operator acts on the m-fold tensor product of A. The intrinsic tensorial form of the theorem--invariant coefficients multiplying tensor powers--is inherited from the contraction structure rather than imposed ad hoc. We provide explicit verification for two-dimensional space and a dimension-independent proof using Laplace expansion combined with Newton-Girard identities. This framework clarifies why the theorem\u0027s structure depends on ambient dimension and suggests extensions to higher-order tensors where classical characteristic polynomial methods fail.",
    "arxiv_id": "2601.06136",
    "authors": [
      "Xiao Wang"
    ],
    "categories": [
      "math.RA",
      "math.AG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "The Geometric Origin of the Cayley-Hamilton Theorem: A Constructive Proof via Dimensional Syzygy",
    "url": "https://arxiv.org/abs/2601.06136",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "eb51b9f1-5f00-4979-9530-4cdd53249bd1",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}