[
  {
    "id": "ARR-2026-0JTZHGRV1C8G98TM-v1",
    "type": "article",
    "title": "Machine-Checked Extended Gauge Invariance at an SU(2) Crossing: Four-Edge Wilson Holonomy, Haar-Preserving Quotient Coordinates, and Reduction to the Ward Chart",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          2
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-0JTZHGRV1C8G98TM",
    "version": "v1",
    "genre": "Archived manuscript",
    "URL": "https://airr.science/papers/ARR-2026-0JTZHGRV1C8G98TM/versions/v1/",
    "note": "Archived manuscript; ARR-2026-0JTZHGRV1C8G98TM v1. Not peer reviewed by AIRR. Historical import; not assessed by AIRR.",
    "abstract": "We formalize extended gauge invariance at a simple four-edge SU(2) crossing and connect the geometric edge chart to the two-coordinate chart used by a machine-checked crossing Ward identity. On SU(2)^4 we define the two opposite-edge right actions from the abstract Makeenko--Migdal theorem, prove that they are commuting product-Haar-preserving actions, and show that their common parameter composes to ordinary vertex gauge invariance. The four-edge Wilson word tr2(a3^-1 beta a2 a4^-1 alpha a1) is proved invariant under both half-actions. We construct the explicit quotient r(a)=(a2 a4^-1,a1 a3^-1), a canonical section, and prove existence and uniqueness of the universal factorization for every extended-gauge-invariant complex function. The complete map from the cyclic four-edge chart to physical and gauge coordinates is proved to preserve literal four-fold Haar measure in one public endpoint. Finally, the four-edge Wilson word is identified exactly with the prior two-coordinate crossing word evaluated on r(a). The Lean producer has 56 public declarations in 488 physical lines; all 36 theorems depend only on propext, Classical.choice, and Quot.sound, with no local proof escape. No heat-kernel area derivative or full Makeenko--Migdal equation is claimed."
  }
]