{
  "schema": "groups.qft.org/entity/2",
  "versions": {
    "schema": "2.0.0",
    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "canonicalURL": "https://groups.qft.org/atlas/groups/standard-model-z1/",
  "entity": {
    "id": "group:standard-model-z1",
    "slug": "standard-model-z1",
    "name": "Standard Model compact covering product",
    "symbol": "SU(3) × SU(2) × U(1)",
    "category": "lie",
    "level": "executable",
    "aliases": [
      "SM Z1",
      "Standard Model quotient 1"
    ],
    "searchTerms": [
      "Standard Model",
      "hypercharge",
      "global form",
      "diagonal center",
      "SU5"
    ],
    "tags": [
      "Compact",
      "Connected",
      "Product quotient"
    ],
    "summary": "A compact connected twelve-dimensional global group with algebra su(3) ⊕ su(2) ⊕ u(1), quotient order 1, and fundamental group Z. Usual Standard Model matter descends in the stated Q=6Y convention.",
    "definition": "The compact product SU(3) × SU(2) × U(1), with U(1) period 2π and integer hypercharge Q=6Y. It is a covering group of the three nontrivial specified quotients, but it is not the universal cover.",
    "definitionEvidence": [
      {
        "source": "block-baez-huerta",
        "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
      },
      {
        "source": "tong",
        "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
      },
      {
        "source": "conrad-pi1-lie",
        "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
      }
    ],
    "properties": [
      {
        "key": "dimension",
        "label": "Real dimension",
        "value": "12",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "assumptions": []
      },
      {
        "key": "rank",
        "label": "Rank",
        "value": "4",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "assumptions": []
      },
      {
        "key": "compact",
        "label": "Compact",
        "value": "Yes",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "assumptions": []
      },
      {
        "key": "connected",
        "label": "Connected",
        "value": "Yes",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "assumptions": []
      },
      {
        "key": "pi1",
        "label": "Fundamental group",
        "value": "Z",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "assumptions": []
      },
      {
        "key": "quotient-order",
        "label": "Specified quotient order",
        "value": "1",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "assumptions": []
      },
      {
        "key": "center",
        "label": "Abstract center",
        "value": "U(1) × Z_6",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "derivation": {
          "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.",
          "inputs": [
            "source:conrad-pi1-lie",
            "Γd projects injectively to both the diagonal finite center and the U(1) factor."
          ]
        },
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "representations",
        "title": "Matter and the embedded kernel",
        "paragraphs": [
          {
            "text": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/1) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.",
            "evidence": [
              {
                "source": "block-baez-huerta",
                "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
              },
              {
                "source": "tong",
                "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
              },
              {
                "source": "conrad-pi1-lie",
                "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
              }
            ],
            "assumptions": []
          },
          {
            "text": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.",
            "evidence": [
              {
                "source": "block-baez-huerta",
                "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
              },
              {
                "source": "tong",
                "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
              },
              {
                "source": "conrad-pi1-lie",
                "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "topology",
        "title": "Topology and the universal cover",
        "paragraphs": [
          {
            "text": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_1 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_1. The compact product used as the quotient domain already has fundamental group Z.",
            "evidence": [
              {
                "source": "block-baez-huerta",
                "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
              },
              {
                "source": "tong",
                "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
              },
              {
                "source": "conrad-pi1-lie",
                "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "conventions",
        "title": "Hypercharge and SU(5)",
        "paragraphs": [
          {
            "text": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.",
            "evidence": [
              {
                "source": "block-baez-huerta",
                "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
              },
              {
                "source": "tong",
                "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
              },
              {
                "source": "conrad-pi1-lie",
                "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-algebra",
        "target": "algebra:standard-model-r",
        "label": "The common gauge algebra",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ]
      },
      {
        "type": "has-central-quotient",
        "target": "group:standard-model-z2",
        "label": "Diagonal quotient of order 2",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ]
      },
      {
        "type": "has-central-quotient",
        "target": "group:standard-model-z3",
        "label": "Diagonal quotient of order 3",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ]
      },
      {
        "type": "has-central-quotient",
        "target": "group:standard-model-z6",
        "label": "Diagonal quotient of order 6",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ]
      }
    ],
    "capabilities": [
      "product-gauge-groups",
      "local-operators",
      "operator-enumeration"
    ],
    "kind": "group",
    "data": {
      "category": "lie",
      "dimension": 12,
      "compact": true,
      "connected": true,
      "abelian": false,
      "simple": false,
      "algebraId": "algebra:standard-model-r",
      "displayTex": "SU(3)\\times SU(2)\\times U(1)",
      "quotientOrder": 1,
      "conventionId": "product-su-u1-center-v1",
      "quotient": {
        "continuous": false,
        "dimension": 0,
        "order": 1,
        "finiteOrder": 1,
        "invariantFactors": [],
        "latex": "1",
        "elementScope": "Complete finite subgroup.",
        "elements": [
          {
            "centers": [
              0,
              0
            ],
            "phase": {
              "numerator": "0",
              "denominator": "1",
              "text": "0",
              "latex": "0"
            },
            "order": 1
          }
        ],
        "generators": [],
        "generatorMeaning": "One generating set; this is not an identification of each generator with a particular invariant factor.",
        "u1ProjectionOrder": 1
      },
      "fundamentalGroup": {
        "freeRank": 1,
        "torsionOrder": 1,
        "torsionInvariantFactors": [],
        "latex": "\\mathbb Z",
        "phaseZeroSubgroup": [
          {
            "centers": [
              0,
              0
            ],
            "phase": {
              "numerator": "0",
              "denominator": "1",
              "text": "0",
              "latex": "0"
            },
            "order": 1
          }
        ],
        "derivation": "Lift the selected subgroup to the universal cover (product SU factors) × R. Its torsion is exactly the subgroup at U(1) phase zero; the real projection is an infinite cyclic lattice."
      },
      "capabilityPresets": {
        "product-gauge-groups": {
          "title": "Standard Model matter with quotient order 1",
          "ranks": "3,2",
          "matter": "Q: 1,0 | 1 | 1\nu_c: 0,1 | 0 | -4\nd_c: 0,1 | 0 | 2\nL: 0,0 | 1 | -3\ne_c: 0,0 | 0 | 6\nH: 0,0 | 1 | 3",
          "generators": ""
        },
        "local-operators": {
          "setup": "{\"schema\":\"groups.qft.org/physical-setup/1\",\"name\":\"Weinberg operator with diagonal quotient order 1\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":1,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":3},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1}],\"operator\":{\"powers\":{\"L\":2,\"H\":2},\"derivatives\":0,\"equivalence\":\"algebraic\",\"coefficientField\":\"C\"}}"
        },
        "operator-enumeration": {
          "inventory": "{\"schema\":\"groups.qft.org/operator-enumeration/1\",\"name\":\"Standard Model with diagonal quotient order 1\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":1,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"Q\",\"name\":\"Q\",\"kind\":\"left-weyl\",\"su3\":[1,0],\"su2\":1,\"charge\":1,\"flavors\":1},{\"id\":\"Qdag\",\"name\":\"Q†\",\"kind\":\"right-weyl\",\"su3\":[0,1],\"su2\":1,\"charge\":-1,\"flavors\":1},{\"id\":\"uc\",\"name\":\"uᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":-4,\"flavors\":1},{\"id\":\"ucdag\",\"name\":\"uᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":4,\"flavors\":1},{\"id\":\"dc\",\"name\":\"dᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":2,\"flavors\":1},{\"id\":\"dcdag\",\"name\":\"dᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":-2,\"flavors\":1},{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1},{\"id\":\"Ldag\",\"name\":\"L†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"ec\",\"name\":\"eᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":6,\"flavors\":1},{\"id\":\"ecdag\",\"name\":\"eᶜ†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":-6,\"flavors\":1},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"Hdag\",\"name\":\"H†\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1}],\"enumeration\":{\"minDimension\":1,\"maxDimension\":6,\"includeIdentity\":false,\"symmetries\":[]}}"
        }
      }
    }
  },
  "claims": [
    {
      "id": "group:standard-model-z1#definition",
      "subject": "group:standard-model-z1",
      "key": "definition",
      "label": "Definition",
      "value": "The compact product SU(3) × SU(2) × U(1), with U(1) period 2π and integer hypercharge Q=6Y. It is a covering group of the three nontrivial specified quotients, but it is not the universal cover.",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#dimension",
      "subject": "group:standard-model-z1",
      "key": "dimension",
      "label": "Real dimension",
      "value": "12",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#rank",
      "subject": "group:standard-model-z1",
      "key": "rank",
      "label": "Rank",
      "value": "4",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#compact",
      "subject": "group:standard-model-z1",
      "key": "compact",
      "label": "Compact",
      "value": "Yes",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#connected",
      "subject": "group:standard-model-z1",
      "key": "connected",
      "label": "Connected",
      "value": "Yes",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#pi1",
      "subject": "group:standard-model-z1",
      "key": "pi1",
      "label": "Fundamental group",
      "value": "Z",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#quotient-order",
      "subject": "group:standard-model-z1",
      "key": "quotient-order",
      "label": "Specified quotient order",
      "value": "1",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#center",
      "subject": "group:standard-model-z1",
      "key": "center",
      "label": "Abstract center",
      "value": "U(1) × Z_6",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "derivation": {
        "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.",
        "inputs": [
          "source:conrad-pi1-lie",
          "Γd projects injectively to both the diagonal finite center and the U(1) factor."
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#representations-1",
      "subject": "group:standard-model-z1",
      "key": "representations-1",
      "label": "Matter and the embedded kernel",
      "value": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/1) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#representations-2",
      "subject": "group:standard-model-z1",
      "key": "representations-2",
      "label": "Matter and the embedded kernel",
      "value": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#topology-1",
      "subject": "group:standard-model-z1",
      "key": "topology-1",
      "label": "Topology and the universal cover",
      "value": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_1 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_1. The compact product used as the quotient domain already has fundamental group Z.",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#conventions-1",
      "subject": "group:standard-model-z1",
      "key": "conventions-1",
      "label": "Hypercharge and SU(5)",
      "value": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#relation-1",
      "subject": "group:standard-model-z1",
      "key": "relation-1",
      "label": "The common gauge algebra",
      "value": "algebra:standard-model-r",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "relationship": "has-algebra",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#relation-2",
      "subject": "group:standard-model-z1",
      "key": "relation-2",
      "label": "Diagonal quotient of order 2",
      "value": "group:standard-model-z2",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "relationship": "has-central-quotient",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#relation-3",
      "subject": "group:standard-model-z1",
      "key": "relation-3",
      "label": "Diagonal quotient of order 3",
      "value": "group:standard-model-z3",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "relationship": "has-central-quotient",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "group:standard-model-z1#relation-4",
      "subject": "group:standard-model-z1",
      "key": "relation-4",
      "label": "Diagonal quotient of order 6",
      "value": "group:standard-model-z6",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "relationship": "has-central-quotient",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "sources": [
    {
      "id": "tong",
      "title": "Tong · Line operators in the Standard Model",
      "authors": [
        "David Tong"
      ],
      "url": "https://arxiv.org/abs/1705.01853",
      "kind": "paper",
      "locatorHint": "Introduction and Section 3: the Z6 quotient, integer-normalized hypercharge, and the additional choice of line operators.",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "block-baez-huerta",
      "title": "The Algebra of Grand Unified Theories",
      "authors": [
        "John C. Baez",
        "John Huerta"
      ],
      "url": "https://arxiv.org/abs/0904.1556",
      "kind": "paper",
      "locatorHint": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.",
      "licenseNote": "Linked primary source supporting original derivations and calculations. No upstream text, software, or representation dataset is redistributed."
    },
    {
      "id": "conrad-pi1-lie",
      "title": "Brian Conrad, Fundamental Groups of Lie Groups",
      "authors": [
        "Brian Conrad"
      ],
      "url": "https://math.stanford.edu/~conrad/210CPage/handouts/pi1stuff.pdf",
      "kind": "lecture-notes",
      "locatorHint": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup",
      "licenseNote": "Linked primary source; the product-group specialization is derived in the local method page."
    }
  ],
  "review": {
    "status": "pending-expert-review",
    "label": "Expert review pending",
    "currentReviews": 0,
    "staleReviews": 0,
    "fingerprint": "441ce501c0056bce82299a7fc9ee6c09a769992f22eff223e3202fb70e825517",
    "reviewers": []
  }
}