{
  "schema": "groups.qft.org/physical-setup-examples/1",
  "versions": {
    "schema": "2.0.0",
    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "scope": "Eleven worked fixed monomials, with editable field species, 4D Lorentzian spacetime, a specified global gauge group, complex coefficients, and algebraic equivalence. Both setup and calculation files can be imported into the local-operator workbench and recomputed.",
  "setupSchema": "groups.qft.org/physical-setup/1",
  "calculationSchema": "groups.qft.org/physical-calculation/1",
  "jsonSchema": "/data/physical-setup.schema.json",
  "examples": [
    {
      "id": "weinberg-3",
      "title": "Weinberg operator · three flavors",
      "summary": "L²H² has six complex invariants at canonical dimension five: a symmetric 3 × 3 lepton-flavor tensor, with one Higgs doublet.",
      "expectedCount": "6",
      "guide": "/physics/weinberg-operator/",
      "route": "/tools/local-operators/?example=weinberg-3",
      "setup": "/data/physical-setups/weinberg-3.json",
      "calculation": "/data/physical-calculations/weinberg-3.json"
    },
    {
      "id": "weinberg-1",
      "title": "Weinberg operator · one flavor",
      "summary": "The one-flavor L²H² sector has one invariant. Joint spin, weak-isospin and fermion statistics are essential to its nonzero value.",
      "expectedCount": "1",
      "guide": "/physics/weinberg-operator/",
      "route": "/tools/local-operators/?example=weinberg-1",
      "setup": "/data/physical-setups/weinberg-1.json",
      "calculation": "/data/physical-calculations/weinberg-1.json"
    },
    {
      "id": "up-yukawa-3",
      "title": "Up-type Yukawa · three flavors",
      "summary": "Q uᶜ H has nine complex coefficients at dimension four. The two independent three-flavor fermion spaces give an unrestricted 3 × 3 Yukawa matrix.",
      "expectedCount": "9",
      "guide": "/physics/yukawa-and-four-fermion/",
      "route": "/tools/local-operators/?example=up-yukawa-3",
      "setup": "/data/physical-setups/up-yukawa-3.json",
      "calculation": "/data/physical-calculations/up-yukawa-3.json"
    },
    {
      "id": "higgs-quartic",
      "title": "Higgs quartic",
      "summary": "With one Higgs doublet and an explicit conjugate variable, H²(H†)² has one invariant at dimension four.",
      "expectedCount": "1",
      "guide": "/reference/local-operator-counting/",
      "route": "/tools/local-operators/?example=higgs-quartic",
      "setup": "/data/physical-setups/higgs-quartic.json",
      "calculation": "/data/physical-calculations/higgs-quartic.json"
    },
    {
      "id": "higgs-sextic",
      "title": "Higgs sextic",
      "summary": "H³(H†)³ has one invariant at dimension six. Its uniqueness is a statement about the symmetric powers of a single Higgs doublet.",
      "expectedCount": "1",
      "guide": "/reference/local-operator-counting/",
      "route": "/tools/local-operators/?example=higgs-sextic",
      "setup": "/data/physical-setups/higgs-sextic.json",
      "calculation": "/data/physical-calculations/higgs-sextic.json"
    },
    {
      "id": "sterile-majorana-3",
      "title": "Sterile Weyl mass · three flavors",
      "summary": "N² has six invariants at dimension three. A gauge-singlet Weyl mass matrix is symmetric in its three flavor indices.",
      "expectedCount": "6",
      "guide": "/learn/fields-lorentz-and-statistics/",
      "route": "/tools/local-operators/?example=sterile-majorana-3",
      "setup": "/data/physical-setups/sterile-majorana-3.json",
      "calculation": "/data/physical-calculations/sterile-majorana-3.json"
    },
    {
      "id": "weak-doublet-mass-1",
      "title": "Neutral weak doublet · one flavor",
      "summary": "A single charge-zero left-handed weak doublet has no algebraic mass invariant: the candidate contraction vanishes by Fermi statistics.",
      "expectedCount": "0",
      "guide": "/learn/fields-lorentz-and-statistics/",
      "route": "/tools/local-operators/?example=weak-doublet-mass-1",
      "setup": "/data/physical-setups/weak-doublet-mass-1.json",
      "calculation": "/data/physical-calculations/weak-doublet-mass-1.json"
    },
    {
      "id": "weak-doublet-mass-2",
      "title": "Neutral weak doublet · two flavors",
      "summary": "Two charge-zero left-handed weak doublets admit one antisymmetric flavor contraction at dimension three.",
      "expectedCount": "1",
      "guide": "/learn/fields-lorentz-and-statistics/",
      "route": "/tools/local-operators/?example=weak-doublet-mass-2",
      "setup": "/data/physical-setups/weak-doublet-mass-2.json",
      "calculation": "/data/physical-calculations/weak-doublet-mass-2.json"
    },
    {
      "id": "weak-doublet-mass-3",
      "title": "Neutral weak doublet · three flavors",
      "summary": "Three charge-zero left-handed weak doublets admit three antisymmetric flavor contractions at dimension three.",
      "expectedCount": "3",
      "guide": "/learn/fields-lorentz-and-statistics/",
      "route": "/tools/local-operators/?example=weak-doublet-mass-3",
      "setup": "/data/physical-setups/weak-doublet-mass-3.json",
      "calculation": "/data/physical-calculations/weak-doublet-mass-3.json"
    },
    {
      "id": "qqql-3",
      "title": "QQQL · three flavors",
      "summary": "Q³L has 57 independent complex invariants at dimension six for three quark and three lepton flavors, before adding its Hermitian-conjugate sector.",
      "expectedCount": "57",
      "guide": "/physics/yukawa-and-four-fermion/",
      "route": "/tools/local-operators/?example=qqql-3",
      "setup": "/data/physical-setups/qqql-3.json",
      "calculation": "/data/physical-calculations/qqql-3.json"
    },
    {
      "id": "opposite-chirality",
      "title": "Opposite chirality · no derivative",
      "summary": "A left-handed and a right-handed singlet Weyl field give a Lorentz vector, not a scalar. Their derivative-free bilinear therefore has count zero.",
      "expectedCount": "0",
      "guide": "/learn/fields-lorentz-and-statistics/",
      "route": "/tools/local-operators/?example=opposite-chirality",
      "setup": "/data/physical-setups/opposite-chirality.json",
      "calculation": "/data/physical-calculations/opposite-chirality.json"
    }
  ]
}