{
  "title": "Brst Bv Anomalies Dependency Map",
  "question": "Which one-way hypotheses connect a regular gauge system to the Koszul–Tate and BRST resolutions, a proper BV master action, admissible gauge fixing, a renormalized quantum master identity, separate local and global anomaly tests, and conditional perturbative observables?",
  "takeaway": "Regular equations, complete identities, and positive-antifield-number acyclicity produce the Koszul–Tate resolution; gauge generators and higher brackets extend it to BRST; a nondegenerate odd pairing and proper classical master action permit regular gauge fixing; a compatible renormalization prescription and trivial local obstruction permit the quantum master identity order by order, while determinant-line holonomy is an additional global test and neither cancellation result implies continuum positivity or convergence.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "regular gauge data\\\\stationary ideal $I$, complete Noether identities\\\\reducibility tower $Z_1,Z_2,\\ldots$"
    },
    {
      "id": "n1",
      "label": "Koszul--Tate resolution\\\\$(\\mathcal K,\\delta)$, $H_{k>0}(\\delta)=0$\\\\$H_0(\\delta)=C^\\infty(\\Sigma)$"
    },
    {
      "id": "n2",
      "label": "BRST complex\\\\$s=\\delta+\\gamma+s_1+\\cdots$, $s^2=0$\\\\$H^0(s)$ models derived invariants"
    },
    {
      "id": "n3",
      "label": "proper classical BV theory\\\\odd symplectic form of degree $-1$\\\\$(S,S)=0$ and complete antifield couplings"
    },
    {
      "id": "n4",
      "label": "regular gauge fixing and renormalization\\\\Lagrangian slice avoids zero modes\\\\local causal products and admissible counterterms"
    },
    {
      "id": "n5",
      "label": "conditional quantum result\\\\renormalized QME if $[\\mathcal A]=0$\\\\formal local observable algebra"
    },
    {
      "id": "branch",
      "label": "local descent: $s a_d^1+d a_{d-1}^2=0$\\\\global phase: $\\operatorname{hol}(\\operatorname{Det}D)=e^{2\\pi i\\xi}$\\\\both trivial plus analytic estimates still needed for a positive continuum theory"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "resolve",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "extend",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "master",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "fix gauge",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "solve QME",
      "logical_status": "licensed implication"
    },
    {
      "from": "n5",
      "to": "branch",
      "label": "independent tests",
      "logical_status": "licensed implication"
    }
  ]
}