{
  "title": "Brst Bv Anomalies Failure Map",
  "question": "Which concrete omitted hypotheses invalidate a Koszul–Tate resolution, a BV gauge-fixing-independence claim, restoration of the quantum master identity, global fermion-phase consistency, or promotion of anomaly cancellation to a positive continuum theory?",
  "takeaway": "A missed Noether identity leaves positive-antifield-number homology; an incomplete antifield action leaves the BV differential non-nilpotent; a Faddeev–Popov zero mode or boundary term invalidates global gauge-fixing independence; a nontrivial local BRST class cannot be removed by an exact counterterm; the SU(2) mod-two index defeats a purely local anomaly test; and absent cutoff-uniform and BRST-positivity estimates block nonperturbative existence even when all known anomaly sums vanish.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checks",
      "label": "complete gauge resolution\\\\controlled BRST complex"
    },
    {
      "id": "a1",
      "row": "claim checks",
      "label": "proper master action and regular Lagrangian family\\\\classical BV reduction and conditional gauge-fixing independence"
    },
    {
      "id": "a2",
      "row": "claim checks",
      "label": "local causal renormalization and trivial $H^{1,d}(s\\mid d)$ obstruction\\\\quantum master identity order by order"
    },
    {
      "id": "a3",
      "row": "claim checks",
      "label": "trivial determinant holonomy plus analytic construction hypotheses\\\\global phase consistency and only then a candidate physical theory"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "omit one reducibility generator or regularity condition\\\\a positive-degree cycle survives and the quotient is not resolved"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "drop a higher antifield coupling or cross a Gribov horizon\\\\$s^2\\neq0$ or a boundary term spoils canonical gauge-fixing independence"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "take $[\\mathcal A]\\neq0$ in $H^{1,d}(s\\mid d)$\\\\no admissible local counterterm restores the quantum master identity"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "use one $SU(2)$ doublet, or provide charge sums without regulator-removal and positivity bounds\\\\Pfaffian holonomy is nontrivial, or physical Hilbert-space existence remains unproved"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "choose gauge fixing",
      "logical_status": "ordered diagnostic sequence, not a proof that later hypotheses follow from earlier ones"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "renormalize",
      "logical_status": "ordered diagnostic sequence, not a proof that later hypotheses follow from earlier ones"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "global tests",
      "logical_status": "ordered diagnostic sequence, not a proof that later hypotheses follow from earlier ones"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "omit resolution input",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "omit properness or slice regularity",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "retain a local obstruction",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "omit global or analytic input",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}