{
  "controller": "khovanskii",
  "name": "Iterated-exponential Khovanskii finiteness (depth 2)",
  "kind": "theorem",
  "label": "Depth-2 shown here (proof + zero-count witness). Depth-3 is ALSO proven clean (MachLib.IterExpDepth3Bound.chain3_khovanskii_bound_unconditional); the \u2200N generalization is in progress (lemma-1 + the reduce seam are done, machine-checked). General-depth PfaffianFunction.zero_bound STILL cites the classical Khovanskii axiom \u2014 only depths 1\u20133 are counted, not quoted.",
  "source": {
    "lean": "MachLib.ChainExp2NoZeros",
    "theorem": "MachLib.ChainExp2NoZeros.chain2_khovanskii_bound_unconditional"
  },
  "emitted": [],
  "proof": {
    "theorem": "MachLib.ChainExp2NoZeros.chain2_khovanskii_bound_unconditional",
    "claim": "a polynomial in the tower (x, e\u02e3, e^{e\u02e3}) that is nonzero at even one interior point has only FINITELY many zeros on (a,b) \u2014 the count derived from Rolle's theorem, not cited from Khovanskii",
    "trail_file": "proof/chain2_khovanskii_bound_unconditional.axioms.txt",
    "clean": true,
    "forbidden_axioms_found": [],
    "rederived": "2026-07-02T04:37:24Z",
    "source_artifact": "MachLib.ChainExp2NoZeros   (machlib module; the theorem's own #print axioms)",
    "reverify": "make verify-proof",
    "tier": "REPLAY (re-derive: TOOLCHAIN \u2014 Lean)"
  },
  "sim": {
    "polynomial": "(x^3 - 3x) * exp(exp(x))",
    "interval": "(-2.0, 2.0)",
    "refinement_counts": [
      [
        300,
        3
      ],
      [
        3000,
        3
      ],
      [
        30000,
        3
      ]
    ],
    "samples": 480,
    "trace_csv": "trace.csv",
    "plot_png": "khovanskii_zeros.png",
    "check": {
      "quantity": "zeros of (x\u00b3\u22123x)\u00b7e^(e^x) on (-2.0,2.0)",
      "value": 3,
      "relation": "=",
      "bound": 3,
      "holds": true,
      "context": "count stable under 100\u00d7 grid refinement \u21d2 finite; e^(e^x) never vanishes so adds no zeros \u2014 consistent with the machine-checked bound"
    },
    "tier": "LOCAL"
  },
  "hardware": {
    "tier": "none",
    "note": "\u2014 (a pure finiteness theorem; the sim is a numerical zero-count witness, not hardware)"
  }
}