{
 "what": "certified critical-point count for the T4 equilibrium and its potential",
 "statement": "EVERY function in the certified enclosure ball (radius 2.367e-13, l1_nu with nu=1.02) around the T4 candidate density has EXACTLY 2 strict local maxima and 2 strict local minima on the torus; the potential has exactly 1 well(s). Peaks exceed wells: gain-weighted well-counting beats flat well-counting — the equilibrium re-weights a harmonic the potential already contains (it does not \"invent structure\").",
 "verdict": "VERIFIED",
 "source": {
  "cert": "certs/terra-recert-t4.json",
  "verdict": "VERIFIED",
  "r": 2.367300631563991e-13,
  "nu": 1.02,
  "N": 96
 },
 "instance": {
  "sigma": 0.002,
  "gamma": 0.01,
  "A1": 0.003,
  "A2": 0.00045,
  "A3": 0,
  "N": 96,
  "nu": 1.02,
  "G": 1536,
  "a": 0.5,
  "exactBinary64Hex": {
   "sigma": "0x1.0624dd2f1a9fcp-9",
   "gamma": "0x1.47ae147ae147bp-7",
   "A1": "0x1.89374bc6a7efap-9",
   "A2": "0x1.d7dbf487fcb92p-12",
   "A3": "0x0.0p+0"
  },
  "note": "the theorem is about these exact binary64 values, not the decimal strings that name them (TERRA-PORT.md coefficient condition)"
 },
 "ballPads": {
  "d1": 2.7630878844114078e-11,
  "d2": 1.290145079290798e-8,
  "d3": 0.000010164778628132367
 },
 "m": {
  "label": "density m*",
  "maxima": 2,
  "minima": 2,
  "chain": [
   0,
   0.41812416342412373,
   0.5
  ],
  "curv": [
   "+",
   "-",
   "+"
  ],
  "regions": [
   {
    "kind": "curv",
    "lo": 0,
    "hi": 0.11164160865359836,
    "sign": "+",
    "margin": 2.7417292308121426,
    "grid": 256
   },
   {
    "kind": "slope",
    "lo": 0.11164160865359836,
    "hi": 0.2370788576808355,
    "sign": "+",
    "margin": 0.5115238981802818,
    "grid": 256
   },
   {
    "kind": "curv",
    "lo": 0.2370788576808355,
    "hi": 0.44282277123099484,
    "sign": "-",
    "margin": 0.18902239631209813,
    "grid": 256
   },
   {
    "kind": "slope",
    "lo": 0.44282277123099484,
    "hi": 0.46738552220375773,
    "sign": "-",
    "margin": 0.011416751488943364,
    "grid": 256
   },
   {
    "kind": "curv",
    "lo": 0.46738552220375773,
    "hi": 0.5,
    "sign": "+",
    "margin": 0.21308021341464092,
    "grid": 256
   }
  ],
  "minMargin": 0.011416751488943364,
  "assertions": {
   "contiguous": true,
   "curvatureAlternates": true,
   "slopeMatchesChain": true,
   "countFrom": "certified region signs only"
  }
 },
 "V": {
  "label": "potential V",
  "maxima": 1,
  "minima": 1,
  "chain": [
   0,
   0.5
  ],
  "curv": [
   "-",
   "+"
  ],
  "regions": [
   {
    "kind": "curv",
    "lo": 0,
    "hi": 0.1286581438374038,
    "sign": "-",
    "margin": 0.07848193578033713,
    "grid": 256
   },
   {
    "kind": "slope",
    "lo": 0.1286581438374038,
    "hi": 0.27865814383740384,
    "sign": "-",
    "margin": 0.016522216303896928,
    "grid": 256
   },
   {
    "kind": "curv",
    "lo": 0.27865814383740384,
    "hi": 0.5,
    "sign": "+",
    "margin": 0.04666696728809235,
    "grid": 256
   }
  ],
  "minMargin": 0.016522216303896928,
  "assertions": {
   "contiguous": true,
   "curvatureAlternates": true,
   "slopeMatchesChain": true,
   "countFrom": "certified region signs only"
  }
 },
 "peaks": 2,
 "wells": 1,
 "peaksExceedWells": true,
 "rigor": {
  "countFrom": "certified region signs only (float is the proposer, never the authority)",
  "coefficientProducts": "outward interval arithmetic, 2pi enclosed, trig argument rounding chain bounded",
  "cellPad": "(L + ballPadNextDeriv) * h/2 + ballPad — the ball Lipschitz is folded in"
 },
 "meta": {
  "date": "2026-09-01",
  "git": "5c1b0c2"
 }
}