{
 "what": "the face-dimension theorem k = |shared| - cons + z, with its evidence",
 "statement": "For the two-population network equilibrium face, the exact tangent dimension (null space of the stacked restricted conservation rows, computed over Q) equals |shared| - cons + z, where z counts exit-free components of the shared subgraph. COROLLARY, verified both ways on every instance below: the combinatorial shortcut |shared| - cons is correct iff z = 0 and undercounts by exactly z otherwise. SCOPE: combinatorics on the constraint matrices, exact over Q, not an enclosure; the dimension of the set, never the point.",
 "verdict": "VERIFIED",
 "originEnsemble": {
  "seed": 20260901,
  "draws": 4000,
  "tested": 3933,
  "shortcutFailures": 572,
  "zPositive": 572,
  "note": "the frontier bench generator replayed call-for-call \u2014 its published count is re-derived here, not quoted"
 },
 "freshEnsemble": {
  "seed": 5320260901,
  "draws": 4000,
  "tested": 3925,
  "shortcutFailures": 571,
  "zPositive": 571
 },
 "failingInstances": {
  "note": "every instance where the shortcut errs, fully specified; deficit == z on all",
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   "deficitIsZ": true
  },
  {
   "L": 4,
   "k": 1,
   "shortcut": 0,
   "z": 1,
   "formulaExact": true,
   "deficitIsZ": true
  },
  {
   "L": 5,
   "k": 1,
   "shortcut": 0,
   "z": 1,
   "formulaExact": true,
   "deficitIsZ": true
  },
  {
   "L": 6,
   "k": 1,
   "shortcut": 0,
   "z": 1,
   "formulaExact": true,
   "deficitIsZ": true
  },
  {
   "L": 7,
   "k": 1,
   "shortcut": 0,
   "z": 1,
   "formulaExact": true,
   "deficitIsZ": true
  },
  {
   "L": 8,
   "k": 1,
   "shortcut": 0,
   "z": 1,
   "formulaExact": true,
   "deficitIsZ": true
  }
 ],
 "originInstance": {
  "k": 6,
  "z": 0,
  "shortcutAgrees": true,
  "note": "the 15-edge instance whose z = 0 made the shortcut look like a law"
 },
 "provenance": {
  "conjecture": "sin-mfg/research/geometry/face_general.py (read-only origin)",
  "settledAt": "frontier-apps/experiments/terra-faces (no git, no record \u2014 evidence rebuilt here)",
  "implementation": "instruments/facelaw/facelaw.py, authored fresh"
 },
 "meta": {
  "date": "2026-09-01",
  "git": "dc8b193",
  "ms": 1448
 }
}