[
  {
    "equation": "x + ",
    "type": "non_expression",
    "expected": false,
    "reason": "Incomplete binary operator; missing second operand."
  },
  {
    "equation": "3 * (2 + 1) = 9",
    "type": "identity",
    "expected": true,
    "reason": "Arithmetic identity; should pass consistency gate."

  },
  {
    "equation": "$",
    "type": "ambiguous_delimiter",
    "expected": false,
    "reason": "Single dollar sign without content; unclear if TeX math or punctuation."
  },
  {
    "equation": "sin(x) + log(y)",
    "type": "expression",
    "expected": true,
    "reason": "Standard multi-term expression."
  },
  {
    "equation": "x^2 + y^2 = z^2",
    "type": "identity",
    "expected": true,
    "reason": "Pythagorean theorem; valid identity."
  },
  {
    "equation": "10 / 0",
    "type": "undefined_operation",
    "expected": false,
    "reason": "Division by zero in a discrete context without limit definition."
  },
  {
    "equation": "x + y = x + y",
    "type": "tautology",
    "expected": true,
    "reason": "Trivial identity; should pass consistency gate."
  },
  {
    "equation": "(a + b) * c",
    "type": "expression",
    "expected": true,
    "reason": "Standard nested expression."
  },
  {
    "equation": "x \\cdot y \\cdot z",
    "type": "expression",
    "expected": true,
    "reason": "Dot notation; should be accepted by the consistency gate."

  },
  {
    "equation": "2 + 3 = 6",
    "type": "dimension_mismatch",
    "expected": true,
    "reason": "Well-formed structure; arithmetic truth is secondary to consistency."

  }
]