{
  "scope": "Finite example models only. All binary words through length5 paired in a transition-monoid-tagged rope model; Dyck2 streaming words through length7 with retained-version/fork controls; seven same-view distinct-future stack examples. The models do not implement a local Dyck join, quantify over arbitrary representations, or formally prove the universal theorem.",
  "regular_join_cases": 3969,
  "Dyck_streaming_cases": 21845,
  "same_view_distinct_future_examples": [
    {
      "radius": 0,
      "x": "[",
      "y": "(",
      "suffix": "]"
    },
    {
      "radius": 1,
      "x": "[(",
      "y": "((",
      "suffix": ")]"
    },
    {
      "radius": 2,
      "x": "[((",
      "y": "(((",
      "suffix": "))]"
    },
    {
      "radius": 3,
      "x": "[(((",
      "y": "((((",
      "suffix": ")))]"
    },
    {
      "radius": 4,
      "x": "[((((",
      "y": "(((((",
      "suffix": "))))]"
    },
    {
      "radius": 5,
      "x": "[(((((",
      "y": "((((((",
      "suffix": ")))))]"
    },
    {
      "radius": 6,
      "x": "[((((((",
      "y": "(((((((",
      "suffix": "))))))]"
    }
  ],
  "growing_integer_Dyck1_control": {
    "word_length_limit_per_operand": 4,
    "pairs": 961,
    "scope": "Unit-cost unbounded integers, outside the fixed finite-label theorem."
  },
  "finite_state_only_right_operand_control": "epsilon and b reach the same DFA state from q0, but a and ab have different acceptance",
  "formal_verification": "No proof assistant; universal argument is written separately."
}