{
  "scope": "Formal available-word sets for one two-parameter cache with K<=2; all binary-letter append histories through depth 4 for d=1..4, branching from retained versions; symbolic fresh-output counts for those dimensions. No proof-assistant verification or unbounded theorem check.",
  "joint_template_search": [
    {
      "K": 0,
      "available_word_sets": 1,
      "joint_update_exists": false
    },
    {
      "K": 1,
      "available_word_sets": 25,
      "joint_update_exists": false
    },
    {
      "K": 2,
      "available_word_sets": 550,
      "joint_update_exists": true
    }
  ],
  "joint_minimum": 2,
  "prefix_checks": [
    {
      "d": 1,
      "append_calls_in_depth_four_tree": 30,
      "symbolic_lower_bounds_by_position": [
        1
      ]
    },
    {
      "d": 2,
      "append_calls_in_depth_four_tree": 340,
      "symbolic_lower_bounds_by_position": [
        2,
        2
      ]
    },
    {
      "d": 3,
      "append_calls_in_depth_four_tree": 1554,
      "symbolic_lower_bounds_by_position": [
        3,
        3,
        2
      ]
    },
    {
      "d": 4,
      "append_calls_in_depth_four_tree": 4680,
      "symbolic_lower_bounds_by_position": [
        4,
        4,
        3,
        2
      ]
    }
  ]
}