{
  "step": "C456",
  "theorem": "For integer x,H, E_H^2(x) is integer iff529 divides H-x. For a complete integer-endpoint two-E path, H-x=529(x-y)/96; H is integral iff96 divides x-y.",
  "word_gate_counts": [
    {
      "first_pivot": 2756,
      "second_pivot": 2756,
      "effective_pivot": "2756",
      "effective_pivot_sourced": true,
      "complete_paths": 1,
      "integer_output": 1,
      "sourced_output": 1,
      "complete": 1,
      "nonzero_integer_output": 0,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 2756,
      "second_pivot": 3620,
      "effective_pivot": "3170",
      "effective_pivot_sourced": true,
      "complete_paths": 1,
      "integer_output": 1,
      "sourced_output": 1,
      "complete": 1,
      "nonzero_integer_output": 0,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 2758,
      "second_pivot": 2758,
      "effective_pivot": "2758",
      "effective_pivot_sourced": true,
      "complete_paths": 1,
      "integer_output": 1,
      "sourced_output": 1,
      "complete": 1,
      "nonzero_integer_output": 0,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 2816,
      "second_pivot": 2816,
      "effective_pivot": "2816",
      "effective_pivot_sourced": true,
      "complete_paths": 1,
      "integer_output": 1,
      "sourced_output": 1,
      "complete": 1,
      "nonzero_integer_output": 0,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 2818,
      "second_pivot": 2818,
      "effective_pivot": "2818",
      "effective_pivot_sourced": true,
      "complete_paths": 1,
      "integer_output": 1,
      "sourced_output": 1,
      "complete": 1,
      "nonzero_integer_output": 0,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 3620,
      "second_pivot": 2756,
      "effective_pivot": "3206",
      "effective_pivot_sourced": true,
      "complete_paths": 2,
      "integer_output": 2,
      "sourced_output": 2,
      "complete": 2,
      "nonzero_integer_output": 1,
      "nonzero_sourced_output": 1,
      "nonzero_complete": 1
    },
    {
      "first_pivot": 3620,
      "second_pivot": 3620,
      "effective_pivot": "3620",
      "effective_pivot_sourced": true,
      "complete_paths": 1,
      "integer_output": 1,
      "sourced_output": 1,
      "complete": 1,
      "nonzero_integer_output": 0,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 3622,
      "second_pivot": 2758,
      "effective_pivot": "3208",
      "effective_pivot_sourced": true,
      "complete_paths": 0,
      "integer_output": 2,
      "sourced_output": 2,
      "complete": 0,
      "nonzero_integer_output": 1,
      "nonzero_sourced_output": 1,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 3622,
      "second_pivot": 3622,
      "effective_pivot": "3622",
      "effective_pivot_sourced": true,
      "complete_paths": 1,
      "integer_output": 1,
      "sourced_output": 1,
      "complete": 1,
      "nonzero_integer_output": 0,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 3680,
      "second_pivot": 2816,
      "effective_pivot": "3266",
      "effective_pivot_sourced": true,
      "complete_paths": 0,
      "integer_output": 2,
      "sourced_output": 1,
      "complete": 0,
      "nonzero_integer_output": 1,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 3680,
      "second_pivot": 3680,
      "effective_pivot": "3680",
      "effective_pivot_sourced": true,
      "complete_paths": 1,
      "integer_output": 1,
      "sourced_output": 1,
      "complete": 1,
      "nonzero_integer_output": 0,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 3682,
      "second_pivot": 2818,
      "effective_pivot": "3268",
      "effective_pivot_sourced": true,
      "complete_paths": 0,
      "integer_output": 2,
      "sourced_output": 1,
      "complete": 0,
      "nonzero_integer_output": 1,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    },
    {
      "first_pivot": 3682,
      "second_pivot": 3682,
      "effective_pivot": "3682",
      "effective_pivot_sourced": true,
      "complete_paths": 1,
      "integer_output": 1,
      "sourced_output": 1,
      "complete": 1,
      "nonzero_integer_output": 0,
      "nonzero_sourced_output": 0,
      "nonzero_complete": 0
    }
  ],
  "integer_output_cases": [
    {
      "first_pivot": 2756,
      "second_pivot": 2756,
      "H": 2756,
      "input": 2756,
      "middle": "2756",
      "output": 2756,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 2756,
      "second_pivot": 3620,
      "H": 3170,
      "input": 3170,
      "middle": "3206",
      "output": 3170,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 2758,
      "second_pivot": 2758,
      "H": 2758,
      "input": 2758,
      "middle": "2758",
      "output": 2758,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 2816,
      "second_pivot": 2816,
      "H": 2816,
      "input": 2816,
      "middle": "2816",
      "output": 2816,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 2818,
      "second_pivot": 2818,
      "H": 2818,
      "input": 2818,
      "middle": "2818",
      "output": 2818,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 3620,
      "second_pivot": 2756,
      "H": 3206,
      "input": 2148,
      "middle": "2020",
      "output": 1956,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 2
    },
    {
      "first_pivot": 3620,
      "second_pivot": 2756,
      "H": 3206,
      "input": 3206,
      "middle": "3170",
      "output": 3206,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 3620,
      "second_pivot": 3620,
      "H": 3620,
      "input": 3620,
      "middle": "3620",
      "output": 3620,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 3622,
      "second_pivot": 2758,
      "H": 3208,
      "input": 2150,
      "middle": "2022",
      "output": 1958,
      "sourced_output": true,
      "complete": false,
      "K_input_multiple": 2
    },
    {
      "first_pivot": 3622,
      "second_pivot": 2758,
      "H": 3208,
      "input": 3208,
      "middle": "3172",
      "output": 3208,
      "sourced_output": true,
      "complete": false,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 3622,
      "second_pivot": 3622,
      "H": 3622,
      "input": 3622,
      "middle": "3622",
      "output": 3622,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 3680,
      "second_pivot": 2816,
      "H": 3266,
      "input": 2737,
      "middle": "2655",
      "output": 2641,
      "sourced_output": false,
      "complete": false,
      "K_input_multiple": 1
    },
    {
      "first_pivot": 3680,
      "second_pivot": 2816,
      "H": 3266,
      "input": 3266,
      "middle": "3230",
      "output": 3266,
      "sourced_output": true,
      "complete": false,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 3680,
      "second_pivot": 3680,
      "H": 3680,
      "input": 3680,
      "middle": "3680",
      "output": 3680,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 3682,
      "second_pivot": 2818,
      "H": 3268,
      "input": 2739,
      "middle": "2657",
      "output": 2643,
      "sourced_output": false,
      "complete": false,
      "K_input_multiple": 1
    },
    {
      "first_pivot": 3682,
      "second_pivot": 2818,
      "H": 3268,
      "input": 3268,
      "middle": "3232",
      "output": 3268,
      "sourced_output": true,
      "complete": false,
      "K_input_multiple": 0
    },
    {
      "first_pivot": 3682,
      "second_pivot": 3682,
      "H": 3682,
      "input": 3682,
      "middle": "3682",
      "output": 3682,
      "sourced_output": true,
      "complete": true,
      "K_input_multiple": 0
    }
  ],
  "displacement_checks": [
    {
      "path": [
        2756,
        2756,
        2756
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    },
    {
      "path": [
        3170,
        3206,
        3170
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    },
    {
      "path": [
        2735,
        2733,
        2731
      ],
      "displacement": 4,
      "effective_gap": "529/24",
      "gap_forced_by_displacement": "529/24"
    },
    {
      "path": [
        2758,
        2758,
        2758
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    },
    {
      "path": [
        2620,
        2608,
        2520
      ],
      "displacement": 100,
      "effective_gap": "13225/24",
      "gap_forced_by_displacement": "13225/24"
    },
    {
      "path": [
        2816,
        2816,
        2816
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    },
    {
      "path": [
        2818,
        2818,
        2818
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    },
    {
      "path": [
        2148,
        2020,
        1956
      ],
      "displacement": 192,
      "effective_gap": "1058",
      "gap_forced_by_displacement": "1058"
    },
    {
      "path": [
        3206,
        3170,
        3206
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    },
    {
      "path": [
        3620,
        3620,
        3620
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    },
    {
      "path": [
        2746,
        2670,
        2582
      ],
      "displacement": 164,
      "effective_gap": "21689/24",
      "gap_forced_by_displacement": "21689/24"
    },
    {
      "path": [
        3622,
        3622,
        3622
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    },
    {
      "path": [
        3105,
        3055,
        3081
      ],
      "displacement": 24,
      "effective_gap": "529/4",
      "gap_forced_by_displacement": "529/4"
    },
    {
      "path": [
        3680,
        3680,
        3680
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    },
    {
      "path": [
        3682,
        3682,
        3682
      ],
      "displacement": 0,
      "effective_gap": "0",
      "gap_forced_by_displacement": "0"
    }
  ],
  "totals": {
    "integer_output": 17,
    "sourced_output": 15,
    "complete": 11,
    "nonzero_integer_output": 4,
    "nonzero_sourced_output": 2,
    "nonzero_complete": 1
  }
}
