{
  "schema": "er33t.elementary-rule-identities.v1",
  "date": "2026-09-30",
  "kind": "exhaustive local enumeration with analytical induction",
  "encoding": "i = 4L + 2C + R; output = (rule >> i) & 1; outputs listed from 111 to 000",
  "rows": [
    {
      "input_LCR": "000",
      "rule110_output": 0,
      "complement_input_LCR": "111",
      "rule137_on_complement": 1,
      "conjugacy_holds": true,
      "rule42_output": 0,
      "rule42_gate_holds": true
    },
    {
      "input_LCR": "001",
      "rule110_output": 1,
      "complement_input_LCR": "110",
      "rule137_on_complement": 0,
      "conjugacy_holds": true,
      "rule42_output": 1,
      "rule42_gate_holds": true
    },
    {
      "input_LCR": "010",
      "rule110_output": 1,
      "complement_input_LCR": "101",
      "rule137_on_complement": 0,
      "conjugacy_holds": true,
      "rule42_output": 0,
      "rule42_gate_holds": true
    },
    {
      "input_LCR": "011",
      "rule110_output": 1,
      "complement_input_LCR": "100",
      "rule137_on_complement": 0,
      "conjugacy_holds": true,
      "rule42_output": 1,
      "rule42_gate_holds": true
    },
    {
      "input_LCR": "100",
      "rule110_output": 0,
      "complement_input_LCR": "011",
      "rule137_on_complement": 1,
      "conjugacy_holds": true,
      "rule42_output": 0,
      "rule42_gate_holds": true
    },
    {
      "input_LCR": "101",
      "rule110_output": 1,
      "complement_input_LCR": "010",
      "rule137_on_complement": 0,
      "conjugacy_holds": true,
      "rule42_output": 1,
      "rule42_gate_holds": true
    },
    {
      "input_LCR": "110",
      "rule110_output": 1,
      "complement_input_LCR": "001",
      "rule137_on_complement": 0,
      "conjugacy_holds": true,
      "rule42_output": 0,
      "rule42_gate_holds": true
    },
    {
      "input_LCR": "111",
      "rule110_output": 0,
      "complement_input_LCR": "000",
      "rule137_on_complement": 1,
      "conjugacy_holds": true,
      "rule42_output": 0,
      "rule42_gate_holds": true
    }
  ],
  "local_cases": 8,
  "rule137_identity": "F137(1-L,1-C,1-R)=1-F110(L,C,R)",
  "rule42_identity": "F42(L,C,R)=R AND NOT(L AND C)",
  "global_induction": "At time 0 take complementary complete states and complementary boundary/background values. If the states are complementary at time t, each local neighborhood is complementary. The exhaustive identity gives complementary outputs at every site. Thus the states remain complementary at time t + 1, and induction gives the relation for all synchronous updates under the specified transformation.",
  "universality_scope": "The identity transfers an existing encoded computation bijectively. It does not independently prove universality, classify Rule 42, or establish an all-time property of Rule 30.",
  "provider_calls_for_proof": 0,
  "sources": [
    {
      "title": "Cook (2004): Universality in Elementary Cellular Automata",
      "url": "https://doi.org/10.25088/complexsystems.15.1.1"
    },
    {
      "title": "Wolfram Atlas: equivalent elementary rules",
      "url": "https://atlas.wolfram.com/01/01/views/10/TableView.html"
    }
  ]
}