{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R786",
  "content_hash": null,
  "slug": "threshold-six-three-claim-five-and-circuit",
  "type": "claim",
  "title": "Five AND gates suffice",
  "summary": "Two three-bit counters and a short carry calculation give a five-AND threshold circuit.",
  "relevance": "For Multiplicative complexity of the six-bit threshold-at-least-three function, record threshold-six-three-claim-five-and-circuit (“Five AND gates suffice”) records a bound, answer, status fact, or structural consequence. The record states: Two three-bit counters and a short carry calculation give a five-AND threshold circuit.",
  "relevance_source": "recorded",
  "body": "Define\n\\[\n\\begin{aligned}\ns_0&=x_1\\oplus x_2\\oplus x_3,&\na_0&=((x_1\\oplus x_2)(x_1\\oplus x_3))\\oplus x_1,\\\\\ns_1&=x_4\\oplus x_5\\oplus x_6,&\na_1&=((x_4\\oplus x_5)(x_4\\oplus x_6))\\oplus x_4,\\\\\na_2&=s_0s_1,&&\\\\\na_3&=((a_0\\oplus a_1)(a_0\\oplus a_2))\\oplus a_0,&&\\\\\na_4&=(s_0\\oplus s_1)(a_0\\oplus a_1\\oplus a_2).&&\n\\end{aligned}\n\\]\nThe output is \\(a_3\\oplus a_4\\). Each displayed product is one two-input AND, giving five in total.\n\nHere \\(s_i\\) and \\(a_i\\) are the low bit and carry for each initial triple. After combining them, \\(s_0\\oplus s_1\\), \\(a_0\\oplus a_1\\oplus a_2\\), and \\(a_3\\) are the three bits of the Hamming weight. On the attainable weights 0 through 6, the predicate for at least 3 is the high bit XOR the product of the low and middle bits. That is exactly \\(a_3\\oplus a_4\\).",
  "status": "supported",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all 64 inputs to the displayed six-input XOR-AND straight-line program",
    "bounds": {
      "input_rows": {
        "min": 64,
        "max": 64
      },
      "and_gates": {
        "min": 5,
        "max": 5
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/j.tcs.2008.01.030",
      "locator": "Five-AND circuit derived and exhaustively checked by TheoremDB entry research on 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.tcs.2008.01.030",
    "locator": "Five-AND circuit derived and exhaustively checked by TheoremDB entry research on 2026-07-25"
  },
  "relations": [
    {
      "slug": "R784",
      "title": "The certified multiplicative-complexity interval is 3 to 5",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R782",
      "title": "Exhaustive verifier for the five-AND circuit",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "threshold-at-least-three-six-multiplicative-complexity",
      "title": "threshold at least three six multiplicative complexity",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}