{
  "target": "GF(2^4) constant multipliers mod x^4+x+1 (MCM / fixed-coefficient datapath primitive)",
  "engine": "SAT exact synthesis (z3) with optimality certificate + exhaustive re-verify",
  "results": [
    {
      "constant": 1,
      "min_gates": 0,
      "naive_xor": 0,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=0 SAT; r'<0 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 2,
      "min_gates": 1,
      "naive_xor": 1,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=1 SAT; r'<1 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 3,
      "min_gates": 4,
      "naive_xor": 5,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=4 SAT; r'<4 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 4,
      "min_gates": 2,
      "naive_xor": 2,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=2 SAT; r'<2 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 5,
      "min_gates": 4,
      "naive_xor": 6,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=4 SAT; r'<4 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 6,
      "min_gates": 5,
      "naive_xor": 5,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=5 SAT; r'<5 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 7,
      "min_gates": 5,
      "naive_xor": 9,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=5 SAT; r'<5 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 8,
      "min_gates": 3,
      "naive_xor": 3,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=3 SAT; r'<3 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 9,
      "min_gates": 1,
      "naive_xor": 1,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=1 SAT; r'<1 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 10,
      "min_gates": 4,
      "naive_xor": 8,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=4 SAT; r'<4 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 11,
      "min_gates": 4,
      "naive_xor": 6,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=4 SAT; r'<4 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 12,
      "min_gates": 4,
      "naive_xor": 5,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=4 SAT; r'<4 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 13,
      "min_gates": 2,
      "naive_xor": 3,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=2 SAT; r'<2 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 14,
      "min_gates": 4,
      "naive_xor": 8,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=4 SAT; r'<4 all UNSAT (machine-checked lower bound)"
    },
    {
      "constant": 15,
      "min_gates": 3,
      "naive_xor": 6,
      "proved_minimal": true,
      "verified": true,
      "certificate": "r=3 SAT; r'<3 all UNSAT (machine-checked lower bound)"
    }
  ],
  "total_min_gates": 46,
  "total_unshared_naive_xor": 68,
  "reduction_vs_unshared_naive_pct": 32.4,
  "all_proved_minimal": true,
  "elapsed_s": 9.6,
  "headline": "PROVEN OPTIMAL: for each of the 15 constants, no smaller 2-input-gate circuit exists (machine-checked, r-1 UNSAT). This beats ANY construction, including CSE-optimized.",
  "baseline_caveat": "total_unshared_naive_xor counts INDEPENDENT per-output XOR trees with NO common-subexpression sharing -- a WEAK baseline. reduction_vs_unshared_naive_pct is illustrative only, NOT a claim of beating a real (CSE-optimized or published) baseline. The load-bearing claim is proven-optimality, not the percentage.",
  "ownability_note": "A specific proven-minimal constant-multiplier circuit is patent-eligible (a machine). Honest scope: these are small/known primitives -- the value is the PROVEN-OPTIMAL engine on a real deployed family, not a claim to beat a published SOTA. UNPINNED; not booked into the wireless base."
}