{
  "component": "BabyBear field F_p + degree-4 extension F_{p^4} (port spec component (a), FOUNDATION)",
  "precompile": "0x0000000000000000000000000000000000000ae8",
  "scope": "port spec section 2; the field-arithmetic foundation the whole STARK verifier is built on",
  "validates": "F_p and F_{p^4} field axioms; Fermat inverse vs independent extended-Euclid bignum; generator has order p-1; two-adic generator order 2^k; W is a QNR so x^4-W is irreducible (real field); across Python/Node/Java",
  "does_not_validate": "any real STARK proof (this is only component (a) of six); and that GENERATOR=31 / W=11 / two_adic_generator(27) are Plonky3's EXACT constants ([VERIFY]/[MEASURE] vs p3-baby-bear at pinned SP1 v6.1.0)",
  "top_level_verifier": "unchanged, still fail-closed (returns EMPTY for every input)",
  "constants": {
    "P": 2013265921,
    "P_expr": "2^31 - 2^27 + 1 = 15*2^27 + 1 = 0x78000001",
    "generator": 31,
    "two_adicity": 27,
    "two_adic_generator_27": 440564289,
    "W": 11
  },
  "cross_language": [
    "java",
    "node",
    "python"
  ],
  "flags": [
    "[VERIFY] GENERATOR = 31 is Plonky3 p3-baby-bear's chosen multiplicative generator (order = p-1 is PROVEN here; the identity as Plonky3's is [VERIFY])",
    "[VERIFY] W = 11 is Plonky3's BabyBear BinomialExtensionField<_,4> non-residue (x^4-11 irreducibility is PROVEN here; the identity as Plonky3's W is [VERIFY]/[MEASURE])",
    "[VERIFY] two_adic_generator(27) exact value vs Plonky3 (order 2^27 is PROVEN here)",
    "[MEASURE] byte-for-byte conformance of Fp/Fp4 outputs vs p3-baby-bear / p3-field unit vectors at the pinned revision (needs the pinned Plonky3 source)"
  ],
  "notes": [
    "multiplicative generator 31 proven order = p-1 = 2013265920 [VERIFY it is Plonky3's]",
    "two_adic_generator(27) = 440564289, order proven 2^27 [VERIFY exact Plonky3 value]",
    "W = 11: 11^((p-1)/2) == p-1 (QNR) and p == 1 mod 4  =>  x^4 - 11 irreducible (Lidl-Niederreiter Thm 3.75)  =>  F_(p^4) is a genuine field [VERIFY it is Plonky3's W]",
    "ord(W=11) = 671088640 = 2^27 * 5; ord(x) = 4*ord(W) = 2684354560 divides 4(p-1)",
    "cross-language implementations compared: java, node, python"
  ],
  "passed": 396242,
  "failed": 0,
  "total": 396242
}