{
  "scope": "Counterexample to exhaustiveness of the intended seven parameter families in Theorem 2.6; no thinness verdict.",
  "arithmetic": "exact rational and integer",
  "p": "x**13 + x**12 - x**7 - x**6 + x + 1",
  "q": "x**13 - 3*x**12 + 4*x**11 - 5*x**10 + 7*x**9 - 8*x**8 + 8*x**7 - 8*x**6 + 8*x**5 - 7*x**4 + 5*x**3 - 4*x**2 + 3*x - 1",
  "p_factors": "Phi_2 * Phi_36",
  "q_factors": "Phi_1^3 * Phi_3 * Phi_6 * Phi_18",
  "degree": 13,
  "spectrum_gcd": "1",
  "orbit_determinant": "-4608",
  "observability_determinant": "1",
  "form_integer_scale": "72",
  "inertia_positive_negative_zero": [
    12,
    1,
    0
  ],
  "gram_first_row": [
    "-2",
    "-4",
    "-8",
    "-13",
    "-20",
    "-28",
    "-36",
    "-44",
    "-52",
    "-59",
    "-64",
    "-68",
    "-72"
  ],
  "irreducibility_witness": "C-I has image span(u) and kernel ker(e_n^T); B-orbit of u and dual B-orbit of e_n^T both span.",
  "primitivity_witness": "Prime degree 13 forces any nontrivial block system to have 1-dimensional blocks. Then B would be monomial and hence semisimple, contrary to its cyclic minimal polynomial having (x-1)^3.",
  "unmatched_p_shapes": [
    {
      "family": "M1",
      "p": "x**13 - 2*x**12 + 2*x**11 - 2*x**10 + 2*x**9 - 2*x**8 + 2*x**7 - 2*x**6 + 2*x**5 - 2*x**4 + 2*x**3 - 2*x**2 + 2*x - 1"
    },
    {
      "family": "M2",
      "p": "(x + 1)*(x**12 + 1)"
    },
    {
      "family": "M3",
      "p": "(x + 1)**2*(x**11 + 1)"
    },
    {
      "family": "N1:k=1,ell=13",
      "p": "x**13 - 2*x**12 + 2*x**11 - 2*x**10 + 2*x**9 - 2*x**8 + 2*x**7 - 2*x**6 + 2*x**5 - 2*x**4 + 2*x**3 - 2*x**2 + 2*x - 1"
    },
    {
      "family": "N1:k=13,ell=1",
      "p": "x**13 - x**12 + x**11 - x**10 + x**9 - x**8 + x**7 - x**6 + x**5 - x**4 + x**3 - x**2 + x - 1"
    },
    {
      "family": "N2:k=1,ell=12",
      "p": "x**13 - x**12 + x - 1"
    },
    {
      "family": "N2:k=2,ell=6",
      "p": "x**13 - x**12 + x**7 - x**6 + x - 1"
    },
    {
      "family": "N2:k=3,ell=4",
      "p": "x**13 - x**12 + x**9 - x**8 + x**5 - x**4 + x - 1"
    },
    {
      "family": "N2:k=4,ell=3",
      "p": "x**13 - x**12 + x**10 - x**9 + x**7 - x**6 + x**4 - x**3 + x - 1"
    },
    {
      "family": "N2:k=6,ell=2",
      "p": "x**13 - x**12 + x**11 - x**10 + x**9 - x**8 + x**7 - x**6 + x**5 - x**4 + x**3 - x**2 + x - 1"
    },
    {
      "family": "N2:k=12,ell=1",
      "p": "x**13 - 1"
    },
    {
      "family": "N3:k=1,ell=12",
      "p": "x**13 + x**12 + x + 1"
    },
    {
      "family": "N3:k=2,ell=6",
      "p": "x**13 + x**12 + x**7 + x**6 + x + 1"
    },
    {
      "family": "N3:k=3,ell=4",
      "p": "x**13 + x**12 + x**9 + x**8 + x**5 + x**4 + x + 1"
    },
    {
      "family": "N3:k=4,ell=3",
      "p": "x**13 + x**12 + x**10 + x**9 + x**7 + x**6 + x**4 + x**3 + x + 1"
    },
    {
      "family": "N3:k=6,ell=2",
      "p": "x**13 + x**12 + x**11 + x**10 + x**9 + x**8 + x**7 + x**6 + x**5 + x**4 + x**3 + x**2 + x + 1"
    },
    {
      "family": "N3:k=12,ell=1",
      "p": "x**13 + 2*x**12 + 2*x**11 + 2*x**10 + 2*x**9 + 2*x**8 + 2*x**7 + 2*x**6 + 2*x**5 + 2*x**4 + 2*x**3 + 2*x**2 + 2*x + 1"
    },
    {
      "family": "N4:k=1,ell=11",
      "p": "x**13 + 2*x**12 + x**11 + x**2 + 2*x + 1"
    },
    {
      "family": "N4:k=11,ell=1",
      "p": "x**13 + 3*x**12 + 4*x**11 + 4*x**10 + 4*x**9 + 4*x**8 + 4*x**7 + 4*x**6 + 4*x**5 + 4*x**4 + 4*x**3 + 4*x**2 + 3*x + 1"
    }
  ],
  "allowed_comparisons": "P/Q exchange and simultaneous multiplication of roots by -1; any scalar shift into another integral pair is real by the nonzero integer trace, hence is +/-1.",
  "script_sha256": "56a485363272257d201342e0b50083c8d1fe04ac02a747e30e0a5b35270d031a"
}