[1] coefficient formulas (exact integer expansion of (1 + b t + c t^2)^k, k = 2..21)
    k = 7: [t^1] = {(1, 0): 7}  [t^4] = {(0, 2): 21, (2, 1): 105, (4, 0): 35}
           [t^7] = {(1, 3): 140, (3, 2): 210, (5, 1): 42, (7, 0): 1}  [t^13] = {(1, 6): 7}
    phi_4 (k=7) = [21, 105, 35]  phi_7 (k=7) = [140, 210, 42, 1]  phi_5 (k=9) = [252, 504, 126]  phi_9 (k=9) = [630, 1680, 756, 72, 1]
    formula for phi_n and rule (4) agree with the expansion for all k <= 21, n <= 2k: True
[2] Theorem 1.2 (k = 7), exact
    f(I_-) = (-22 + -231/25*s)   f(I_+) = (-22 + 231/25*s)
    q^3 = f(I_+)/f(I_-) = (-15053/14053 + 420/14053*s)
    k=7: coefficients t^0..t^14 of the cleared identity with nonzero normal form: []
    q^6 - 1 = (q^3)^2 - 1 = (58212000/197486809 + -12644520/197486809*s)
    f(I_+)^2 - f(I_-)^2 = (0 + -20328/25*s)  (= (462 s/25)(-44))
[3] Example 4.3 (k = 9), exact
    phi_9(-2 - s) = (434 + 384*s)   phi_9(-2 + s) = (434 + -384*s)
    k=9: coefficients t^0..t^18 of the cleared identity with nonzero normal form: []
    q^4 = phi_9(I1)/phi_9(I0) = (-120817/26639 + 83328/26639*s)
[4] Theorem 1.3: (C1)-(C3) for odd k in [7, 201] modulo the prime 2^61 - 1; exactly over Q for k <= 61
    modular criterion: 98 values of k, failures: []
    exact over Q: 28 values of k, failures: []
    controls: k = 5 gives deg phi_3 = 1 (construction not available); a planted common factor is detected
[5] Theorem 1.4: interval certificates (test of Lemma 5.1, exact rational majorants)
    certificate          k  s   max|F(c0)|   rho=max row bound   max|phi(X)-c0| bound   r          min|c_i2|   result
    cert_k5_d2_s4_A      5  4  3.49e-60    9.98e-15           1.18e-44              7.89e-31   0.487      PASS
        controls (must FAIL): shift:FAIL, fixed:FAIL, trunc:FAIL
    cert_k6_d2_s5_A      6  5  1.29e-59    2.67e-13           3.16e-43              7.89e-31   0.342      PASS
        controls (must FAIL): shift:FAIL, fixed:FAIL, trunc:FAIL
    cert_k6_d2_s5_B      6  5  5.53e-59    1.44e-13           1.70e-43              7.89e-31   0.291      PASS
        controls (must FAIL): shift:FAIL, fixed:FAIL, trunc:FAIL
    cert_k7_d2_s6_A      7  6  1.59e-58    3.53e-12           4.17e-42              7.89e-31   0.101      PASS
        controls (must FAIL): shift:FAIL, fixed:FAIL, trunc:FAIL
    cert_k8_d2_s6_A      8  6  2.83e-59    2.94e-12           3.48e-42              7.89e-31   0.257      PASS
        controls (must FAIL): shift:FAIL, fixed:FAIL, trunc:FAIL
    cert_k9_d2_s7_A      9  7  2.74e-59    1.29e-13           1.52e-43              7.89e-31   0.341      PASS
        controls (must FAIL): shift:FAIL, fixed:FAIL, trunc:FAIL
[6] numerical sanity checks (double precision; not part of any proof)
    k = 7: 48 choices of (s, w, beta, q, p); largest |[t^n](sum of the six 7th powers) - [t^n] t|, n = 0..14: 4.5e-14; min |sigma| = 1.7146
    sample choice (s > 0, w = e^(2 pi i/3), real q): beta = 1.6686890836 i, q = -0.8824824684, p = 0.4360670215, A = -8.1018773317 i, sigma = -13.0499070329 i
    the six quadratics a + b t + c t^2 (homogeneous form a y^2 + b xy + c x^2), principal 7th roots:
      a = +0.57735089 +0.13177657 i   b = -0.21989413 +0.96341912 i   c = +0.57735089 +0.13177657 i
      a = +0.79848228 +0.00000000 i   b = +0.34819179 +0.00000000 i   c = -0.70464662 +0.00000000 i
      a = +0.59054263 -0.04425505 i   b = -0.89033315 -0.42876185 i   c = -0.33359732 -0.48929740 i
      a = +0.76300796 -0.23535678 i   b = -0.07747996 +0.33946190 i   c = +0.51654252 +0.47928142 i
      a = +0.55126208 -0.21635442 i   b = +0.61612965 -0.77260215 i   c = -0.08826261 +0.58558417 i
      a = +0.76300796 +0.23535678 i   b = -0.07747996 -0.33946190 i   c = +0.51654252 -0.47928142 i
    k = 9: q^4 = -0.1116187648; |sigma| = 1318.1617; largest coefficient error of t = sum of eight 9th powers: 2.9e-15
ALL CHECKS PASSED
