(1) Examples 7.1 and 7.2
  7.1  x11 = 1/20: B = 133/160, sum = 63/160, sign(sqrt B + sqrt x33 - 1) = 1
  7.1  x11 = 1/4: B = 21/32, sum = 19/32, sign(sqrt B + sqrt x33 - 1) = -1
  7.1  x11 = 9/20: B = 391/480, sum = 127/160, sign(sqrt B + sqrt x33 - 1) = 1
  7.2  x112 = 1/100: P = 83/800, Q = 249/256, sum = 79/400, sign(sqrt P + sqrt Q - 1) = 1
  7.2  x112 = 9/32: P = 3/8, Q = 1/8, sum = 15/32, sign(sqrt P + sqrt Q - 1) = -1
  7.2  x112 = 221/400: P = 517/800, Q = 1551/14144, sum = 37/50, sign(sqrt P + sqrt Q - 1) = 1
(2) Proposition 6.1, corner patterns E*(s)
  2x2: 20 random rational theta: P_X(theta^s) = 0 and completion formula exact
  3x3: 20 random rational theta: P_X(theta^s) = 0 and completion formula exact
  2x2x2: 20 random rational theta: P_X(theta^s) = 0 and completion formula exact
  2x3x4: 20 random rational theta: P_X(theta^s) = 0 and completion formula exact
  3x3x3: 20 random rational theta: P_X(theta^s) = 0 and completion formula exact
  2x2x2x2: 20 random rational theta: P_X(theta^s) = 0 and completion formula exact
  2x3x2x3x2: 20 random rational theta: P_X(theta^s) = 0 and completion formula exact
(3) Proposition 5.1, consistency equations of the pinned families
  2x2x2: (A(1+rho)+X'_1) prod_(j>=2) (A+X_j) = A^(n-1) exact for 20 random theta
  3x3x3: (A(1+rho)+X'_1) prod_(j>=2) (A+X_j) = A^(n-1) exact for 20 random theta
  2x3x4: (A(1+rho)+X'_1) prod_(j>=2) (A+X_j) = A^(n-1) exact for 20 random theta
  2x2x2x2: (A(1+rho)+X'_1) prod_(j>=2) (A+X_j) = A^(n-1) exact for 20 random theta
  3x2x4x2x3: (A(1+rho)+X'_1) prod_(j>=2) (A+X_j) = A^(n-1) exact for 20 random theta
  3x3: -alpha t^2 + (1 + alpha beta - x33) t - beta = 0 at t = a_1, exact for 20 random theta
  3x5: -alpha t^2 + (1 + alpha beta - x33) t - beta = 0 at t = a_1, exact for 20 random theta
  6x4: -alpha t^2 + (1 + alpha beta - x33) t - beta = 0 at t = a_1, exact for 20 random theta
(4) Remark 7.3: pinned admissible patterns without finitely completable entries
  2x2x2: pinned admissible patterns certified full-dimensional: 24; without finitely completable entry outside E: 0; first example None
  2x2x3: pinned admissible patterns certified full-dimensional: 228; without finitely completable entry outside E: 12; first example ['111', '122', '212', '223']
  2x2x2x2: pinned admissible patterns certified full-dimensional: 1240; without finitely completable entry outside E: 64; first example ['1111', '1122', '1212', '2221']
  2x3x3: pinned admissible patterns certified full-dimensional: 4032; without finitely completable entry outside E: 72; first example ['111', '122', '133', '212', '223']
  example {111,122,212,223} in 2x2x3: admissible, pinned elements 122 and 212, Jacobian determinant nonzero

ALL AUDIT CHECKS PASSED
