==============================================================================
W1  3x3 matrix
E = ['11', '12', '21', '33']
theta0 = [['1/2', '1/4', '1/4'], ['1/2', '3/8', '1/8']]
x0 = p(theta0)  [completable] = {11: 1/4, 12: 3/16, 21: 1/8, 33: 1/32}  (sum 19/32)
   x0 is an interior point of the completable region: B=21/32, sqrt(B)+sqrt(x33) < 1 exactly
   y_minus = {11: 0, 12: 3/16, 21: 1/8, 33: 1/32}  (sum 11/32) -> NOT completable (x11=0 but x12,x21>0: zero row/column property)
   y_plus = {11: 1/2, 12: 3/16, 21: 1/8, 33: 1/32}  (sum 27/32) -> NOT completable (B=55/64, sqrt(B)+sqrt(x33)>1 exactly)
   midpoint of y_minus, y_plus = x0 (completable)  => complement NOT convex

   y_minus = {11: 1/100, 12: 3/16, 21: 1/8, 33: 1/32}  (sum 283/800) -> NOT completable (B=2133/800, sqrt(B)+sqrt(x33)>1 exactly)
   y_plus = {11: 49/100, 12: 3/16, 21: 1/8, 33: 1/32}  (sum 667/800) -> NOT completable (B=33333/39200, sqrt(B)+sqrt(x33)>1 exactly)
   midpoint of y_minus, y_plus = x0 (completable)  => complement NOT convex

==============================================================================
W1' 3x3 matrix, triage numbers
E = ['11', '12', '21', '33']
theta0 = [['1/2', '1/4', '1/4'], ['1/2', '3/8', '1/8']]
x0 = p(theta0)  [completable] = {11: 1/4, 12: 3/16, 21: 1/8, 33: 1/32}  (sum 19/32)
   x0 is an interior point of the completable region: B=21/32, sqrt(B)+sqrt(x33) < 1 exactly
   y_minus = {11: 1/20, 12: 3/16, 21: 1/8, 33: 1/32}  (sum 63/160) -> NOT completable (B=133/160, sqrt(B)+sqrt(x33)>1 exactly)
   y_plus = {11: 9/20, 12: 3/16, 21: 1/8, 33: 1/32}  (sum 127/160) -> NOT completable (B=391/480, sqrt(B)+sqrt(x33)>1 exactly)
   midpoint of y_minus, y_plus = x0 (completable)  => complement NOT convex

==============================================================================
W2  2x2x2 tensor
E = ['111', '112', '222']
theta0 = [['1/2', '1/2'], ['3/4', '1/4'], ['1/4', '3/4']]
x0 = p(theta0)  [completable] = {111: 3/32, 112: 9/32, 222: 3/32}  (sum 15/32)
   x0 is an interior point of the completable region: P=3/8, Q=1/8, sqrt P + sqrt Q < 1 exactly
   y_minus = {111: 3/32, 112: 0, 222: 3/32}  (sum 3/16) -> NOT completable (x112=0 but x111,x222>0)
   y_plus = {111: 3/32, 112: 9/16, 222: 3/32}  (sum 3/4) -> NOT completable (P=21/32, Q=7/64, sqrt P + sqrt Q > 1 exactly)
   midpoint of y_minus, y_plus = x0 (completable)  => complement NOT convex

   y_minus = {111: 3/32, 112: 1/100, 222: 3/32}  (sum 79/400) -> NOT completable (P=83/800, Q=249/256, sqrt P + sqrt Q > 1 exactly)
   y_plus = {111: 3/32, 112: 221/400, 222: 3/32}  (sum 37/50) -> NOT completable (P=517/800, Q=1551/14144, sqrt P + sqrt Q > 1 exactly)
   midpoint of y_minus, y_plus = x0 (completable)  => complement NOT convex

==============================================================================
W3  3x3x2 tensor
E = ['111', '121', '211', '331', '332']
theta0 = [['1/2', '1/4', '1/4'], ['1/2', '3/8', '1/8'], ['1/2', '1/2']]
x0 = p(theta0)  [completable] = {111: 1/8, 121: 3/32, 211: 1/16, 331: 1/64, 332: 1/64}  (sum 5/16)
   x0 is an interior point of the completable region: B=21/32, sqrt(B)+sqrt(x33) < 1 exactly [after reduction to W1; c1 is a continuous function of x]
   y_minus = {111: 0, 121: 3/32, 211: 1/16, 331: 1/64, 332: 1/64}  (sum 3/16) -> NOT completable (x111=0 but x121,x211>0)
   y_plus = {111: 1/4, 121: 3/32, 211: 1/16, 331: 1/64, 332: 1/64}  (sum 7/16) -> NOT completable (B=55/64, sqrt(B)+sqrt(x33)>1 exactly) [after reduction to W1]
   midpoint of y_minus, y_plus = x0 (completable)  => complement NOT convex

   y_minus = {111: 1/100, 121: 3/32, 211: 1/16, 331: 1/64, 332: 1/64}  (sum 79/400) -> NOT completable (B=2407/1600, sqrt(B)+sqrt(x33)>1 exactly) [after reduction to W1]
   y_plus = {111: 6/25, 121: 3/32, 211: 1/16, 331: 1/64, 332: 1/64}  (sum 171/400) -> NOT completable (B=10769/12800, sqrt(B)+sqrt(x33)>1 exactly) [after reduction to W1]
   midpoint of y_minus, y_plus = x0 (completable)  => complement NOT convex

==============================================================================
W4  trivial witness (entry > 1), pattern W2, e = 112: x0_e = 343/512 ; y_plus_e = 2*x0_e = 343/256 > 1: True

ALL EXACT CHECKS PASSED
