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(1) unpinned patterns: sign(G) vs brute-force oracle, and random-segment convexity test
  E*(2x2x2) [KKKR Ex.3.4]                          sign agree  38, disagree 0, skipped  2; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 1.62e-03
  E*(2x3x3)                                        sign agree  36, disagree 0, skipped  4; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 1.53e-02
  E*(2x2x2x2)                                      sign agree  40, disagree 0, skipped  0; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 5.26e-03
  E*(3x3) hook                                     sign agree  39, disagree 0, skipped  1; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 1.55e-02
  two row-stars 2x4 {11,12,23,24}                  sign agree  37, disagree 0, skipped  3; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 4.31e-02
  diagonal 3x3x3 (KR Thm 4.1)                      sign agree  39, disagree 0, skipped  1; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 1.79e-03
  section 2x2x3 {111,222,123} (non-KKKR count)     sign agree  37, disagree 0, skipped  3; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 4.20e-02
  mixed 3x3x3 {111,112,123,233,333} (non-KKKR)     sign agree  40, disagree 0, skipped  0; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 1.73e-02
  mixed 3x3x3 {111,112,223,333} (non-KKKR)         sign agree  38, disagree 0, skipped  2; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 4.00e-02
  star-forest 3x4 {11,12,13,24,34} (non-KKKR)      sign agree  36, disagree 0, skipped  4; segments tested 25, violations 0, min[G(w)-min(G(y),G(z))] = 6.42e-03

  E* closed form check (G via convex program vs 1-D formula inf_A A*prod(1+X_j/A)):
    2x2x2: max |difference| over 20 points = 4.25e-08
    2x3x3: max |difference| over 20 points = 4.98e-08
    2x2x2x2: max |difference| over 20 points = 4.78e-08
    3x4: max |difference| over 20 points = 3.31e-08

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(2) pinned patterns: numerical search for the Theorem A(ii) witness inside Delta_E
  2x2x2: pinned patterns tested 24, numerical witness inside Delta_E found for 24
    last example: E=['122', '211', '222'], pinned e=222, x0=[0.0513, 0.2133, 0.0648], completable t-interval ~[0.0400,0.1735] inside [0,0.7354]; endpoints t=0.0373, 0.1762
  3x3x3: pinned patterns tested 10, numerical witness inside Delta_E found for 10
    last example: E=['112', '121', '122', '133', '232', '312'], pinned e=112, x0=[0.0211, 0.1232, 0.024, 0.0372, 0.0104, 0.0052], completable t-interval ~[0.0053,0.0379] inside [0,0.8001]; endpoints t=0.0046, 0.0385

summary: sign disagreements 0 ; convexity violations 0
