indep_exact.py -- independent verification run 2; seed 20260930; exact Fractions

[1] Lemma 4.2 (trace) and Lemma 4.3 (PQ+QP=tr(PQ)I) on random data
   done

[2] S_n exactly at random integer points (termwise n<=9, subset DP n<=12)
   n= 2 inst=0 S_n(DP)=289/1024  S_n(termwise)=289/1024  control unsigned sum != 0: True
   n= 2 inst=1 S_n(DP)=1225/6561  S_n(termwise)=1225/6561  control unsigned sum != 0: True
   n= 2 inst=2 S_n(DP)=484/169  S_n(termwise)=484/169  control unsigned sum != 0: True
   n= 3 inst=0 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 3 inst=1 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 3 inst=2 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: False
   n= 4 inst=0 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 4 inst=1 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 4 inst=2 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 5 inst=0 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 5 inst=1 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 5 inst=2 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 6 inst=0 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 6 inst=1 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 6 inst=2 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 7 inst=0 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 7 inst=1 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 7 inst=2 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 8 inst=0 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 8 inst=1 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 8 inst=2 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 9 inst=0 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 9 inst=1 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n= 9 inst=2 S_n(DP)=0  S_n(termwise)=0  control unsigned sum != 0: True
   n=10 inst=0 S_n(DP)=0  control unsigned sum != 0: True
   n=10 inst=1 S_n(DP)=0  control unsigned sum != 0: True
   n=10 inst=2 S_n(DP)=0  control unsigned sum != 0: True
   n=11 inst=0 S_n(DP)=0  control unsigned sum != 0: True
   n=12 inst=0 S_n(DP)=0  control unsigned sum != 0: True

[3] Controls with other weight functions (should be nonzero for n = 1,2 mod 4; zero for n = 0,3 mod 4 by reversal)
   n= 3  sym!=0:False  [AA]/rand-antisym!=0:False  rand-antisym/[A'A']!=0:False   (reversal predicts zero: True)
   n= 4  sym!=0:False  [AA]/rand-antisym!=0:False  rand-antisym/[A'A']!=0:False   (reversal predicts zero: True)
   n= 5  sym!=0:True  [AA]/rand-antisym!=0:True  rand-antisym/[A'A']!=0:True   (reversal predicts zero: False)
   n= 6  sym!=0:True  [AA]/rand-antisym!=0:True  rand-antisym/[A'A']!=0:True   (reversal predicts zero: False)
   n= 7  sym!=0:False  [AA]/rand-antisym!=0:False  rand-antisym/[A'A']!=0:False   (reversal predicts zero: True)
   n= 8  sym!=0:False  [AA]/rand-antisym!=0:False  rand-antisym/[A'A']!=0:False   (reversal predicts zero: True)
   n= 9  sym!=0:True  [AA]/rand-antisym!=0:True  rand-antisym/[A'A']!=0:True   (reversal predicts zero: False)
   n=10  sym!=0:True  [AA]/rand-antisym!=0:True  rand-antisym/[A'A']!=0:True   (reversal predicts zero: False)
   n=11  sym!=0:False  [AA]/rand-antisym!=0:False  rand-antisym/[A'A']!=0:False   (reversal predicts zero: True)
   cycle-notation reading of sgn gives (-1)^(n-1) * unsigned sum (nonzero) -- covered by [2] unsigned control

[3b] Informational: with the denominators removed, sum sgn(pi) prod[A A] = tr(M_1 s_{n-1}(M_2..M_n))
   n=3: numerator-only signed sum = -1687224  (Amitsur-Levitzki s_4=0 on M_2 forces 0 for n>=5)
   n=4: numerator-only signed sum = 0  (Amitsur-Levitzki s_4=0 on M_2 forces 0 for n>=5)
   n=5: numerator-only signed sum = 0  (Amitsur-Levitzki s_4=0 on M_2 forces 0 for n>=5)
   n=6: numerator-only signed sum = 0  (Amitsur-Levitzki s_4=0 on M_2 forces 0 for n>=5)
   n=7: numerator-only signed sum = 0  (Amitsur-Levitzki s_4=0 on M_2 forces 0 for n>=5)
   n=8: numerator-only signed sum = 0  (Amitsur-Levitzki s_4=0 on M_2 forces 0 for n>=5)

[4] Theorem 1.4(a),(b) with random traceless integer matrices (exact)
   n=3: sum == PT(123) tr(M2M3) I: True; tr(M1 S) for traceless M1 = 0, for general M1 = -63/400
   n=3: sum == PT(123) tr(M2M3) I: True; tr(M1 S) for traceless M1 = 0, for general M1 = -1/3520
   n=4: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=4: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=5: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=5: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=6: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=6: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=7: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=7: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=8: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=8: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True
   n=9: sum = ((Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1))); controls nonzero: general matrices True, random weights True

[5] Remark 5.1 (bilinear version)
   n=3: value -33275/18144 == -PT(123)[A2A3]^2[XY] = -33275/18144: True; nonzero: True
   n=4: value 0
   n=5: value 0
   n=6: value 0
   n=7: value 0
   n=8: value 0

[6] Proposition 5.2 for all T, and the shuffle decomposition of sum_sigma sgn(sigma|T) sigma
   n=3: 4 subsets T checked
   n=4: 8 subsets T checked
   n=5: 16 subsets T checked
   n=6: 32 subsets T checked
   n=7: 64 subsets T checked
   n=8: 128 subsets T checked
   (for T a nonempty proper subset of C the identity is thus a direct consequence of Lemma 3.1)

[7] Lemma 3.1 (all splits u,v for n<=7), Corollary 3.2 (DDM) for n<=7, Lemma 2.1(b)
   n=3: 2 shuffle identities, DDM expansion equal: True
   n=4: 12 shuffle identities, DDM expansion equal: True
   n=5: 72 shuffle identities, DDM expansion equal: True
   n=6: 480 shuffle identities, DDM expansion equal: True
   n=7: 3600 shuffle identities, DDM expansion equal: True
   normalisation B1=(0,1), B_i=(1,t_i), PT=-O and identity (3.1) checked for n<=7

[8] Proposition 4.1 with y_c = a_c I + N_c, N_c strictly upper triangular 3x3 (commutators central)
   n=3: sum == PT(123)[y2,y3]: True
   n=4: sum is zero: True;  control with generic 3x3 matrices nonzero: True
   n=5: sum is zero: True;  control with generic 3x3 matrices nonzero: True
   n=6: sum is zero: True;  control with generic 3x3 matrices nonzero: True
   n=7: sum is zero: True;  control with generic 3x3 matrices nonzero: True
   n=8: sum is zero: True;  control with generic 3x3 matrices nonzero: True

[9] Corollary 1.3: P_n at integer points (generic: P_n = D_n S_n; degenerate A': P_n = 0)
   n=3: P_n generic = 0, degenerate = [0, 0, 0]
   n=4: P_n generic = 0, degenerate = [0, 0, 0]
   n=5: P_n generic = 0, degenerate = [0, 0, 0]
   n=6: P_n generic = 0, degenerate = [0, 0, 0]
   n=7: P_n generic = 0, degenerate = [0, 0, 0]
   n=8: P_n generic = 0, degenerate = [0, 0, 0]

FAILURES: 0  []
elapsed 5.5s
