Part A: Theorem 1 (the automaton A_2, n = 2, C = 2)
  A1  625 letters, 8 live transitions, all other transitions lead to the absorbing state dead: ok
  A2  DFS over the full alphabet (3197500 transitions examined): the 2047 accepted words of length <= 21 are exactly the words of Lemma 3.1: ok
  A3  d = 1: all 5 tridiagonal matrices with entries in {-2..2}: 0 recognized = the matrices M_s with 2N+2 = 1: ok
  A3  d = 2: all 625 tridiagonal matrices with entries in {-2..2}: 1 recognized = the matrices M_s with 2N+2 = 2: ok
  A3  d = 3: all 78125 tridiagonal matrices with entries in {-2..2}: 0 recognized = the matrices M_s with 2N+2 = 3: ok
  A3  d = 4: all 9765625 tridiagonal matrices with entries in {-2..2}: 2 recognized = the matrices M_s with 2N+2 = 4: ok
  A4  all 2047 recognized matrices of dimension <= 22: tridiagonal, entries in {-2..2}, window word reproduced, equal to M_s, dim ker = [#U = #D] (exact over Q): ok
  A5  c_d = binom(2k,k) for d = 4k+2 and 0 otherwise (d <= 22) = coefficients of x^2(1-4x^4)^(-1/2); S_60 < sqrt(3)/6 < S_60 + 4^-61/3 (checked as 12 S^2 < 1 < 12 (S + tail)^2); 12t^2 - 1 has no rational root: ok
Part B: Section 4 (values of F_A)
  B1  Proposition 4.1, r = 1: DFS over the 625 letters (8125 transitions): recognized matrices of dimension 2..8 are diag(0,lam,...,lam), lam = 1, 2, kernel 1; brute force d <= 3 agrees; C q^2/(1-q) = p/s for six rationals: ok
  B2  Example 4.4 (L = {U,D,I}^*, c = (1,-1,0)): DFS over the 625 letters (4099375 transitions): the 3280 accepted words of length <= 15 are the window words of the M_s; kernels [#U = #D]; c_(2N+2) = T_N (N <= 7); F(1/2) = 1/sqrt(5) enclosed: ok
  B3  Proposition 4.2 kernel rule with odd labels ((4,1), (3,6); c = (2,-1)) on all 2047 words of length <= 10: dim ker M_s = [c(s) = 0]: ok
Part C: Theorem 2 (T in M_2(Q[Z/2 wr Z]))
  C1  T built in Q[Z/2 wr Z] from pi_0, pi_1, chi, alpha, beta, S = t^2 equals the expanded form of the note; chi^2 = chi alpha = chi beta = chi; 4T has integer coefficients: ok
  C2  the reduction Z/2 wr Z -> G_K = Z/2 wr Z/K respects all 12544 products of a sample set, K <= 5: ok
  C2  K = 1: |G_K| = 2: dim ker(left mult. by 4T) = 3 = dim ker(right mult.) on Q[G_K]^2 (exact); normalized: 3/2
  C2  K = 2: |G_K| = 8: dim ker(left mult. by 4T) = 10 = dim ker(right mult.) on Q[G_K]^2 (exact); normalized: 5/4
  C2  K = 3: |G_K| = 24: dim ker(left mult. by 4T) = 8 = dim ker(right mult.) on Q[G_K]^2 (exact); normalized: 1/3
  C2  K = 4: |G_K| = 64: dim ker(left mult. by 4T) = 24 = dim ker(right mult.) on Q[G_K]^2 (exact); normalized: 3/8
  C2  K = 5: |G_K| = 160: dim ker(left mult. by 4T) = 32 = dim ker(right mult.) on Q[G_K]^2 (exact); normalized: 1/5
  C3  K =  1: (1/|G_K|) sum_x dim ker theta_x(T) =          3/2  = direct group-ring value: ok
  C3  K =  2: (1/|G_K|) sum_x dim ker theta_x(T) =          5/4  = direct group-ring value  = formula of Proposition 6.2: ok
  C3  K =  3: (1/|G_K|) sum_x dim ker theta_x(T) =          1/3  = direct group-ring value: ok
  C3  K =  4: (1/|G_K|) sum_x dim ker theta_x(T) =          3/8  = direct group-ring value  = formula of Proposition 6.2: ok
  C3  K =  5: (1/|G_K|) sum_x dim ker theta_x(T) =          1/5  = direct group-ring value: ok
  C3  K =  6: (1/|G_K|) sum_x dim ker theta_x(T) =         5/24  = formula of Proposition 6.2: ok
  C3  K =  7: (1/|G_K|) sum_x dim ker theta_x(T) =       71/448: ok
  C3  K =  8: (1/|G_K|) sum_x dim ker theta_x(T) =       19/128  = formula of Proposition 6.2: ok
  C3  K =  9: (1/|G_K|) sum_x dim ker theta_x(T) =     325/2304: ok
  C3  K = 10: (1/|G_K|) sum_x dim ker theta_x(T) =     327/2560  = formula of Proposition 6.2: ok
  C3  K = 11: (1/|G_K|) sum_x dim ker theta_x(T) =   1475/11264: ok
  C3  K = 12: (1/|G_K|) sum_x dim ker theta_x(T) =     731/6144  = formula of Proposition 6.2: ok
  C3  K = 13: (1/|G_K|) sum_x dim ker theta_x(T) =   6631/53248: ok
  C3  K = 14: (1/|G_K|) sum_x dim ker theta_x(T) =   3305/28672  = formula of Proposition 6.2: ok
  C4  P = 1 (K = 2): all 4 configurations: no even/odd coupling; segment blocks = 4 M_s entry for entry; kernels [#U = #D]; marker-free kernels [no U] + [no D]: ok
  C4  P = 2 (K = 4): all 16 configurations: no even/odd coupling; segment blocks = 4 M_s entry for entry; kernels [#U = #D]; marker-free kernels [no U] + [no D]: ok
  C4  P = 3 (K = 6): all 64 configurations: no even/odd coupling; segment blocks = 4 M_s entry for entry; kernels [#U = #D]; marker-free kernels [no U] + [no D]: ok
  C4  P = 4 (K = 8): all 256 configurations: no even/odd coupling; segment blocks = 4 M_s entry for entry; kernels [#U = #D]; marker-free kernels [no U] + [no D]: ok
  C4  P = 5 (K = 10): all 1024 configurations: no even/odd coupling; segment blocks = 4 M_s entry for entry; kernels [#U = #D]; marker-free kernels [no U] + [no D]: ok
  C4  P = 6 (K = 12): all 4096 configurations: no even/odd coupling; segment blocks = 4 M_s entry for entry; kernels [#U = #D]; marker-free kernels [no U] + [no D]: ok
  C4  P = 7 (K = 14): all 16384 configurations: no even/odd coupling; segment blocks = 4 M_s entry for entry; kernels [#U = #D]; marker-free kernels [no U] + [no D]: ok
  C4  total: 21844 configurations, 36409 segment blocks
  C5  T_N = sum_k binom(N,2k) binom(2k,k) = #{s : #U = #D} (N <= 10) = coefficients of (1-2y-3y^2)^(-1/2) (N <= 40); S_300 < sqrt(5)/20 < S_300 + (3/4)^301/4 (as 80 S^2 < 1 < 80 (S + tail)^2); 80t^2 - 1 has no rational root; the error identity of Remark 6.3 holds exactly for P <= 60: ok
      d_10 = 0.127734375000000   d_10 - S_300 = 1.593e-02   (3/4)T_(P-1)4^-P = 1.392e-02   2^(1-P)/P = 1.250e-02
      d_20 = 0.112326431274414   d_20 - S_300 = 5.230e-04   (3/4)T_(P-1)4^-P = 2.245e-03   2^(1-P)/P = 1.953e-04
      d_40 = 0.111810817546575   d_40 - S_300 = 7.419e-06   (3/4)T_(P-1)4^-P = 8.799e-05   2^(1-P)/P = 9.537e-08
      d_80 = 0.111803407718414   d_80 - S_300 = 8.843e-09   (3/4)T_(P-1)4^-P = 1.958e-07   2^(1-P)/P = 4.547e-14
      d_120 = 0.111803398890696   d_120 - S_300 = 1.571e-11   (3/4)T_(P-1)4^-P = 5.055e-10   2^(1-P)/P = 2.891e-20
TOTAL checks: 14386, failures: 0; running time 59.4 s
ALL CHECKS PASSED
