check_note.py -- the exact program of the note (record OWR-4428-012, note B)
  group laws: e_j^p = 1, t e_j t^-1 = e_{j+1}; rho_j idempotent with e_j rho_j = rho_j; pi_j, pi'_j complementary (p = 2, 3, 5)   [0.0 s]

Part A: the 5-bit operator T of Theorem A (p = 2, r = 5, shift t^5)
  A.1 T built in Q[G_2] from its definition: chi, alpha-1, beta-1 pairwise orthogonal idempotents, chi alpha = chi beta = chi; at the 32 characters of the lamps 0..4 they are the indicators of M, U, D; counts (|M|, |U|, |D|, |I|) = (8, 5, 5, 14), weights (w_M, w_U, w_D, w_I) = (8, 5, 5, 14), probabilities q = (1/4, 5/32, 5/32, 7/16); simplified form holds; terms per entry [36, 4, 4, 36]; largest denominator 32, so 32 T is integral   [0.0 s]
  A.2 phi_K is a homomorphism (random products); kernels of left and of right multiplication by 32 T on Q[G_K]^2 computed in the group ring without Fourier transform (exact ranks) agree: d_1 = 3/2, d_2 = 1, d_3 = 1/3, d_4 = 11/32, d_5 = 23/16, d_6 = 23/128, d_7 = 1/7   [113.1 s]
  A.3 fibre operators theta_x(32 T) for all x in (Z/2)^K, built from the group-ring coefficients (entries computed in Q(zeta_2), all rational), K in 1..15: d_1 = 3/2, d_2 = 1, d_3 = 1/3, d_4 = 11/32, d_5 = 23/16, d_6 = 23/128, d_7 = 1/7, d_8 = 157/1024, d_9 = 331/2304, d_10 = 537/1024, d_11 = 3313/22528, d_12 = 6793/49152, d_13 = 7307/53248, d_14 = 15641/114688, d_15 = 14227/49152; equal to the group-ring values for K in 1..7 and to Proposition 3.8 for K = 5, 10, 15 and to the values recorded by the programs of the package   [131.5 s]
  A.4 all 33824 configurations with K = 5 P, P <= 3, every class: no entries between classes, rows as in Lemma 3.2, 125480 segment blocks each equal to 32 M_s, closed in rows and columns and partitioning the class; kernel of a class part = number of balanced segments (with markers) or [no U] + [no D] (without); the class kernels add up to dim ker theta_x   [131.5 s]
  A.5 300 random windows of 60 positions of the line (class 0, no reduction): 36000 rows as in Lemma 3.2 and 4113 segment blocks between consecutive markers equal to 32 M_s   [132.3 s]
  A.6 W_0..W_7 = [1, 14, 246, 4844, 100966, 2172324, 47679036, 1060903704] (brute force over {U,D,I}^N with weights 5, 5, 14 = binomial form); the specialized formula (8) for d_5P equals Proposition 3.8 for P <= 12; v^2 = q_M^4/((1-q_I)^2 - 4 q_U q_D) = 1/56, v = (1/4) sqrt(2/7); d_5 = 1/4 + 19/16 = 23/16 as in Section 4.1   [132.3 s]

Part B: the family T^(n) of Theorem C for n = 2 (r = 4) and n = 4 (r = 6)
  B(n=2).1 T^(2) built in Q[G_2] from its definition: chi, alpha-1, beta-1 pairwise orthogonal idempotents, chi alpha = chi beta = chi; at the 16 characters of the lamps 0..3 they are the indicators of M, U, D; counts (|M|, |U|, |D|, |I|) = (4, 2, 2, 8), weights (w_M, w_U, w_D, w_I) = (4, 2, 2, 8), probabilities q = (1/4, 1/8, 1/8, 1/2); simplified form holds; terms per entry [14, 4, 4, 14]; largest denominator 8, so 16 T is integral   [132.3 s]
  B(n=2).2 phi_K is a homomorphism (random products); kernels of left and of right multiplication by 16 T on Q[G_K]^2 computed in the group ring without Fourier transform (exact ranks) agree: d_1 = 3/2, d_2 = 7/4, d_3 = 1/3, d_4 = 3/2, d_5 = 1/5, d_6 = 9/32, d_7 = 85/448   [226.4 s]
  B(n=2).3 fibre operators theta_x(16 T) for all x in (Z/2)^K, built from the group-ring coefficients (entries computed in Q(zeta_2), all rational), K in 1..16: d_1 = 3/2, d_2 = 7/4, d_3 = 1/3, d_4 = 3/2, d_5 = 1/5, d_6 = 9/32, d_7 = 85/448, d_8 = 37/64, d_9 = 761/4608, d_10 = 483/2560, d_11 = 3533/22528, d_12 = 251/768, d_13 = 16057/106496, d_14 = 9481/57344, d_15 = 1215/8192, d_16 = 1889/8192; equal to the group-ring values for K in 1..7 and to Proposition 3.8 for K = 4, 8, 12, 16 and to the values recorded by the programs of the package   [250.2 s]
  B(n=2).4 all 69904 configurations with K = 4 P, P <= 4, every class: no entries between classes, rows as in Lemma 3.2, 274960 segment blocks each equal to 16 M_s, closed in rows and columns and partitioning the class; kernel of a class part = number of balanced segments (with markers) or [no U] + [no D] (without); the class kernels add up to dim ker theta_x   [250.2 s]
  B(n=4).1 T^(4) built in Q[G_2] from its definition: chi, alpha-1, beta-1 pairwise orthogonal idempotents, chi alpha = chi beta = chi; at the 64 characters of the lamps 0..5 they are the indicators of M, U, D; counts (|M|, |U|, |D|, |I|) = (16, 11, 11, 26), weights (w_M, w_U, w_D, w_I) = (16, 11, 11, 26), probabilities q = (1/4, 11/64, 11/64, 13/32); simplified form holds; terms per entry [68, 4, 4, 68]; largest denominator 64, so 64 T is integral   [250.3 s]
  B(n=4).2 phi_K is a homomorphism (random products); kernels of left and of right multiplication by 64 T on Q[G_K]^2 computed in the group ring without Fourier transform (exact ranks) agree: d_1 = 3/2, d_2 = 7/4, d_3 = 3/2, d_4 = 9/16, d_5 = 1/5, d_6 = 45/32   [251.5 s]
  B(n=4).3 fibre operators theta_x(64 T) for all x in (Z/2)^K, built from the group-ring coefficients (entries computed in Q(zeta_2), all rational), K in 1..12: d_1 = 3/2, d_2 = 7/4, d_3 = 3/2, d_4 = 9/16, d_5 = 1/5, d_6 = 45/32, d_7 = 39/224, d_8 = 109/512, d_9 = 445/1536, d_10 = 117/640, d_11 = 3115/22528, d_12 = 2041/4096; equal to the group-ring values for K in 1..6 and to Proposition 3.8 for K = 6, 12 and to the values recorded by the programs of the package   [254.6 s]
  B(n=4).4 all 4160 configurations with K = 6 P, P <= 2, every class: no entries between classes, rows as in Lemma 3.2, 12384 segment blocks each equal to 64 M_s, closed in rows and columns and partitioning the class; kernel of a class part = number of balanced segments (with markers) or [no U] + [no D] (without); the class kernels add up to dim ker theta_x   [254.6 s]
  B.5 counts (4, 2, 2, 8) for n = 2 and (16, 11, 11, 26) for n = 4; v^2 = 2^n/(64(2^n-1)): sqrt(3)/12 and sqrt(15)/30; for n = 2 the largest denominator is 8   [254.6 s]

Part C: the operators T_p of Theorem D (r = 2, shift t^2) for p = 3 and p = 5
  C(p=3).1 T_3 built in Q[G_3] from its definition: chi, alpha-1, beta-1 pairwise orthogonal idempotents, chi alpha = chi beta = chi; at the 9 characters of the lamps 0..1 they are the indicators of M, U, D; counts (|M|, |U|, |D|, |I|) = (1, 1, 1, 1), weights (w_M, w_U, w_D, w_I) = (4, 2, 2, 1), probabilities q = (4/9, 2/9, 2/9, 1/9); simplified form holds; terms per entry [18, 9, 9, 18]; largest denominator 9, so 9 T is integral   [254.6 s]
  C(p=3).2 phi_K is a homomorphism (random products); kernels of left and of right multiplication by 9 T on Q[G_K]^2 computed in the group ring without Fourier transform (exact ranks) agree: d_1 = 4/3, d_2 = 10/9, d_3 = 38/81, d_4 = 29/81   [257.4 s]
  C(p=3).3 fibre operators theta_x(9 T) for all x in (Z/3)^K, built from the group-ring coefficients (entries computed in Q(zeta_3), all rational), K in 1..10: d_1 = 4/3, d_2 = 10/9, d_3 = 38/81, d_4 = 29/81, d_5 = 382/1215, d_6 = 214/729, d_7 = 4342/15309, d_8 = 3449/13122, d_9 = 47342/177147, d_10 = 76666/295245; equal to the group-ring values for K in 1..4 and to Proposition 3.8 for K = 2, 4, 6, 8, 10 and to the values recorded by the programs of the package   [282.1 s]
  C(p=3).4 all 66429 configurations with K = 2 P, P <= 5, every class: no entries between classes, rows as in Lemma 3.2, 287864 segment blocks each equal to 9 M_s, closed in rows and columns and partitioning the class; kernel of a class part = number of balanced segments (with markers) or [no U] + [no D] (without); the class kernels add up to dim ker theta_x   [282.1 s]
  C(p=5).1 T_5 built in Q[G_5] from its definition: chi, alpha-1, beta-1 pairwise orthogonal idempotents, chi alpha = chi beta = chi; at the 25 characters of the lamps 0..1 they are the indicators of M, U, D; counts (|M|, |U|, |D|, |I|) = (1, 1, 1, 1), weights (w_M, w_U, w_D, w_I) = (16, 4, 4, 1), probabilities q = (16/25, 4/25, 4/25, 1/25); simplified form holds; terms per entry [50, 25, 25, 50]; largest denominator 25, so 25 T is integral   [282.1 s]
  C(p=5).2 phi_K is a homomorphism (random products); kernels of left and of right multiplication by 25 T on Q[G_K]^2 computed in the group ring without Fourier transform (exact ranks) agree: d_1 = 6/5, d_2 = 26/25, d_3 = 242/375   [297.0 s]
  C(p=5).3 fibre operators theta_x(25 T) for all x in (Z/5)^K, built from the group-ring coefficients (entries computed in Q(zeta_5), all rational), K in 1..6: d_1 = 6/5, d_2 = 26/25, d_3 = 242/375, d_4 = 297/625, d_5 = 7442/15625, d_6 = 21802/46875; equal to the group-ring values for K in 1..3 and to Proposition 3.8 for K = 2, 4, 6 and to the values recorded by the programs of the package   [304.0 s]
  C(p=5).4 all 16275 configurations with K = 2 P, P <= 3, every class: no entries between classes, rows as in Lemma 3.2, 61632 segment blocks each equal to 25 M_s, closed in rows and columns and partitioning the class; kernel of a class part = number of balanced segments (with markers) or [no U] + [no D] (without); the class kernels add up to dim ker theta_x   [304.0 s]
  C.5 the expansion displayed in Theorem D holds for p = 3, 5; v^2 = (p-1)^5/(p^4(p+3)) = 16/243 and 128/625 (4 sqrt(3)/27, 8 sqrt(2)/25); 200 random windows of 40 positions of the line for p = 3: 16000 rows as in Lemma 3.2, 3423 segment blocks equal to 9 M_s   [304.3 s]
  C.6 other encodings for p = 3: marker 00, U 01, D 10, I 11: v^2 = 1/729; marker 00, U 11, D 01, I 10: v^2 = 1/1377; for each, the fibre values d_2, d_4, d_6 equal Proposition 3.8   [304.5 s]

Part D: the general template, counter matrices, series, values, enclosures, error, design
  D1 24 random partitions of {0,1}^r into M (non-empty), U, D, I with p in {2, 3}, r in {1, 2, 3}: the operator built from the eps_b passes the checks of (1), and its fibre values d_rP equal Proposition 3.8 in all 80 cases (K = rP <= 9)   [306.1 s]
  D2 all 9841 counter matrices M_s, s in {U,D,I}^N, N <= 8: dim ker M_s = [#U = #D] (exact), and for balanced s the kernel contains (A_0, B_0, ..., A_N, B_N)   [306.4 s]
  D3 for T, T^(2), T^(4), T_3, T_5: omega_N by enumeration (N <= 6) = binomial form = coefficients of ((1 - q_I y)^2 - 4 q_U q_D y^2)^(-1/2) (N <= 40); (1-q_I)^2 - 4 q_U q_D = q_M^2 + 2 q_M (q_U+q_D) + (q_U-q_D)^2, so v <= q_M and the series at y = 1 converges   [306.4 s]
  D4 rational enclosures from 160 terms with the tail bound q_M (1-q_M)^160 (checked by squaring): T: (1/4)sqrt(2/7) in [0.133630620956212, 0.133630620956212]; T^(2): sqrt(3)/12 in [0.144337567297406, 0.144337567297406]; T^(4): sqrt(15)/30 in [0.129099444873580, 0.129099444873580]; T_3: 4sqrt(3)/27 in [0.256600119639833, 0.256600119639833]; T_5: 8sqrt(2)/25 in [0.452548339959390, 0.452548339959390]; the bound |d_rP - v| <= q_M (1-q_M)^P + ((q_D+q_I)^P + (q_U+q_I)^P)/P holds for P <= 40   [306.7 s]
  D5 signs of d_2P(T_p) - v_p for P = 1..40 (decided with enclosures from 400 terms): p=2: ++++++++++++++++++++++++++++++++++++++++; p=3: ++++++++++++++++++++++++++++++++++++++++; p=5: +++-+-+-+-+-++++++++++++++++++++++++++++; p=7: +++-+-+-+-+-+-+-+-+-+-+-++++++++++++++++; p=11: +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-   [309.2 s]
  D6 (p-1)(p+3) lies strictly between p^2 and (p+1)^2 and is not a square for 2 <= p <= 100000; for n = 2..12 the family counts satisfy n_M + 2 n_U + n_I = 2^(n+2), n_I + 2 n_U = 3 2^n, n_I - 2 n_U = 4, v^2 = 2^n/(64(2^n-1)), and 2^n - 1 is not a square   [309.2 s]
  D7 counts (m, n_U, n_D) with p = 2 and r bits giving (1/4)sqrt(2/7): none for r <= 4; r = 5: [(8, 5, 5)]; r = 6: [(16, 5, 13), (16, 10, 10), (16, 13, 5)]; r = 7: [(32, 10, 26), (32, 20, 20), (32, 26, 10)]   [309.2 s]

ALL CHECKS PASSED (29577739 checks)   [309.2 s]
