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fcedaa0 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 | #!/usr/bin/env python3
"""Exact audit of a fixed-five-dimensional holonomy completion example.
All arithmetic is rational. The base is S^1 x R^4, the flat connection is
d + N dt, and N is a nilpotent Jordan shift. No surreal or differential
geometry package is required. The general statements are proved separately.
"""
from fractions import Fraction as Q
from math import factorial
from pathlib import Path
import json
def identity(n):
return [[Q(int(i == j)) for j in range(n)] for i in range(n)]
def multiply(a, b):
return [[sum(a[i][k] * b[k][j] for k in range(len(b)))
for j in range(len(b[0]))] for i in range(len(a))]
def power(a, j):
result = identity(len(a))
for _ in range(j):
result = multiply(result, a)
return result
def rank(a):
a = [list(row) for row in a]
i = 0
for j in range(len(a[0])):
p = next((p for p in range(i, len(a)) if a[p][j]), None)
if p is None:
continue
a[i], a[p] = a[p], a[i]
pivot = a[i][j]
a[i] = [x/pivot for x in a[i]]
for p in range(len(a)):
if p != i:
scale = a[p][j]
a[p] = [x-scale*y for x, y in zip(a[p], a[i])]
i += 1
if i == len(a):
break
return i
def determinant(a):
a = [list(row) for row in a]
result = Q(1)
for j in range(len(a)):
p = next((p for p in range(j, len(a)) if a[p][j]), None)
if p is None:
return Q(0)
if p != j:
a[j], a[p] = a[p], a[j]
result = -result
pivot = a[j][j]
result *= pivot
for p in range(j+1, len(a)):
scale = a[p][j]/pivot
for k in range(j, len(a)):
a[p][k] -= scale*a[j][k]
return result
def jordan_shift(n):
return [[Q(int(j == i+1)) for j in range(n)] for i in range(n)]
def exp_negative_nilpotent(n, epsilon=Q(1)):
N = jordan_shift(n)
result = [[Q(0) for _ in range(n)] for _ in range(n)]
for p in range(n):
term = power(N, p)
coefficient = (-epsilon)**p / factorial(p)
for i in range(n):
for j in range(n):
result[i][j] += coefficient*term[i][j]
return result
def orbit_matrix(n, start, epsilon=Q(1)):
T = exp_negative_nilpotent(n, epsilon)
vector = [[Q(int(i == start))] for i in range(n)]
columns = []
for j in range(n):
columns.append([row[0] for row in vector])
vector = multiply(T, vector)
return [list(row) for row in zip(*columns)]
def log_unipotent(T):
n = len(T)
S = [[T[i][j]-Q(int(i == j)) for j in range(n)] for i in range(n)]
result = [[Q(0) for _ in range(n)] for _ in range(n)]
for p in range(1, n):
term = power(S, p)
coefficient = Q((-1)**(p+1), p)
for i in range(n):
for j in range(n):
result[i][j] += coefficient*term[i][j]
return result
def wedge(a, b):
"""Exterior monomial product, returning (sign, sorted indices)."""
if set(a) & set(b):
return 0, ()
sign = (-1)**sum(x > y for x in a for y in b)
return sign, tuple(sorted(a+b))
def main():
# Coordinates 0=t, 1=x1, ..., 4=x4.
B_indices = (0, 2, 3, 4)
derivative_sign, F_indices = wedge((1,), B_indices)
# B=-x1*v*dt^dx2^dx3^dx4, so the coefficient derivative is -v.
assert -derivative_sign == 1 and F_indices == (0, 1, 2, 3, 4)
assert wedge((0,), B_indices)[0] == 0 # N dt wedge B = 0.
assert wedge((0,), (0,))[0] == 0 # A wedge A = 0.
cases = []
for n in range(1, 13):
N = jordan_shift(n)
assert all(x == 0 for row in power(N, n) for x in row)
orbit = orbit_matrix(n, n-1)
det = determinant(orbit)
assert det == 1 and rank(orbit) == n
for i in range(n):
for j in range(n):
p = n-1-i
assert orbit[i][j] == Q((-j)**p, factorial(p))
partial_ranks = []
for s in range(n):
r = rank(orbit_matrix(n, s))
assert r == s+1
partial_ranks.append(r)
cases.append({'coefficient_rank': n,
'pointwise_curvature_rank': 1,
'holonomy_saturated_rank': rank(orbit),
'cyclic_orbit_determinant': str(det),
'nonterminal_vector_orbit_ranks': partial_ranks})
scaled_cases = []
prefix_checks = 0
for n in range(1, 9):
N = jordan_shift(n)
v = [[Q(int(i == n-1))] for i in range(n)]
for epsilon in (Q(0), Q(1, 7), Q(-2, 3)):
T = exp_negative_nilpotent(n, epsilon)
orbit = orbit_matrix(n, n-1, epsilon)
det = determinant(orbit)
expected_det = epsilon**(n*(n-1)//2)
assert det == expected_det
assert rank(orbit) == (n if epsilon else 1)
assert log_unipotent(T) == [[-epsilon*x for x in row] for row in N]
if epsilon:
for r in range(n):
# For one Jordan block all invariant proper spaces are
# these prefixes; the classification is proved in prose.
P = [[Q(int(i == j and i < r)) for j in range(n)]
for i in range(n)]
assert multiply(P, multiply(N, P)) == multiply(N, P)
assert multiply(P, multiply(T, P)) == multiply(T, P)
projection = multiply(P, v)
assert all(row[0] == 0 for row in projection)
error_squared = sum((v[i][0]-projection[i][0])**2
for i in range(n))
assert error_squared == 1
prefix_checks += 1
scaled_cases.append({'coefficient_rank': n,
'epsilon': str(epsilon),
'orbit_determinant': str(det),
'expected_epsilon_power': str(expected_det),
'orbit_rank': rank(orbit),
'finite_logarithm_identity': True})
result = {
'status': 'all exact checks passed',
'arithmetic': 'fractions.Fraction',
'base': 'S^1 x R^4 (dimension five)',
'connection': 'd + N dt; t has period one',
'monodromy': 'exp(-N)',
'potential': '-x1 v dt wedge dx2 wedge dx3 wedge dx4',
'curvature': 'v dt wedge dx1 wedge dx2 wedge dx3 wedge dx4',
'flatness_and_wedge_sign_checks': 'passed',
'cases': cases,
'scaled_connection_cases': scaled_cases,
'proper_invariant_prefix_and_unit_error_checks': prefix_checks,
'prefix_scope': 'Invariant-space classification is proved separately; the verifier checks every prefix in the tested dimensions.',
'scope': 'Finite exact certificates for the separately proved arbitrary-rank family.'
}
target = Path(__file__).with_name('holonomy_completion_results.json')
target.write_text(json.dumps(result, indent=2)+'\n')
print(json.dumps(result, indent=2))
if __name__ == '__main__':
main()
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