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8e19377 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 | #!/usr/bin/env python3
"""Exact finite checks for marginal kernels and separable coding obstructions.
Only standard-library integers and fractions. These checks do not prove
all-dimension statements or literature priority; proofs are in the manuscript.
"""
from fractions import Fraction as F
from itertools import product, combinations
from math import prod
import json
import random
def points(shape):
return list(product(*(range(q) for q in shape)))
def dot(a, b):
return sum((x * y for x, y in zip(a, b)), F(0))
def rank(rows):
if not rows:
return 0
a = [list(map(F, row)) for row in rows]
nr, nc = len(a), len(a[0])
pivot = 0
for j in range(nc):
found = next((i for i in range(pivot, nr) if a[i][j]), None)
if found is None:
continue
a[pivot], a[found] = a[found], a[pivot]
value = a[pivot][j]
a[pivot] = [x / value for x in a[pivot]]
for i in range(pivot + 1, nr):
if a[i][j]:
factor = a[i][j]
a[i] = [x - factor * y for x, y in zip(a[i], a[pivot])]
pivot += 1
if pivot == nr:
break
return pivot
def marginal_matrix(shape):
xs = points(shape)
rows = []
descriptors = []
for i in range(len(shape)):
other = [j for j in range(len(shape)) if j != i]
for assignment in product(*(range(shape[j]) for j in other)):
row = [int(all(x[j] == a for j, a in zip(other, assignment))) for x in xs]
rows.append(row)
descriptors.append((i, other, assignment))
return rows, descriptors
def matvec(a, x):
return [dot(row, x) for row in a]
def transpose_matvec(a, x):
return [sum((a[i][j] * x[i] for i in range(len(a))), F(0))
for j in range(len(a[0]))]
def one_factor_basis(q):
# Orthogonal integer Helmert contrasts; no square roots are approximated.
out = [[F(1)] * q]
for a in range(1, q):
out.append([F(1 if x < a else -a if x == a else 0) for x in range(q)])
for i, u in enumerate(out):
for v in out[:i]:
assert dot(u, v) == 0
return out
def tensor_vector(shape, basis, labels):
return [F(prod(basis[i][labels[i]][x[i]] for i in range(len(shape))))
for x in points(shape)]
def center_coordinate(shape, values, coordinate):
xs = points(shape)
lookup = dict(zip(xs, values))
out = []
for x in xs:
total = F(0)
for a in range(shape[coordinate]):
y = list(x)
y[coordinate] = a
total += lookup[tuple(y)]
out.append(total / shape[coordinate])
return out
def observable_from_marginals(shape, sums):
"""Reconstruct (I-P_K)z using inclusion-exclusion of averaging maps.
The input consists only of actual codimension-one sums; no original z
or hidden tensor coefficients are available to this decoder.
"""
a, descriptions = marginal_matrix(shape)
xs = points(shape)
one_coordinate_averages = []
for i in range(len(shape)):
entries = {}
for value, (coord, other, assignment) in zip(sums, descriptions):
if coord == i:
entries[assignment] = value / shape[i]
other = [j for j in range(len(shape)) if j != i]
one_coordinate_averages.append([entries[tuple(x[j] for j in other)] for x in xs])
out = [F(0)] * len(xs)
for size in range(1, len(shape) + 1):
for subset in combinations(range(len(shape)), size):
value = one_coordinate_averages[subset[0]]
for i in subset[1:]:
value = center_coordinate(shape, value, i)
sign = 1 if size % 2 else -1
out = [x + sign * y for x, y in zip(out, value)]
assert matvec(a, out) == sums
return out
def check_spectrum():
reports = []
count = 0
for shape in ((2, 2), (3, 3), (4, 4), (2, 3, 4), (3, 3, 3)):
a, _ = marginal_matrix(shape)
basis = [one_factor_basis(q) for q in shape]
null_count = 0
spectral_multiplicity = {}
for labels in product(*(range(q) for q in shape)):
v = tensor_vector(shape, basis, labels)
expected_eigenvalue = sum(q for q, label in zip(shape, labels) if label == 0)
lhs = transpose_matvec(a, matvec(a, v))
assert lhs == [expected_eigenvalue * x for x in v]
spectral_multiplicity[expected_eigenvalue] = spectral_multiplicity.get(
expected_eigenvalue, 0) + 1
null_count += expected_eigenvalue == 0
count += 1
expected_nullity = prod(q - 1 for q in shape)
actual_rank = rank(a)
assert null_count == expected_nullity
assert actual_rank == prod(shape) - expected_nullity
assert min(e for e in spectral_multiplicity if e) == min(shape)
reports.append({"shape": list(shape), "N": prod(shape),
"rank": actual_rank, "kernel_dimension": expected_nullity,
"gram_eigenvalues_and_multiplicities": spectral_multiplicity})
return {"exact_eigenvector_cases": count, "shapes": reports}
def check_separable_channels():
reports = []
for shape in ((2, 2), (3, 3), (4, 4), (2, 3, 4), (3, 4, 4), (3, 3, 3)):
a, _ = marginal_matrix(shape)
xs = points(shape)
for m in range(1, min(shape) + 1):
stacked = []
coordinate_ranks = []
for q in shape:
coordinate_ranks.append(rank([[(x + 1) ** j for x in range(q)]
for j in range(m)]))
for j in range(m):
# All coordinate factors are strictly positive Vandermonde rows.
mask = [prod((x[i] + 1) ** j for i in range(len(shape))) for x in xs]
assert min(mask) > 0
stacked.extend([[F(value) * weight for value, weight in zip(row, mask)]
for row in a])
expected_nullity = prod(q - r for q, r in zip(shape, coordinate_ranks))
actual_nullity = prod(shape) - rank(stacked)
assert actual_nullity == expected_nullity
assert expected_nullity == prod(max(q - m, 0) for q in shape)
reports.append({"shape": list(shape), "channels": m,
"coordinate_ranks": coordinate_ranks,
"joint_nullity": actual_nullity})
# Repeating a proportional channel adds no information.
repeated = a + [[3 * x for x in row] for row in a]
assert rank(repeated) == rank(a)
# Highlighted strict obstructions and their exact endpoints.
summary = {(tuple(r['shape']), r['channels']): r['joint_nullity'] for r in reports}
assert summary[((3, 3), 2)] == 1 and summary[((3, 3), 3)] == 0
assert summary[((4, 4), 3)] == 1 and summary[((4, 4), 4)] == 0
assert summary[((2, 3, 4), 2)] == 0
return {"stacked_rank_cases": len(reports),
"proportional_channel_counterexamples": 6, "cases": reports}
def check_scalar_completion():
rng = random.Random(20102026)
completed = probe_count = orthogonality_checks = null_probe_tests = 0
examples = []
for shape in ((2, 2), (3, 3), (4, 4), (2, 3, 4), (3, 3, 3)):
a, _ = marginal_matrix(shape)
basis = [one_factor_basis(q) for q in shape]
hidden = [tensor_vector(shape, basis, labels)
for labels in product(*(range(1, q) for q in shape))]
expected_d = prod(q - 1 for q in shape)
assert len(hidden) == expected_d
for i, u in enumerate(hidden):
assert matvec(a, u) == [0] * len(a)
for v in hidden[:i]:
assert dot(u, v) == 0
orthogonality_checks += 1
for repetition in range(4):
z = [F(rng.randrange(1, 100), rng.randrange(1, 20))
for _ in range(prod(shape))]
sums = matvec(a, z)
visible = observable_from_marginals(shape, sums)
for h in hidden:
assert dot(h, visible) == 0
total = sum(sums[:prod(shape) // shape[0]], F(0))
assert total == sum(z, F(0))
out = list(visible)
for h in hidden:
alpha = F(1, 1 + max(map(abs, h)))
mask = [1 + alpha * x for x in h]
assert min(mask) > 0 and max(mask) < 2
assert sum(mask, F(0)) == prod(shape)
measured = dot(mask, z)
coefficient = (measured - total) / (alpha * dot(h, h))
out = [x + coefficient * y for x, y in zip(out, h)]
probe_count += 1
assert out == z
completed += 1
# Omitting one orthogonal completion leaves a concrete positive
# indistinguishable pair, illustrating the general rank lower bound.
omitted = repetition % len(hidden)
delta = min(z) / (2 * max(map(abs, hidden[omitted])))
alternative = [x + delta * y for x, y in zip(z, hidden[omitted])]
assert min(alternative) > 0 and alternative != z
assert matvec(a, alternative) == sums
for i, h in enumerate(hidden):
if i != omitted:
assert dot(h, alternative) == dot(h, z)
null_probe_tests += 1
examples.append({"shape": list(shape), "necessary_and_sufficient_extra_scalars": expected_d})
return {"positive_tensor_reconstructions": completed,
"positive_mask_scalar_probes": probe_count,
"orthogonality_checks": orthogonality_checks,
"missing_probe_nonuniqueness_examples": null_probe_tests,
"dimensions": examples}
def check_bounded_fourier_completion():
"""Exact fourth-root-of-unity cases of the general Fourier construction.
Shapes use only 2 and 4, so real/imaginary character values are integers;
no trigonometric approximation or complex floating point is involved.
"""
reports = []
noise_equalities = 0
rho = F(1, 2)
roots = ((1, 0), (0, 1), (-1, 0), (0, -1))
for shape in ((2, 2), (4, 4), (2, 4, 4), (4, 4, 4)):
xs = points(shape)
a, _ = marginal_matrix(shape)
contrasts = []
for frequency in product(*(range(1, q) for q in shape)):
conjugate = tuple((-k) % q for k, q in zip(frequency, shape))
if frequency > conjugate:
continue
real, imag = [], []
for x in xs:
phase = sum((4 // q) * k * xi for q, k, xi in zip(shape, frequency, x)) % 4
re, im = roots[phase]
real.append(F(re))
imag.append(F(im))
contrasts.append(real)
if frequency != conjugate:
contrasts.append(imag)
assert len(contrasts) == prod(q - 1 for q in shape)
for i, h in enumerate(contrasts):
assert matvec(a, h) == [0] * len(a)
assert dot(h, h) in (prod(shape), F(prod(shape), 2))
for other in contrasts[:i]:
assert dot(h, other) == 0
z = [F(2 + (i * 7) % 23, 5) for i in range(prod(shape))]
visible = observable_from_marginals(shape, matvec(a, z))
total = sum(z, F(0))
reconstructed = list(visible)
for h in contrasts:
mask = [1 + rho * x for x in h]
assert min(mask) >= F(1, 2) and max(mask) <= F(3, 2)
coefficient = (dot(mask, z) - total) / (rho * dot(h, h))
reconstructed = [x + coefficient * y for x, y in zip(reconstructed, h)]
assert reconstructed == z
# Single-coordinate noise on a non-self-conjugate mode attains the
# uniform sqrt(2/N)/rho bound. Binary-only shapes have the stronger one.
target = min(contrasts, key=lambda h: dot(h, h))
epsilon = F(1, 1000)
perturbation = [epsilon * x / (rho * dot(target, target)) for x in target]
actual_error_squared = dot(perturbation, perturbation)
exact_bound_squared = epsilon ** 2 / (rho ** 2 * dot(target, target))
assert actual_error_squared == exact_bound_squared
if dot(target, target) == F(prod(shape), 2):
assert actual_error_squared == 2 * epsilon ** 2 / (rho ** 2 * prod(shape))
noise_equalities += 1
reports.append({"shape": list(shape), "positive_probes": len(contrasts),
"minimum_contrast_norm_squared": str(dot(target, target)),
"recovered_exactly": True})
return {"exact_fourier_cases": len(reports),
"sharp_noise_equalities": noise_equalities, "cases": reports}
def run():
return {"status": "PASS", "arithmetic": "exact integers/Fractions",
"spectrum": check_spectrum(),
"separable_channels": check_separable_channels(),
"scalar_completion": check_scalar_completion(),
"bounded_fourier_completion": check_bounded_fourier_completion(),
"scope": "Finite checks support proofs; no priority or physical-sensor optimality claim."}
if __name__ == '__main__':
print(json.dumps(run(), indent=2, sort_keys=True))
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