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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 | #!/usr/bin/env python3
"""Exact finite tests for temperature-complete tomography and its trades.
Python standard library only. Powers of positive integer z stand for the
unnormalized measurements at inverse temperatures k*beta0. No numerical
Laplace inversion or root finder is used. Proofs are in trades_notes.md.
"""
from collections import Counter, defaultdict
from fractions import Fraction
from itertools import combinations, product
import json
def grid(shape):
return list(product(*(range(q) for q in shape)))
def lines(shape):
points = grid(shape)
out = []
for axis in range(len(shape)):
groups = defaultdict(list)
for p in points:
groups[p[:axis] + p[axis + 1:]].append(p)
out.extend(tuple(group) for _, group in sorted(groups.items()))
return out
def signature(values, points, coordinate_lines):
at = dict(zip(points, values))
return tuple(tuple(sorted(at[p] for p in line)) for line in coordinate_lines)
def is_rectangular_checkerboard(h, shape):
support = [p for p, value in h.items() if value]
levels = [sorted({p[i] for p in support}) for i in range(len(shape))]
if any(len(level) != 2 for level in levels):
return False
if set(support) != set(product(*levels)):
return False
reference = tuple(level[0] for level in levels)
amplitude = h[reference]
return all(h[p] == amplitude * (-1) ** sum(
p[i] == levels[i][1] for i in range(len(shape))) for p in support)
def complete_line_zero(interior_values, shape):
"""Expand free interior coefficients in tensor (e_a-e_last) bases."""
n = len(shape)
interior_points = list(product(*(range(q - 1) for q in shape)))
coefficients = dict(zip(interior_points, interior_values))
h = {}
for p in grid(shape):
missing = [i for i in range(n) if p[i] == shape[i] - 1]
total = 0
choices = [range(shape[i] - 1) if i in missing else (p[i],)
for i in range(n)]
for a in product(*choices):
total += coefficients[a]
h[p] = (-1) ** len(missing) * total
return h
def check_line_zero_enumeration():
shape = (3, 3, 3)
coordinate_lines = lines(shape)
total = nonzero = signed = minimum_support = 0
minimum_positive = 99
support_histogram = Counter()
for interior in product((-1, 0, 1), repeat=8):
h = complete_line_zero(interior, shape)
assert all(sum(h[p] for p in line) == 0 for line in coordinate_lines)
total += 1
if not any(h.values()):
continue
nonzero += 1
support = sum(value != 0 for value in h.values())
positives = sum(value > 0 for value in h.values())
negatives = sum(value < 0 for value in h.values())
assert support >= 8 and positives >= 4 and negatives >= 4
minimum_positive = min(minimum_positive, positives)
support_histogram[support] += 1
if max(map(abs, h.values())) <= 1:
signed += 1
if support == 8:
assert is_rectangular_checkerboard(h, shape)
minimum_support += 1
return {"shape": list(shape), "interior_arrays": total,
"nonzero_line_zero_tensors": nonzero,
"signed_nonzero_tensors": signed,
"minimum_support_tensors": minimum_support,
"minimum_positive_support": minimum_positive,
"support_histogram": dict(sorted(support_histogram.items()))}
def check_colored_fibers(shape, alphabet):
points = grid(shape)
coordinate_lines = lines(shape)
fibers = defaultdict(list)
total = 0
for values in product(alphabet, repeat=len(points)):
fibers[signature(values, points, coordinate_lines)].append(values)
total += 1
pair_checks = minimal_trade_pairs = bounded_multiplicity_fields = 0
threshold = 2 ** (len(shape) - 1)
for fiber in fibers.values():
for values in fiber:
if max(Counter(values).values()) < threshold:
assert len(fiber) == 1
bounded_multiplicity_fields += 1
for a, b in combinations(fiber, 2):
changed = [p for p, va, vb in zip(points, a, b) if va != vb]
assert len(changed) >= 2 ** len(shape)
moved_colors = set()
for color in set(a) | set(b):
h = {p: int(va == color) - int(vb == color)
for p, va, vb in zip(points, a, b)}
if any(h.values()):
moved_colors.add(color)
assert sum(value > 0 for value in h.values()) >= threshold
assert Counter(a)[color] >= threshold
assert all(sum(h[p] for p in line) == 0
for line in coordinate_lines)
if len(changed) == 2 ** len(shape):
assert len(moved_colors) == 2
color = next(iter(moved_colors))
h = {p: int(va == color) - int(vb == color)
for p, va, vb in zip(points, a, b)}
assert is_rectangular_checkerboard(h, shape)
minimal_trade_pairs += 1
pair_checks += 1
return {"shape": list(shape), "alphabet": list(alphabet),
"fields": total, "fibers": len(fibers),
"ambiguous_fibers": sum(len(fiber) > 1 for fiber in fibers.values()),
"equal_data_pair_checks": pair_checks,
"minimal_trade_pair_checks": minimal_trade_pairs,
"multiplicity_threshold": threshold,
"bounded_multiplicity_fields_certified_unique": bounded_multiplicity_fields}
def elementary_from_powers(powers):
e = [Fraction(1)]
for m in range(1, len(powers) + 1):
e.append(sum(((-1) ** (k - 1) * e[m - k] * powers[k - 1]
for k in range(1, m + 1)), Fraction(0)) / m)
return e
def polynomial_of_roots(values):
coefficients = [Fraction(1)]
for z in values:
following = [Fraction(0)] * (len(coefficients) + 1)
for j, coefficient in enumerate(coefficients):
following[j] += coefficient
following[j + 1] -= z * coefficient
coefficients = following
return coefficients
def check_newton_moments():
cases = multiset_fibers = 0
for length in range(2, 6):
seen = {}
for values in product(range(1, 5), repeat=length):
powers = tuple(sum(z ** k for z in values) for k in range(1, length + 1))
elementary = elementary_from_powers(powers)
polynomial = [(-1) ** k * e for k, e in enumerate(elementary)]
assert polynomial == polynomial_of_roots(values)
multiset = tuple(sorted(values))
assert powers not in seen or seen[powers] == multiset
seen[powers] = multiset
cases += 1
multiset_fibers += len(seen)
return {"line_lengths": [2, 3, 4, 5],
"ordered_line_cases": cases,
"distinct_multisets": multiset_fibers}
def check_sharp_embedded_trades():
cases = moment_checks = 0
for n in range(2, 6):
shape = (3,) * n
points = grid(shape)
coordinate_lines = lines(shape)
a, b = {}, {}
fresh = 100
for p in points:
if all(x < 2 for x in p):
a[p] = 2 if sum(p) % 2 == 0 else 5
b[p] = 5 if sum(p) % 2 == 0 else 2
else:
a[p] = b[p] = fresh
fresh += 1
assert sum(a[p] != b[p] for p in points) == 2 ** n
assert Counter(a.values())[2] == Counter(a.values())[5] == 2 ** (n - 1)
assert signature([a[p] for p in points], points, coordinate_lines) == signature(
[b[p] for p in points], points, coordinate_lines)
for line in coordinate_lines:
for power in range(1, 7):
assert sum(a[p] ** power for p in line) == sum(b[p] ** power for p in line)
moment_checks += 1
cases += 1
# Three distinct cyclic tensors have identical line multisets and are
# therefore invisible at all temperatures. They are not a global two-color
# parity interchange on the ternary domain.
cyclic_cases = 0
for n in (2, 3, 4):
shape = (3,) * n
points = grid(shape)
coordinate_lines = lines(shape)
tensors = [[(2, 5, 11)[(sum(p) + phase) % 3] for p in points]
for phase in range(3)]
signatures = [signature(v, points, coordinate_lines) for v in tensors]
assert signatures[0] == signatures[1] == signatures[2]
assert len(set(map(tuple, tensors))) == 3
cyclic_cases += 1
return {"sharp_rectangular_examples": cases,
"exact_power_projection_checks": moment_checks,
"cyclic_ternary_examples": cyclic_cases}
def run():
return {"status": "PASS", "arithmetic": "exact integers and Fractions",
"line_zero_kernel": check_line_zero_enumeration(),
"colored_fibers": [check_colored_fibers((2, 2, 2), range(3)),
check_colored_fibers((3, 3), range(3))],
"finite_temperature_moments": check_newton_moments(),
"sharp_examples": check_sharp_embedded_trades(),
"primary_prior_art": [
"https://arxiv.org/abs/0704.1730",
"https://arxiv.org/abs/1104.1295",
"https://eccc.weizmann.ac.il/report/2008/105/download/"],
"scope": "Finite checks support the manuscript proofs; no novelty claim follows."}
if __name__ == "__main__":
print(json.dumps(run(), indent=2, sort_keys=True))
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