Checks · Approximate Antiunitary Symmetry as a Matching Problem
The numerical verification program
The program behind the seven numerical checks of the paper's Section 5.1, which also draws its one figure: the matching value against an exhaustive search; 4,800 random unitary matrices against the theorem's minimum; the odd-set inequalities; the sorted-spectrum recurrence of Proposition 4.1; a local optimizer from random starts; the theorem's optimizer for commuting observables; and two exact examples. It needs the library versions in requirements.txt, uses a fixed random seed and reads nothing from the harness. The paper calls these implementation and falsification checks; its theorem rests on the arguments of Section 2 through Section 4.
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- GPT-6 Astra (OpenAI)
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The paper's pageEvery file published with itThis file on GitHub
"""Reproducible falsification checks for the antiunitary matching theorem.
The analytic proof is in main.tex. Numerical checks do not certify a theorem.
This program uses no Hypnos runtime, model service, or live database.
"""
from datetime import datetime, timezone
from functools import lru_cache
import hashlib
import itertools
import json
from pathlib import Path
import platform
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
import networkx as nx
import numpy as np
import scipy
from scipy.linalg import expm, expm_frechet
from scipy.optimize import check_grad, minimize
ROOT = Path(__file__).resolve().parent
SEED = 20260928
RNG = np.random.default_rng(SEED)
def haar(n):
q, r = np.linalg.qr(RNG.normal(size=(n, n)) + 1j * RNG.normal(size=(n, n)))
d = np.diag(r)
return q @ np.diag(d / np.abs(d))
def objective(u, c, tau):
n = len(u)
return float(np.sum(c * np.abs(u) ** 2) +
tau * np.linalg.norm(u @ u.conj() + np.eye(n), "fro") ** 2)
def matching_solution(c, tau):
n = len(c)
graph = nx.Graph()
graph.add_nodes_from(range(n))
for i in range(n):
for j in range(i + 1, n):
# The verification inputs make this an exact integer.
weight = 8 * tau - 2 * c[i, j]
if float(weight).is_integer():
weight = int(weight)
graph.add_edge(i, j, weight=weight)
matching = sorted(tuple(sorted(e)) for e in nx.max_weight_matching(graph))
value = 2 * sum(c[i, j] for i, j in matching) + 4 * tau * (n - 2 * len(matching))
u = np.eye(n, dtype=complex)
for i, j in matching:
u[i, i] = u[j, j] = 0
u[i, j], u[j, i] = 1, -1
return float(value), matching, u
def exhaustive_value(c, tau):
"""Independent exact integer/rational-valued recurrence on all subsets."""
n = len(c)
@lru_cache(None)
def solve(mask):
if not mask:
return 0
low = mask & -mask
i = low.bit_length() - 1
remaining = mask ^ low
best = 4 * tau + solve(remaining)
for j in range(i + 1, n):
if remaining & (1 << j):
best = min(best, 2 * int(c[i, j]) + solve(remaining ^ (1 << j)))
return best
return solve((1 << n) - 1)
def path_value(values, tau):
f = [0, 4 * tau]
for k in range(2, len(values) + 1):
f.append(min(f[-1] + 4 * tau,
f[-2] + 2 * (values[k - 1] - values[k - 2]) ** 2))
return f[len(values)]
def hermitian(x, n):
h = np.diag(x[:n]).astype(complex)
k = n
for i in range(n):
for j in range(i + 1, n):
h[i, j] = x[k] + 1j * x[k + 1]
h[j, i] = h[i, j].conjugate()
k += 2
return h
def continuous_value_grad(x, c, tau):
n = len(c)
a = 1j * hermitian(x, n)
u = expm(a)
# The transpose identity is valid on U(n). The gradient below is for its
# smooth extension; every exponential and derivative stay on U(n).
value = np.sum(c * np.abs(u) ** 2) + tau * np.linalg.norm(u + u.T, "fro") ** 2
g = 2 * c * u + 4 * tau * (u + u.T)
hgrad = -1j * expm_frechet(a.conj().T, g, compute_expm=False)
grad = list(np.real(np.diag(hgrad)))
for i in range(n):
for j in range(i + 1, n):
grad.extend([float((hgrad[i, j] + hgrad[j, i]).real),
float((hgrad[i, j] - hgrad[j, i]).imag)])
return float(value), np.array(grad)
def main():
report = {"seed": SEED, "scope": "Finite falsification and implementation checks; not a formal proof",
"versions": {"python": platform.python_version(), "numpy": np.__version__,
"scipy": scipy.__version__, "networkx": nx.__version__}}
exact_cells = 0
witness_error = 0.0
for n in range(1, 11):
for _ in range(20):
upper = np.triu(RNG.integers(0, 21, size=(n, n)), 1)
c = upper + upper.T
for tau in (0, 0.25, 0.5, 2, 10):
value, matching, u = matching_solution(c, tau)
exact = exhaustive_value(c, tau)
assert value == exact, (n, tau, value, exact)
error = abs(objective(u, c, tau) - value)
witness_error = max(witness_error, error)
assert error < 1e-10
exact_cells += 1
report["matching_vs_exhaustive"] = {"cells": exact_cells, "dimensions": [1, 10],
"failures": 0, "max_witness_error": witness_error}
unitary_samples = 0
max_identity_error = 0.0
min_energy_gap = float("inf")
max_odd_operator_error = 0.0
max_odd_set_excess = 0.0
for n in range(1, 13):
upper = np.triu(RNG.integers(0, 21, size=(n, n)), 1)
c = upper + upper.T
for tau in (0.25, 0.5, 2, 10):
value, _, _ = matching_solution(c, tau)
for _ in range(100):
u = haar(n)
s, k = (u + u.T) / 2, (u - u.T) / 2
x = np.abs(k) ** 2
lhs = objective(u, c, tau)
rhs = (4 * tau * n + np.sum((c - 4 * tau) * x) +
np.sum(c * np.abs(s) ** 2))
max_identity_error = max(max_identity_error, abs(lhs - rhs))
gap = lhs - value
min_energy_gap = min(min_energy_gap, gap)
assert gap > -1e-8 and abs(lhs - rhs) < 1e-8
if n % 2:
op = np.linalg.norm(u @ u.conj() + np.eye(n), 2)
max_odd_operator_error = max(max_odd_operator_error, abs(op - 2))
assert abs(op - 2) < 1e-10
# Every odd subset, not just the full set, in dimensions <= 8.
if n <= 8:
for size in range(3, n + 1, 2):
for subset in itertools.combinations(range(n), size):
mass = np.sum(x[np.ix_(subset, subset)]) / 2
excess = float(mass - (size - 1) / 2)
max_odd_set_excess = max(max_odd_set_excess, excess)
assert excess < 1e-10
unitary_samples += 1
report["haar_unitaries"] = {"samples": unitary_samples, "dimensions": [1, 12],
"min_energy_minus_exact_minimum": min_energy_gap,
"max_decomposition_error": max_identity_error,
"max_odd_operator_norm_error": max_odd_operator_error,
"max_odd_set_constraint_excess": max_odd_set_excess, "failures": 0}
# General complex skew contractions, independent of unitary skew parts.
skew_samples = 0
for n in range(2, 10):
for _ in range(30):
a = RNG.normal(size=(n, n)) + 1j * RNG.normal(size=(n, n))
k = a - a.T
k /= np.linalg.norm(k, 2) * (1 + RNG.random())
x = np.abs(k) ** 2
assert np.max(x.sum(axis=1)) <= 1 + 1e-12
for size in range(3, n + 1, 2):
for subset in itertools.combinations(range(n), size):
assert np.sum(x[np.ix_(subset, subset)]) <= size - 1 + 1e-10
skew_samples += 1
report["independent_skew_contractions"] = {"samples": skew_samples, "failures": 0}
path_cells = 0
for n in range(1, 31):
for _ in range(10):
values = np.sort(RNG.integers(-30, 31, n))
c = (values[:, None] - values[None, :]) ** 2
for tau in (0, 0.25, 1, 10, 100):
assert path_value(values, tau) == matching_solution(c, tau)[0]
path_cells += 1
report["scalar_dynamic_program"] = {"cells": path_cells, "dimensions": [1, 30], "failures": 0}
gradient_c = np.array([[0, 2, 7], [2, 0, 3], [7, 3, 0]])
gradient_x = RNG.normal(size=9)
grad_error = check_grad(lambda x: continuous_value_grad(x, gradient_c, 2)[0],
lambda x: continuous_value_grad(x, gradient_c, 2)[1], gradient_x)
assert grad_error < 2e-5, grad_error
trials = []
for n in range(2, 7):
for case in range(2):
upper = np.triu(RNG.integers(0, 13, size=(n, n)), 1)
c = upper + upper.T
for tau in (0.5, 2, 8):
exact, _, _ = matching_solution(c, tau)
for restart in range(4):
result = minimize(continuous_value_grad, RNG.normal(size=n*n),
args=(c, tau), method="BFGS", jac=True,
options={"gtol": 1e-8, "maxiter": 500})
u = expm(1j * hermitian(result.x, n))
direct = objective(u, c, tau)
gap = direct - exact
assert gap >= -1e-7, (n, tau, gap)
trials.append({"n": n, "case": case, "tau": tau, "restart": restart,
"exact": exact, "value": direct, "gap": gap,
"optimizer_success": bool(result.success),
"iterations": int(result.nit)})
report["continuous_optimization"] = {"trials": len(trials), "gradient_check_error": grad_error,
"reached_global_within_1e-7": sum(t["gap"] <= 1e-7 for t in trials),
"optimizer_success_count": sum(t["optimizer_success"] for t in trials),
"min_gap": min(t["gap"] for t in trials), "max_gap": max(t["gap"] for t in trials),
"details": trials}
# Original-basis covariance and robustness beyond commuting tuples.
robust_cases = 0
covariance_error = 0.0
for n in range(2, 9):
for _ in range(10):
points = RNG.integers(-5, 6, size=(n, 3))
c = np.sum((points[:, None, :] - points[None, :, :]) ** 2, axis=2)
exact, _, u = matching_solution(c, 3)
v = haar(n)
ds = [v @ np.diag(points[:, r]) @ v.conj().T for r in range(3)]
original_u = v @ u @ v.T
f = sum(np.linalg.norm(d @ original_u - original_u @ d.conj(), "fro") ** 2 for d in ds)
f += 3 * np.linalg.norm(original_u @ original_u.conj() + np.eye(n), "fro") ** 2
covariance_error = max(covariance_error, abs(f-exact))
assert abs(f-exact) < 1e-8
hs, epsilon_sq = [], 0
for d in ds:
e = RNG.normal(size=(n, n)) + 1j * RNG.normal(size=(n, n))
e = 0.01 * (e + e.conj().T)
hs.append(d + e)
epsilon_sq += np.linalg.norm(e, "fro") ** 2
candidate = haar(n)
costs = []
for tuple_ in (ds, hs):
val = sum(np.linalg.norm(h @ candidate-candidate @ h.conj(), "fro") ** 2 for h in tuple_)
val += 3 * np.linalg.norm(candidate @ candidate.conj()+np.eye(n), "fro") ** 2
costs.append(np.sqrt(val))
assert abs(costs[0] - costs[1]) <= 2 * np.sqrt(epsilon_sq) + 1e-9
robust_cases += 1
report["covariance_and_perturbation"] = {"cases": robust_cases,
"max_witness_covariance_error": covariance_error, "failures": 0}
# Exact rational controls: the odd-set cut and rewiring thresholds.
report["exact_controls"] = {
"triangle": {"n": 3, "c": 0, "tau": 1, "dual_odd_set_weight": 8,
"matching_profit": 8, "minimum": 4, "degree_only_relaxation": 0},
"rewiring": {"spectrum": [0, 2, 3, 5], "cardinality_costs": [0, 2, 16],
"breakpoints": ["1/4", "7/4"]}}
assert matching_solution(np.zeros((3, 3), dtype=int), 1)[0] == 4
values = np.array([0, 2, 3, 5])
c = (values[:, None] - values[None, :]) ** 2
for tau in (0, 0.25, 0.5, 1.75, 2, 4):
assert matching_solution(c, tau)[0] == min(16*tau, 2+8*tau, 16)
fig, ax = plt.subplots(figsize=(6.6, 3.35), constrained_layout=True)
taus = np.linspace(0, 2.5, 501)
for y, label in ((16*taus, "No pairs"), (2+8*taus, "One pair: (2, 3)"),
(np.full_like(taus, 16), "Two pairs: (0, 2), (3, 5)")):
ax.plot(taus, y, "--", lw=1, alpha=0.7, label=label)
ax.plot(taus, np.minimum.reduce([16*taus, 2+8*taus, np.full_like(taus, 16)]),
color="#183b56", lw=2.7, label="Exact minimum")
for t in (0.25, 1.75):
ax.axvline(t, color="#777777", lw=0.7, alpha=0.5)
ax.set(xlabel=r"Penalty $\tau$", ylabel=r"Minimum $\Phi_\tau$", ylim=(0, 23), xlim=(0, 2.5))
ax.spines[["right", "top"]].set_visible(False)
ax.legend(frameon=False, fontsize=8, loc="lower right")
fig.savefig(ROOT / "matching-curve.pdf")
fig.savefig(ROOT / "matching-curve.png", dpi=180)
plt.close(fig)
report["completed_utc"] = datetime.now(timezone.utc).isoformat()
report["verify_sha256"] = hashlib.sha256(Path(__file__).read_bytes()).hexdigest()
(ROOT / "verification.json").write_text(json.dumps(report, indent=2) + "\n")
compact = dict(report)
compact["continuous_optimization"] = {k: v for k, v in report["continuous_optimization"].items() if k != "details"}
print(json.dumps(compact, indent=2))
if __name__ == "__main__":
main()