fix: remove gauge bias in random_vector() — complex Gaussian sampling

[PFmix]

Problem

random_vector() in set_relationship had two compounding biases:

1. Real-only sampling — np.array([2*random()-1 for _ in range(4)], dtype='complex')
   draws only from the real axis; the imaginary degrees of freedom are never explored.
2. abs() clamp — vec[0] = abs(vec[0]) forces the first component non-negative,
   biasing fallback vectors toward |00⟩ for +ZZ targets.

Together these introduce a gauge anomaly: all-open (s_ij = +1 everywhere) scores
systematically higher in S^tomo than gauge-equivalent all-walled (s_ij = −1) and
unfrustrated Mattis configurations. On a 120-qubit square lattice at STEPS=2, the gap
reaches ~0.075 (~20% relative inflation for all-open).

Root cause

random_vector() is called as a fallback in make_vec when the projected vector
orthogonalises to zero. For the initial |+>^N state, this fallback is structurally
guaranteed for −ZZ edges (analytically: |P_{-ZZ} @ vecs[1]|² = 0 exactly) but never
triggered for +ZZ edges. The sign-asymmetric fallback frequency, combined with the
biased sampling, inflates +ZZ scores relative to −ZZ.

Fix

Replace the two biased lines with a single complex Gaussian draw:

# before
vec = np.array([2 * random() - 1 for _ in range(4)], dtype='complex')
vec[0] = abs(vec[0])

# after
vec = np.random.randn(4) + 1j * np.random.randn(4)

This uniformly covers the full complex unit sphere (after normalize), making the
fallback unbiased with respect to gauge sign. The random import from random is
no longer needed inside random_vector after this change.

Testing

Sweep over N=120 (10×12 square lattice), K=3, STEPS=1–3, N=6 builds per config:


│ STEPS │ original gap │ fixed gap │ % closed │

│ 1     │ 0.0338       │ 0.0331    │ 2 %      │

│ 2     │ 0.0750       │ 0.0510    │ 32 %     │

│ 3     │ 0.0666       │ 0.0370    │ 44 %     │


SVG maze and max-frustrated configurations are unaffected (deltas within σ ≈ 0.006),
as expected for mixed/symmetric sign distributions where the fallback asymmetry averages
out across edges.

A residual gap (~56% of original) remains due to a structural property of eigh on the
initial tensor-product state — the second null-space eigenvector always falls back for
−ZZ but not for +ZZ regardless of sampling distribution. This is documented separately.

It is also a residual issue I did not manage to solve.