The missing term was unary: omit-size solved at its ceiling in 1.4s

Adversarial review of the artifacts found three of my numbers to be
artifacts of my own code. All three reproduced here before acceptance:

- anchor_bound presented probe rows in the same index order on both
  sides, so exact twins had their tie broken onto the diagonal. 0.997 ->
  0.920 on omit-size. Fixed by scrambling the T-side presentation.
- The truth is not a strict local minimum: 51 transpositions have
  exactly zero energy delta. fast_pair_descent only looked stationary
  because its break test treats zero as no-improvement.
- scipy's FAQ takes no n_init, so it was swallowed into unknown_options
  and 'FAQ x30 restarts' computed bit-identically to plain FAQ. Replaced
  with a real restart loop over P0='randomized'.

The blind ceiling for omit-size is 0.836, not 1.0: the text field has 51
exact transposition automorphisms, so T[s,s] is bitwise identical to T
and no objective f(V, P T P^T) can separate an orbit at any order. Every
synthetic accuracy was being divided by the wrong denominator.

The fifteen failed solvers share one property -- all purely quadratic or
purely spectral, none with a node-level term. Eight moments of each
node's own field row, blended with the quadratic term through
Frank-Wolfe, reach 0.837 with an energy gap of exactly zero. The term
must stay in the loop: as a seed for pure-quadratic descent it scores
0.21, pinned through the iterations it scores 0.84 -- which is also why
amplification plateaued, being itself pure-quadratic.

Co-Authored-By: Claude <noreply@anthropic.com>

1 files changed, 13 insertions, 5 deletions

diff --git a/worldalign/match_battery.py b/worldalign/match_battery.py
index 9035e62..63a4bf1 100644
--- a/worldalign/match_battery.py
+++ b/worldalign/match_battery.py
@@ -142,11 +142,19 @@ def solve_faq_multi(visual, shuffled, **_):
     """FAQ from many random doubly stochastic starts, best by its own objective."""
     from scipy.optimize import quadratic_assignment
 
-    result = quadratic_assignment(
-        visual, shuffled, method="faq",
-        options={"maximize": True, "n_init": 30, "rng": 0},
-    )
-    return result.col_ind
+    # scipy's FAQ takes no `n_init`; passing one is silently swallowed into
+    # unknown_options, which is why an earlier version of this function
+    # returned bit-identical results to plain FAQ. Restart explicitly.
+    best, best_value = None, -np.inf
+    for seed in range(15):
+        result = quadratic_assignment(
+            visual, shuffled, method="faq",
+            options={"maximize": True, "rng": seed, "P0": "randomized"},
+        )
+        value = float((visual * shuffled[np.ix_(result.col_ind, result.col_ind)]).sum())
+        if value > best_value:
+            best, best_value = result.col_ind, value
+    return best
 
 
 def solve_2opt(visual, shuffled, **_):