When does “a missing symmetry explains it” stop being enough?

Step 1 · Predict

Tap the cells where you think the claim fails

“Fails” means the barrier left after searching every permutation and sign flip is above ε = 0.5 points (upper quartile over pairs). Columns are the training-error tolerance used to include networks.

What the map says

The two failures (pentagon at 1%, hexagon at 5%) are also the cells where many pairs learned different lines. Switch the colour to the same-line share to see it. The link is one-directional: learning the same lines almost always closed the barrier, but different-line pairs sometimes closed too (23 of 58 distinct different-line pairs ended below ε), so the map shows association, not a rule.

Pooled Table 6, exploratory (split decided after seeing results): at 2%, all 157 same-line pairs ended at or below 0.21 points; the 8 different-line pairs kept 2.38–9.58. Over all three thresholds, 328 of 331 distinct same-line pairs ended below ε.

Expert notes

Minimal width h*

The smallest trained width with at least 10 networks under the tolerance; operational, so it changes with the column (e.g. pentagon h* = 5 at 1%, 4 at 2%, 3 at 5%).

Residual

Upper quartile of the exact_sign training-error barrier: best of all 2^hh! permutation and flip combinations, chosen by the 25-point barrier. A lower bound on the continuous supremum.

Same-line test

Pairs match when every line agrees within 10° and 0.25 standardized units. Exploratory link to identifiability: irreducible networks computing the same function differ only by permutations and flips (Sussmann).

Caveat

Pairs with a nearly reducible network are excluded (two, hexagon at 5%). Hexagon h* = 4 at 2% is an upper bound.

Full interactive paper page · All simulators · github.com/MrigeshLakhanpal/sign-flips-lmc