The rule is parity, and it runs both ways

At a correlation of 0.5, four combinations of a dictionary and an outcome shape. The joint sign flip (X, Y) → (−X, −Y) leaves the bivariate normal alone at every correlation, so a function that changes sign under it is orthogonal to one that does not. A product of two odd functions is even; a product of an odd and an even one is odd. So an odd dictionary removes exactly none of the first and something of the second, and an even dictionary does the reverse — which it does, to machine precision, in both of the two rows that should be zero. This is one rule where there had been two: that a median split's square is constant, and that a polynomial dictionary contains the products a correlation generates.

The smallest share each dictionary removes, over seven outcome shapes, at a correlation of 0.5. A rule balancing the mean of each covariate has a worst case of exactly zero — against the square, and against both products. Adding a median split to it, which is the second thing every trial balances, leaves the worst case at exactly zero, because a median split is odd and so is a mean. Adding the square instead moves it to 6.8%, and the extra functions after that move it to 7.4%. The worst case is decided by which parities the dictionary contains rather than by how many functions are in it.

Each curve is one dictionary, swept over eight tolerances at two hundred units; a point above the line is a set thin enough that walking beats hunting. The curves lie nearly on top of one another, which is the answer to whether the crossing is a fact about the tolerance or about the thinness it produces: the crossings sit between one admissible assignment in 176 and one in 268 for dictionaries of 3 to 6 functions. The mechanism is that a hunt costs exactly 1/p and a walk costs almost the same everywhere — between 82 and 394 evaluations per usable draw across the whole table — so the crossing is wherever 1/p reaches a number that does not move.

Every admissible set of 14 units this table can enumerate, by how much of the assignment space it admits and how many pieces it falls into under single swaps. A walk is uniform on the piece it starts in and never leaves it. The pieces are not fragments: at 522 admissible assignments the set splits into 3 halves of exactly 520 each, and every assignment's complement is in the other half — no sequence of admissible single swaps takes an assignment to its own mirror image. Two-swap proposals reconnect four of the six disconnected sets here — the two they do not are the thinnest, where a two-unit move rarely lands anywhere admissible either — which makes a bigger proposal a correctness repair rather than the speed dial it was measured as.