Sheppard's arcsine, by two routes

Corr(sign X, sign Y) as the covariates' correlation runs from zero to one, drawn twice. One route is a sixty-four-node quadrature of the orthant probability over the correlation — the general construction, which works at any pair of cut points; the other is (2/π) arcsin ρ, which is elementary and works only at the median. They agree to 3.3e-16 at every one of 81 correlations, which is what licenses the quadrature everywhere else. The curve is above the diagonal at small ρ and below it at large: two signs agree with probability ½ + arcsin(ρ)/π, so a correlation of 0.5 gives exactly ⅓ and a correlation of 0.8 gives 0.5903.

The error in ⟨c(X), c(Y)⟩ from truncating Mehler's sum at J orders, for a median split at four correlations, against the exact answer. The truncation leaves the sum over m > J of a_m b_m ρ^m, which Cauchy–Schwarz bounds by ρᴶ⁺¹√(tail·tail) — so the coefficients' tail sets the constant and the correlation sets the rate. At ρ = 0.2 sixteen orders are exact to the last bit a double carries; at 0.5 the error is 7.1e-8; at 0.95 it is still 1.5e-2 and thirty-two orders are needed. The caveat this replaces was the ρ = 1 statement, carried to correlations a trial actually has.

What a balancing rule handed both main effects removes of the pure interaction between them, as the covariates become dependent. For median splits it is exactly zero at every correlation, because sign(x)² = 1: the interaction sign(X)sign(Y) is orthogonal to sign(X) and to sign(Y) whatever ρ is. For the product of the raw covariates it is 4ρ²/(1+ρ²)² — 64.00% by ρ = 0.5, rising to all of it at perfect correlation. A cut away from the median sits between them and is not small: 23.01% at a cut of one. The zero is not a fact about interactions. It is a fact about a dictionary whose functions square to a constant, which a polynomial one does not.