Sharp worst-case factors for randomly pivoted Cholesky and LU

For every positive integer $r$, an explicit one-parameter family of real,
entrywise-positive, positive-definite matrices of order $r+1$ makes the
expected trace error of $r$ randomly pivoted Cholesky steps approach $2^r$
times the optimal rank-$r$ trace error. The same family makes the expected
squared Frobenius error of $r$ randomly pivoted LU steps approach $4^r$ times
the optimal rank-$r$ squared Frobenius error. Thus both previously established
universal factors are sharp as suprema. The same suprema hold on
positive-definite correlation matrices when the dimension is unrestricted. In
particular, a polynomial-in-$r$ factor for same-rank RPCholesky and a $2^r$
factor for same-rank RPLU are impossible without additional hypotheses. The
proof combines scale-separated Cauchy–Binet expansions with an exact
cancellation over pivot histories. A separate $2\times2$ rational example
already disproves the literal $2^r$ RPLU bound at $r=1$.