Tail-envelope guarantees and a clock obstruction

We establish the expected relative-error guarantee of randomly pivoted Cholesky at $\lceil Cr/\varepsilon\rceil$ pivots with the explicit constant $C=3$ when the eigenvalues after the target rank lie below a half-geometric envelope, and with $C=4$ under an inverse-square tail envelope. The leading eigenvalues may have arbitrarily large spread, and all eigenvectors are unrestricted. A further theorem gives a rank-independent constant for every fixed summable power envelope, with its dependence on the envelope exponent displayed explicitly. These results retain a head-product denominator in the determinant-path argument rather than imposing decay on the entire spectrum. We also construct finite rank-one-spike examples showing that, for a continuous-time clock version of the algorithm, no bound $\mathbb E N(t)\le ar+t\tau_r(A)$ on the expected pivot count holds for any finite $a$; this does not refute the guarantee above. A saturation calculation explains why the determinant comparison used here does not reach the diffuse-tail case.