We analyse three KV cache quantization schemes under a fair bit budget: \textbf{KV} (scalar MSE baseline), \textbf{KQV} (WHT + MSE on
K; WHT + MSE + QJL on
V), and \textbf{QKQV} (WHT + MSE + QJL on both). Starting from the Beta distribution on the hypersphere, we trace how QJL on
K inflates inner product variance by
π/2, which softmax amplifies nonlinearly via Jensen's inequality, and we present statistical inference and information metrics to highlight practical differences. Three empirical findings emerge. (1)~At
n=4 (the practically dominant budget), KQV wins on every measure -- KL divergence, geometric
K error, and 6D distance -- across all distributions and ranks tested. (2)~The K--V asymmetry is unconditional: QKQV is consistently worse than KQV in KL divergence at every budget and distribution. (3)~A budget-dependent crossover exists: QKQV achieves better geometric
K reconstruction at
n∈{2,3,5}, KQV at
n∈{4,6}, invariant to rank and tail weight -- an open rate-distortion problem.
KL(pref∥pquant), K-only by construction, bridges K direction error to routing corruption and output collapse. We present a sufficient condition when the Jensen mechanism amplifies superlinearly through the softmax. At
n∈{2,3,5}, QKQV wins geometrically because this assumption does not bind. At
n=4, elevated K error and KL divergence for QKQV strongly suggest the Jensen mechanism is the operative cause of the crossover, providing a new perspective and explanation.