Value Channels Gaussianize Rounding Noise: Tensor Geometry of KV-Cache Quantization

A quantized key-value cache is written once and read by many future, possibly adaptive, queries. We show that, within a factor of two, the uniform read error is governed by one invariant: the expected injective norm of a random sign tensor that pairs the attention family with the Euclidean value space. Three reductions isolate it. Convex order removes the encoder: among conditionally unbiased encoders on fixed grids, independent stochastic rounding is optimal for every convex loss. Duality replaces the query family by its convex hull, and symmetry replaces the bit budget by its orbits. A structure theorem then reduces this invariant, for every attention family and up to absolute constants, to two scalar quantities. It is a Bernoulli counterpart of Chevet's inequality for a Euclidean factor: the error is the worst fixed-query fluctuation, amplified by $\sqrt{d}$ in value dimension $d$, plus a flat-direction width in which the value channels Gaussianize the bounded rounding noise. When each token's values share one grid, Bernoulli and Gaussian noise models therefore differ by a factor of order at most $$\begin{gathered} \max \\ lbrace1,(\log n/d{)}^{1/2} \\ rbrace \end{gathered}$$ on $n$ tokens, and softmax attention with logarithmic-dimensional keys attains this factor in every dimension. Across independent token groups, fluctuations add in ${\ell }_2$ and selections in ${\ell }_1$. Optimal precision is a convex program, rounded within a factor of eight, whose optimum equalizes the contributions to the worst-case error of all coordinates stored above the minimum precision.