Necessary but Not Sufficient: Spectral Tests for Value-Linear Attention Surrogates

We give computable necessary conditions for replacing a fixed attention head by a value-linear recurrent surrogate. The semiseparable spectrum of the causal attention matrix lower-bounds the operator-norm error of any finite-state value-linear recurrence, and retrieval-style blocks yield sharp K and Kd_v state-scaling laws. Empirically, these lower bounds are tight on retrieval-shaped targets but can be loose on smooth or clustered targets, where rank feasibility does not imply trainability or downstream quality. Large pretrained-head surveys suggest that this rank obstruction is often small, including in modern 1B, 7B, and 8B checkpoints. The resulting diagnostic is best used as a rejection test for under-sized recurrent state, not as a certificate that a recurrent replacement will train or improve downstream performance.