Spectral Reuse under Squaring: A Sharp Locality Threshold

Cached curvature responses support reliable update-length decisions only with an inverse-square guarantee; a quadratic-form guarantee alone is insufficient. We identify a sharp transition governed by spectral locality: for a Loewner approximation factor $r$ and log-spectral coupling width $L$, the optimal inverse-amplification norm over all dimensions and condition numbers is finite exactly when $(r-1)e^{L/2}/(r+1)<1$. We derive this optimal norm and construct an asymptotically sharp chain whose amplification grows linearly with dimension at the threshold and exponentially above it. Numerical scans of finite chains show all three regimes. For unrestricted coupling, an exact square-distortion law gives the complementary condition $ϵ\sqrt{\kappa }\to 0$, where $r=1+ϵ$ and $\kappa$ bounds the reference condition number. Together, these bounds give an update-length interval that accounts for asymmetric inputs, damping, and nonzero spectral tails. For example selection on synthetic logistic data with local spectral coupling, this interval reduces Hessian–vector products by a factor of $3.1$ ($595.8$ to $189.3$ per batch) relative to global exact screening. It also reduces selection time by $24\%$ ($1.68$ to $1.28$ ms). On 1024-dimensional digits features, global exact screening reduces total time from $2.25$ to $1.98$ s over 32 training steps with cache refreshes. Direct Cholesky is faster in the evaluated cold-start and banded settings.