= Instantaneous Lorenz error growth at a zero-y crossing
{title2=$|\sigma+r-z|>2\sqrt\sigma$}
At $y=0$, the symmetric Jacobian sum is $\operatorname{diag}_{\rm block}([[-2\sigma,\sigma+r-z],[\sigma+r-z,-2]],-2b)$. Its planar largest eigenvalue is positive exactly under the displayed condition. Thus both $z<\sigma+r-2\sqrt\sigma$ and $z>\sigma+r+2\sqrt\sigma$ permit instantaneous error growth. The vertical direction initially contracts, but skew couplings can rotate errors later; this local criterion is not a claim of persistent growth along a trajectory. For $\sigma=10,r=28,z=26,b=3$, the eigenvalues are $4,-26,-6$ and the growing direction is $(1,2,0)/\sqrt5$.
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