A pointwise Lipschitz lower bound for the joint spectral radius

For every $d\ge1$ and every fixed nonempty compact family $\mathcal M\subseteq\mathbb C^{d\times d}$ there are $C,r>0$ such that every nonempty compact $\mathcal N$ with $d_H(\mathcal M,\mathcal N)<r$ satisfies $\widehat\rho(\mathcal N)\ge\widehat\rho(\mathcal M)-C\,d_H(\mathcal M,\mathcal N)$. This is the pointwise one-sided Lipschitz lower bound for the joint spectral radius. For a product-bounded reference family, a limiting seminorm and a cone argument build words in $\mathcal N$ whose growth rate drops by at most a linear amount. In general, after normalizing to $\widehat\rho(\mathcal M)=1$, nonresonance estimates for block triangular families show that some exterior power $\Lambda^k\mathcal M$ has joint spectral radius one and is product bounded. The estimate for $\Lambda^k\mathcal M$ transfers back through $\widehat\rho(\Lambda^k\mathcal N)\le\widehat\rho(\mathcal N)^k$. The perturbing family need not preserve any invariant subspace of the reference family, and reducible families with unbounded normalized products are included.