Uniform polynomial product bounds and sharp Hölder continuity of the joint spectral radius

For a bounded family of matrices of order $d$, with joint spectral radius one and individual spectral norms at most $L$, every product of length $n$ is bounded by $\Theta_d(Ln)^{d-1}$, where $\Theta_d$ depends only on $d$. The proof uses an approximate extremal norm, a well-rounded coordinate system, and triangular comparison. Diagonal rescaling of a block triangular product is controlled by convex combinations of unitary conjugations. A quantitative comparison, valid at every length and every rescaling threshold, also gives approximate extremal norms with controlled eccentricity. Consequently, the joint spectral radius is uniformly Hölder continuous with exponent $1/d$ on norm-bounded sets of nonempty compact matrix families, over either $\mathbb R$ or $\mathbb C$, with explicit constant $d(2d+1)L^{1-1/d}$. The powers $d-1$ and $1/d$ are sharp in general.