  We study the fine local scaling properties of a class of self-affine fractal
sets called Gatzouras-Lalley carpets. More precisely, we establish a formula
for the Assouad spectrum of all Gatzouras-Lalley carpets as the concave
conjugate of an explicit piecewise-analytic function combined with a simple
parameter change. Our formula implies a number of novel properties for the
Assouad spectrum not previously observed for dynamically invariant sets; in
particular, the Assouad spectrum can be a non-trivial differentiable function
on the entire domain $(0,1)$ and can be strictly concave on open intervals. Our
proof introduces a general framework for covering arguments using techniques
developed in the context of multifractal analysis, including the method of
types from large deviations theory and Lagrange duality from optimisation
theory.
