Compression Is Not Regularisation in Material Field Inference

Recovering a material field from sparse indirect observations is ill-posed, so the parameterisation acts as a prior. Parameterisations are often selected by how compactly they encode the target class. We show in one controlled experiment that this criterion can select the wrong prior. Holding a quantized tensor train's format, capacity, and protocol fixed and changing only the bond cut, bit-interleaved ordering compresses local media up to $15.8\times$ more cheaply than axis-major ordering, yet recovers them less accurately on all six field families. A prior therefore induces two distinct costs: representing the truth, which compression measures, and representing directions that the measurements weakly constrain, which regularisation governs. Evaluated on singular directions of the observation Jacobian, total variation and an $H^1$ penalty cost $1.5$--$42\times$ more on the least observable direction than on the most observable one, whereas the tensor train and separable factorisation cost \emph{less}. This ratio requires no inversion and ranks priors within a problem in advance, with $\rho=-0.84$ against measured recovery error. Structural parameterisation and explicit penalties are complementary, and combining them recovers most of the gap.