Many high-dimensional statistical inference problems are believed to have inherent computational difficulties. Various frameworks have been proposed to provide rigorous evidence of such difficulties, such as lower bounds on constrained computational models (such as low-order functions) and methods grounded in statistical physics based on free-energy landscapes. I’m here. This paper aims to rigorously combine seemingly disparate low-order and free-energy-based approaches. We define a free-energy-based criterion for stiffness and formalize it to the well-established concept of low-order stiffness for a broad class of statistical problems: all Gaussian additive models and specific models with sparsely planted signals. Bind to. By exploiting these strict connections, we can: For Gaussian additive models, the “algebraic” notion of low stiffness means the failure of the “geometric” local MCMC algorithm. We establish that , and provide a new low-order lower bound for sparse linear regression. It is difficult to prove directly. These results provide both conceptual insight into the relationship between different concepts of stiffness and concrete technical tools such as new methods for proving lower bounds of low degree.