A theoretical framework explains why fast computer algorithms fail to detect hidden patterns in high-dimensional statistical problems. People often assume that having enough data to prove a pattern exists means a computer can also find it quickly. Instead, the statistical evidence diverges while remaining invisible when projected onto low-degree polynomial chaos.
In planted models, the likelihood ratio represents the total model evidence. Low-degree polynomial projections act like a coarse filter, capturing broad shapes while blurring away sharp, localized peaks. When the posterior distribution concentrates on an algebraic surface of measure zero, the evidence turns singular and escapes any smooth projection. Polynomial-tailed local scales in shrinkage priors break the annealed second moment calculation across two independent draws.
The author unified statistical-computational gaps across planted clique, tensor estimation, and community detection using Bayesian evidence calculations. The analysis evaluated the degree-D advantage as a truncated divergence governed by the low moments of prior overlap. Standard counterexamples, including noiseless parities solved by Gaussian elimination and structured clustering solved by lattice reduction, confirmed that low-degree algorithms fail specifically when posteriors concentrate on algebraic varieties.
Researchers can now establish the exact validity boundary of low-degree inference heuristics by testing whether posterior distributions remain stable under data perturbations. This framework connects polynomial degree priors to Bayesian double descent, which governs statistical risk at the interpolation depth.