General saturation of the resolved-leading weight in hidden regions

Prove that, for every hidden region satisfying the homogeneity, positive-pinch, hierarchy, and scalefulness conditions, the cancellation of the individually superleading sector causes its first nonzero contribution to occur at the resolved-leading weight.

Background

The Hidden Region Finder separates a superleading cancellation sector from the remaining terms of the Lee–Pomeransky polynomial. The algorithm determines the individual weight of the cancellation-sector monomials and the first resolved weight of the complete polynomial independently, rather than imposing that the cancellation sector saturate the resolved weight.

All certified examples in the paper exhibit the expected saturation: after the cancellation is resolved in suitable local coordinates, the formerly superleading sector contributes at the same weight as the other resolved-leading terms. The authors note, however, that the cancellation sector could in principle first contribute at a higher weight, with the resolved-leading layer supplied entirely by the remaining polynomial. Establishing that this cannot occur would make the hierarchy structure of hidden regions more universal.