  We present the most general sum rules reflecting the cancellation of
ultraviolet divergences in the Higgs potential in weakly-coupled, natural
extensions of the Standard Model. There is a separate sum rule for the
cancellation of the quadratic and logarithmic divergences, and their forms
depend on whether the divergences are canceled by same-spin or opposite-spin
partners. These sum rules can be applied to mass eigenstates and conveniently
used for direct collider tests of naturalness. We study in detail the
feasibility of testing these sum rules in the top sector at a future $100\TeV$
proton collider within two benchmark models, the Little Higgs (LH) and the
Maximally Symmetric Composite Higgs (MSCH). We show how the two ingredients of
the sum rules, the top partner masses and their Yukawa couplings to the Higgs,
can be measured with sufficient accuracy to provide a highly non-trivial
quantitative test of the sum rules. In particular, we study observables
sensitive to the sign of the top partner Yukawa, which is crucial for verifying
the sum rules but is notoriously difficult to measure. We demonstrate that in
the benchmark models under study, a statistically significant discrimination
between the two possible signs of each Yukawa will be feasible with a 30
ab$^{-1}$ data set at $100\TeV$.
