Corollary 44 ([Exact tests and pure weights]). Suppose w is common h -lc. If every common actual valuation satisfies q(f_j)=0 , then every common cone valuation satisfies q⁰( f_j)=0 . The symbols f_j are invariant under G and 𝕋_w . Every …

Apply Proposition 43 ([Sign transfer]) to g=f_j , and combine its upper bound with the defining lower bound. The equality w(f_j)=0 and injectivity of the fine grading imply that f_j has weight zero. The invariant log form ω^0/ h has weight zero, so the pure-weight discrepancy computation of Section gives lc equality for every pure weight. The test ratios then all have value zero. Klt implies that h is strictly positive on every nonzero such weight, so the stated normalization is possible.