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We prove that the stability condition for Fano manifolds defined bySaito-Takahashi, given in terms of the sum of the Ding invariant and the Chowweight, is equivalent to the existence of anticanonically balanced metrics.Combined with the result by Rubinstein-Tian-Zhang, we obtain the followingalgebro-geometric corollary: the $\delta_m$-invariant of Fujita-Odaka satisfies$\delta_m >1$ if and only if the Fano manifold is stable in the sense ofSaito-Takahashi, establishing a Hilbert-Mumford type criterion for $\delta_m>1$. We also extend this result to the K\"ahler-Ricci $g$-solitons and thecoupled K\"ahler-Einstein metrics, and as a by-product we obtain a formula forthe asymptotic slope of the coupled Ding functional in terms of multiple testconfigurations.

Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $\delta_m$-invariant of Fujita-Odaka