Title: $M$-TF equivalences in the real Grothendieck groups

Abstract: For an abelian length category $\mathcal{A}$ with only finitely many isoclasses of simple objects, we have the wall-chamber structure and the TF equivalence in the dual real Grothendeick group $K_0(\mathcal{A})_\mathbb{R}^*=\operatorname{Hom}_\mathbb{R}(K_0(\mathcal{A})_\mathbb{R},\mathbb{R})$, which are defined by semistable subcategories and semistable torsion pairs in $\mathcal{A}$ associated to elements $\theta \in K_0(\mathcal{A})_\mathbb{R}^*$. In this paper, we introduce the $M$-TF equivalence for each object $M \in \mathcal{A}$ as a systematic way to coarsen the TF equivalence. We show that the set $\Sigma(M)$ of the closures of $M$-TF equivalence classes is a finite complete fan in $K_0(\mathcal{A})_\mathbb{R}^*$, and that $\Sigma(M)$ is the normal fan of the Newton polytope $\mathrm{N}(M)$ in $K_0(\mathcal{A})_\mathbb{R}$.