Mapping-space homotopy groups give the derived Hall stabilizer weight

The loop space at $f$ of the restricted arrow core over the fixed target $z$ is the homotopy fiber over $f$ of postcomposition $f\circ -:\mathrm{Map}(x,x{)}^{\simeq }\to \mathrm{Map}(x,z)$, based at ${\mathrm{i}\mathrm{d}}_x$. Here $\mathrm{Map}(x,x{)}^{\simeq }$ is the union of components represented by isomorphisms in $\mathcal{D}(\mathcal{T})$; using the full mapping space would also include noninvertible endomorphisms and would not compute loops in the core. Since this is a union of components, its positive homotopy groups at ${\mathrm{i}\mathrm{d}}_x$ are ${\mathrm{Ext}}^{-i}(x,x)$, while [F2] gives ${\pi }_i\mathrm{Map}(x,z)={\mathrm{Ext}}^{-i}(x,z)$. The long exact sequence in [F3] therefore gives, for each $i\ge 1$, ${\mathrm{Ext}}^{-i-1}(x,x)\to {\mathrm{Ext}}^{-i-1}(x,z)\to {\pi }_{i+1}(C_f)\to {\mathrm{Ext}}^{-i}(x,x)\to {\mathrm{Ext}}^{-i}(x,z)$. Its low-degree tail is ${\mathrm{Ext}}^{-1}(x,x)\to {\mathrm{Ext}}^{-1}(x,z)\to {\pi }_1(C_f)\to \mathrm{Aut}(x)\to {\mathrm{Hom}}_{\mathcal{D}(\mathcal{T})}(x,z)$; the last map has kernel $\mathrm{Aut}(f\mathrm{/}z)$, so the preceding image in $\mathrm{Aut}(x)$ is exactly that stabilizer and the induced map onto it is surjective.

All Ext groups in these sequences are finite and vanish outside a finite range by [F2]. The group $\mathrm{Aut}(f\mathrm{/}z)$ is a subgroup of the finite group $\mathrm{Aut}(x)\subseteq {\mathrm{Ext}}^0(x,x)$. Thus all groups in the displayed exact sequence are finite and the sequence terminates. For a finite exact sequence of finite groups, the product of group orders with alternating exponents is $1$: at each term, its order is the product of the kernel order and image order, and adjacent image factors cancel. Applying this cancellation to step 1.1 yields $\mathrm{\mid }{\pi }_1(C_f){\mathrm{\mid }}^{-1}{\prod }_{j\ge 2}\mathrm{\mid }{\pi }_j(C_f){\mathrm{\mid }}^{(-1{)}^j}=\mathrm{\mid }\mathrm{Aut}(f\mathrm{/}z){\mathrm{\mid }}^{-1}{\prod }_{i>0}\mathrm{\mid }{\mathrm{Ext}}^{-i}(x,z){\mathrm{\mid }}^{(-1{)}^i}{\prod }_{i>0}\mathrm{\mid }{\mathrm{Ext}}^{-i}(x,x){\mathrm{\mid }}^{(-1{)}^{i+1}}$, which is the claimed homotopy weight.