Chamber zeta function and closed galleries in the standard non-uniform complex from ${\mathrm{PGL}}_3$

We introduce the \emph{chamber zeta function} for a complex of groups, defined via an Euler product over primitive tailless chamber galleries, extending the Ihara–Bass framework from weighted graphs to higher-rank settings. Let $\mathcal{B}$ be the Bruhat–Tits building of ${\mathrm{P}\mathrm{G}\mathrm{L}}_3(F)$ for a non-archimedean local field $F$ with residue field ${\mathbb{F}}_q$. For the standard arithmetic quotient $\mathrm{\Gamma }\mathrm{\backslash }\mathcal{B}$ with $\mathrm{\Gamma }={\mathrm{P}\mathrm{G}\mathrm{L}}_3({\mathbb{F}}_q[t])$, we prove an Ihara–Bass type \emph{determinant formula} expressing the chamber zeta function as the reciprocal of a characteristic polynomial of a naturally defined chamber transfer operator. In particular, the chamber zeta function is \emph{rational} in its complex parameter. As an application of the determinant formula, we obtain explicit counting results for closed gallery classes arising from tailless galleries in $\mathcal{B}$, including exact identities and spectral asymptotics governed by the chamber operator.