Asymptotic Factorization of Repeated-Index MZV and t-Value Ratios for Even Arguments

For even integers $s=2m\ge2$, the repeated-index multiple zeta values $\zeta(\{s\}_n)$ and their level--2 analogues $t(\{s\}_n)$ admit closed forms involving factorial factors and algebraic units. While these formulas have been known since the work of Hoffman and Borwein-Bradley-Broadhurst, the structural origin of the unit factors has remained unclear.

In this paper we derive unified combinatorial representations expressing both $\zeta(\{2p\}_n)$ and $t(\{2p\}_n)$ as signed sums of odd $2p$-th roots of unity. We show that all algebraic units appearing in these closed forms arise from a finite dihedral orbit, and that this orbit contains a \emph{unique} dominant element. This uniqueness yields a canonical factorization

\[\tilde A_n(2p)=2^{-2p}\frac{\zeta(\{2p\}_n)}{\zeta(\{2p\}_{n-1})}\frac{t(\{2p\}_{n-1})}{t(\{2p\}_n)}=F_n(2p)\,R_n(2p),\]

where the unit-correction term satisfies $R_n(2p)=1+O(\rho^{-n})$ for some $\rho>1$. As a result, since all remaining units contribute only exponentially small terms, the entire $1/n$ asymptotic expansion of $\tilde A_n(2p)$ is determined solely by the factorial ratio $F_n(2p)$, with no algebraic contribution from the unit part.