Mixed sums of infinite multiple zeta-star values with restricted indices

We study arithmetic sums of infinite multiple zeta-star values subject to upper or lower bounds on their indices. Let $\eta ({\mathcal{D}}_q)$ be the image, under the zeta-star correspondence, of the admissible infinite index sequences with all indices at most $q$, and let $\overline{\eta ({\mathcal{T}}_p)}$ be the closure of the image of the sequences with all indices at least $p$. We prove that $\eta ({\mathcal{D}}_q)+\overline{\eta ({\mathcal{T}}_p)}$ is an interval exactly when $p\le q$; in that case it is $[{\zeta }^{\star }(\{q{\}}^{\mathrm{\infty }})+1,\mathrm{\infty })$. We also show that sums of two upper-bounded sets are half-lines, even with different bounds, whereas sums of two lower-bounded sets are never intervals. The proofs combine a uniform product estimate with natural gap constructions. Lateral thickness handles the diagonal case $p=q\ge 3$, while a non-strict local bridge–gap criterion treats $p=q=2$ directly. Compact truncations and explicit omitted intervals yield the remaining assertions.