Internal congruences modulo powers of $2$ for overpartition tuples with odd parts
Let $\overline{\mathrm{OPT}}_m(n)$ denote the number of overpartition $m$-tuples of $n$ into odd parts. We prove that for every odd $m\ge1$ and every $i\ge3$, \[\sum_{n\ge0}\Bigl(\overline{\mathrm{OPT}}_m\bigl(2^in\bigr)-\overline{\mathrm{OPT}}_m\bigl(2^{i-1}n\bigr)\Bigr)q^n \equiv 2^{\,i+1}\sum_{k\ge0}q^{(2k+1)^2} \pmod{2^{\,i+2}} .\] Thus $\overline{\mathrm{OPT}}_m(2^in)\equiv \overline{\mathrm{OPT}}_m(2^{i-1}n)\pmod{2^{i+1}}$, with equality of $2$-adic valuations exactly at the odd squares. The proof is elementary and uniform in $m$: a single family of integer polynomials, given by a three-term recurrence, governs every $U$-operator identity involved, and a divisibility statement supplies one power of $2$ per iteration.
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