Improve the best-known size (comparator count) of a sorting network on 13 inputs below 45

A comparator network on $n$ channels is a sequence of comparators $(i,j)$, each of which sorts the values on wires $i$ and $j$ into order; it is a sorting network if it sorts every input permutation (equivalently, by the 0/1 principle, every one of the $2^n$ binary inputs). Let $S(n)$ denote the minimum number of comparators in any sorting network on $n$ channels. The values are known exactly for $n \le 12$: $S(9)=25$, $S(10)=29$, $S(11)=35$, $S(12)=39$. For $n=13$ the exact value is unknown; the smallest known sorting network on 13 channels uses 45 comparators, and no matching lower bound is proven. Exhibit a sorting network on 13 channels using at most 44 comparators (improving the upper bound), or prove $S(13) \ge 45$ with a verifiable certificate.

Acceptance. FULLY RESOLVES (upper bound): a list of at most 44 comparators $(i,j)$ on channels $1..13$ that sorts all $2^{13}=8192$ binary inputs - verifiable in milliseconds by applying the network to every 13-bit vector and checking the output is monotone. FULLY RESOLVES (lower bound): a machine-checkable proof (e.g. an UNSAT certificate for the existence of a 44-comparator sorting network) that $S(13) \ge 45$, establishing 45 as optimal. PARTIAL: a 45-comparator network matching the current best, or an improved nontrivial lower bound below 45 with certificate. Provide the comparator list and a verification script.