That helped immensely! My understanding was that each mem file is what each process occupies in memory. But that's wrong. Instead, it's the image of my whole memory, and I can only read the sections which the specific process occupies.

As per your suggestion, I added:

use Fcntl qw(SEEK_SET);
seek $fh, 134512640, SEEK_SET or die "could not seek in mem: $!";
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I got that number from converting from the hex address the maps file in /proc.