and it will evaluate to the empty string

Not quite. The comparison operators and possibly all operators returns triplevar(0, 0.0, "") for false.

$ perl -wE'say 0 + (0 > 1)'
0

$ perl -wE'say 0 + ""'
Argument "" isn't numeric in addition (+) at -e line 1.
0
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which evaluates to $_ previously set to a hash value

Not quite. Scalar assignment evaluates to its LHS, so it evaluates to $max, not $_.

$ perl -E'$_ = 123; ++( $max = $_ ); say $max; say $_;'
124
123
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Of course, they hold the same value at that point, and map promptly copies that value.