Go read perlop.pod. ?: binds tighter than = so the trailing = $k is outside the ?: construct.

   (exists $distinct{$v}) ? $overflow{$k} = $v
                          : $distinct{$v} = $k;
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is

  (  (exists $distinct{$v}) ? $overflow{$k} = $v
                            : $distinct{$v}  )
  = $k;
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which is like

  (exists $distinct{$v}) ? $overflow{$k} = $v = $k
                         : $distinct{$v} = $k
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The correct way to resolve the precedence problem is:

if( $distinct{$v} ) { $overflow{$k}= $v }
               else { $distinct{$v}= $k }
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(:

        - tye (but my friends call me "Tye")