Your explanation of => is not entirely accurate.

if ($month => 10) { ... }
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is equivalent to

if ($month, 10) { ... }
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The => operator does NOT wrap whatever appears on the left into a string. Instead, it allows any expression, but if a bareword is used, it turns the bareword into a string. Any other expression is left as-is, not even coerced into being a string. (Actually rolling a list into a hash will coerce the keys into strings unless the hash has some magic.)