The flaw is the instant you used the deduction theorem. You used it to go from
PA |- “◻C → C” → PA |- “C”
to
PA |- “(◻C->C) → C”
What the induction theorem really says is
T + X |- Y implies T |- “X → Y”
so what you really would have needed to apply the deduction theorm would have been
PA + “◻C → C” |- C
do I win?
The flaw is the instant you used the deduction theorem. You used it to go from
PA |- “◻C → C” → PA |- “C”
to
PA |- “(◻C->C) → C”
What the induction theorem really says is
T + X |- Y implies T |- “X → Y”
so what you really would have needed to apply the deduction theorm would have been
PA + “◻C → C” |- C
do I win?