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Programs are mathematical proofs. If maths cannot be patented, software can't be, either.
Proofs can be represented as programs, not the other way around. Also, USA allows for algorithm parents, and algorithms are maths. While I agree with you, your reasoning is not correct.
No, the proof - program correspondence is in both directions.
Correspondence is quite a weak relation. Very far from one being another.
I'd say if you ask a mathematician, they would disagree with you. But maybe that depends on how far they have gone into maths from common sense
That's why it's also called Curry-Howard isomorphism.
Correspondence is not correlation.
Judges and Justices are not that precise. They aim to preserved public order before anything else. If a whole industry is based on a questionable interpretation of patent, they is a lot of chances that judges would agree on it. Even in countries where you could not patent algorythm, industries patent the documentation, the "software design", the brand name, the illustrations used, and aggregates everything together, to say they own it. And it works.
TL;DR : Class Justice
There are ways to patent maths