Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

=== Premise P3 says: ===

: 

This is a stronger assumption than the existence of a consistent differential algebra of generalized functions. In fact:
* Schwartz proved (1954) that you cannot multiply distributions while maintaining all classical algebraic identities and still have a consistent extension of differentiation.

Colombeau algebras resolve this but sacrifice some classical identities.

Thus P3 contradicts Schwartz' impossibility theorem unless you provide a construction of such an algebra or assert it as an additional axiom known to be incompatible with standard mathematics.

So:

===== Unless you are deliberately working in a nonstandard, non-ZFC mathematical foundation, P3 itself introduces a contradiction. =====
This is allowed if you state it explicitly, but the proof does not.