The square root of 2 is irrational.

Stated the irrationality claim precisely as a negative existential about integer ratios.

Specified standard arithmetic and classical logic.

Identified a direct proof by contradiction as the relevant evidence.

Used a standard reference for the proof and independently reconstructed it.

Recorded the core arithmetic claims.

Checked the proof structure and the quoted source wording.

Reconstructed the contradiction argument in full.

Considered the objection that the proof hides an invalid step.

Checked validity step by step.

No competing explanation of the arithmetic remains.

Noted dependence on classical logic.

Assigned proven relative to the stated axioms.