√2 as the Minimal Irrationality: Algebraic, Geometric, and Ontological Analysis

The number √2 occupies a unique position in mathematics. It is not merely the first proved irrationality, but the minimal irrationality in a strict sense, which we make precise. The work consistently unfolds the minimality of √2 through five independent criteria and shows that all of them stem from a single geometric structure — the right isosceles triangle △₁ₓ₁. Key result. △₁ₓ₁ is the canonical geometric representation of the minimal irrationality. Through it, √2 is connected with φ and π, and is embedded in the fundamental laws of physics. Structure of the work. We give formal definitions of each criterion of minimality, three independent proofs of the irrationality of √2, explicit numerical verifications, and rigorous proofs of the key theorems. Statements not confirmed numerically are marked as hypothe ses.