The Pythagorean Theorem as the Meta-Code of Mathematics (Updated Version)

The work asserts that the Pythagorean theorem is not merely a geometric fact, but a fundamental semantic law of mathematical language. It is shown that all major branches of mathematics (algebra, topology, analysis, number theory) can be interpreted as transformations of a single idea — the balance between symmetry and asymmetry, expressed through the right triangle. Special attention is paid to the right isosceles triangle (1:1:√2) as the minimal "cell of meaning," possessing the properties of orthogonality, self-similarity, and the generation of irrationality. A view is proposed of natural numbers as projections of the geometry of this triangle, and of mathematics as a whole as a game played by the rules of Pythagorean balance. In the concluding section we express the discovered regularities in the language of logical theory (forcing ⊩ and semantic consequence ⊧), showing that the infinium ℑ = △₁ₓ₁ forces mathematics to be exactly what we know it to be.