The Infinium (△₁ₓ₁) as a Universal Generator of Symmetries: Resolving the Structural Contradictions of Mathematics and Physics through 𝔹-Ontology

This work substantiates an alternative view on the foundations of mathematics, in which the primary element is not a structureless point, but the infinium ℑ = △₁ₓ₁ — a right isosceles triangle with legs of 1 and a hypotenuse of √2. It is shown that this object is a universal generator of all types of symmetries, and its internal structure naturally generates both discrete and continuous aspects of mathematical reality. A 𝔹-paradigm is formulated in which all classical symmetry groups — from finite reflection groups to crystallographic and fractal groups — are expressed through compositions of the basic operators of self-similarity, tensor product, and gluing. It is demonstrated how such an approach resolves the fundamental contradictions of classical mathematics: the continuum problem, singularities in physical models, and the undecidability of the continuum hypothesis in ZFC. The appendix provides a formal axiomatics of 𝔹-ontology and a derivation of the spectral gap as a consequence of the irrationality of √2.