The Infinium (△₁ₓ₁) as a Universal Generator of Symmetries: Resolving the Structural Contradictions of Modern Mathematics

This work substantiates an alternative view of the foundations of mathematics, in which the foundation is not the structureless point, but the infinium ℑ = △₁ₓ₁ — a right isosceles triangle with legs 1 and hypotenuse √2. It is shown that this object is a universal generator of all types of symmetries: from its internal group D₃, the self‑similarity operators Φ and Ψ, the tensor product ⊗, and the gluing ⊕ are generated all finite reflection groups, crystallographic groups, fractal symmetries, and continuous Lie groups. It is demonstrated how replacing the point with the infinium removes the fundamental contradictions of classical mathematics — the continuum problem, singularities, the undecidability of the continuum hypothesis — and gives a natural resolution of the Hodge Conjecture. The central generation formula is formulated: Sym(All) ≅ ⟨ D₃, √2, Φ, Ψ, ⊗, ⊕ ⟩. In conclusion it is asserted that the 𝔹‑paradigm does not abolish classical mathematics, but gives it the missing geometric and topological foundation.