The Infinium as a Common Ancestor: Categorical Genealogy of Hilbert Space and Kähler Manifold and the Resolution of Fundamental

We construct a categorical genealogy showing that Hilbert space and Kähler manifold are not independent constructions but share a common ancestor — the infinium ℑ = △₁ₓ₁ (a right isosceles triangle with legs 1 and hypotenuse √2). This object is terminal in the cognitive topos ℰ and generates all of mathematics as the topos of sheaves 𝒯 = Sh(Site(△₁ₓ₁)). From it, via monoidal functors F_H (Hilbertization) and F_K (Kählerization), two fundamental structures arise. We show that their key properties — orthogonality, self-similarity, measure, spectral gap, and the condition dω = 0 — are not postulated but inherited from the common ancestor. This approach resolves the contradiction between discrete and continuous, between algebraic and geometric, and provides a unified view of the nature of mathematical objects.