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

We build a unified categorical scheme in which the fundamental object is the infinium ℑ = △₁ₓ₁ — a right isosceles triangle with legs 1 and hypotenuse √2, serving as the terminal object in the cognitive topos ℰ. From this common ancestor, two fundamental mathematical structures are generated via two functors: the Fock space H(ℑ) and the infinite Kähler manifold K(ℑ). We show that the key properties of both structures — orthogonality, complex structure, measure, self-similarity, and spectral gap — are inherited from the infinium. Such an approach eliminates traditional contradictions between the discrete and the continuous, algebra and geometry, measure and smoothness, and also gives a natural interpretation of the condition dω=0 and the self-adjointness of the Laplacian as consequences of categorical self-similarity.