Motivic Homotopy Theory and Norms in Δ‑Ontology: A Construction Based on the Infinium △₁ₓ₁ (Extended and Deepened Version with Open Questions and Examples)

Abstract

This paper constructs a version of motivic homotopy theory based on Δ‑ontology—a new paradigm of the foundations of mathematics in which the structureless point is replaced by a structural quantum: the right isosceles triangle △₁ₓ₁ with legs 1 and hypotenuse √2. It is shown that the self‑similarity of the infinium naturally generates an interval object isomorphic to the affine line A¹ in the classical theory, and that localization of the category of △‑mosaics with respect to self‑similar equivalences yields a model of the A¹‑homotopy category. Measure in Δ‑ontology is interpreted as a Beilinson regulator, which makes it possible to define motivic cohomology constructively. The motivic idempotent M(ℑ) ⊗ ℚ(1) ≅ M(ℑ) ⊕ M(ℑ), which follows from self‑similarity, is used to construct norms in motivic homotopy. The key new element is the interpretation of the shift operator U on the Hilbert space of △‑mosaics as a prototype of a norm, as well as the connection of the master operator ℋ with a normed Laplacian. This opens the way to a spectral description of motivic homotopy groups and their relation to the zeros of the Riemann zeta function. In addition, physical applications are discussed: quantum gravity, dark matter, and fermion generations receive a natural geometric interpretation within Δ‑ontology. A comparison is made with the classical constructions of Morel–Voevodsky and Bach–Hoyois, revealing the advantages of the new approach: built‑in measure, natural regularization, constructivity, and the unity of the discrete and the continuous. Twelve useful conclusions are formulated that reveal the potential of Δ‑ontology in motivic homotopy theory and related areas. At the end of the paper, open questions and directions for further research are identified, and the simplest examples of computing motivic cohomology in Δ‑ontology are given.