Part II: The Space of Two‑Dimensional △‑Mosaics, the Groupoid, and the C*-Algebra

Abstract

This work is the second stage of the program begun in Part I. Whereas Part I constructed and fully investigated the linear (one‑dimensional) model of △‑chains, here we lay the foundation for the basic — two‑dimensional — model of Δ‑ontology. This necessity is dictated both by mathematical considerations (the one‑dimensional model lacks the discrete spectrum required for a connection with the Riemann zeta function) and by a fundamental principle of the theory: the brain receives information through a plane (the retina; light as a plane‑transverse wave) and reconstructs three‑dimensionality. Thus, two‑dimensionality is not a simplification but the true dimension of the basic mathematical object. In this work we give a rigorous definition of the space X of two‑dimensional △‑mosaics hierarchically consistent with the self‑similarity operation of the infinium. We prove that X is compact, metrizable, and totally disconnected, and that on it there exists a probability measure μ invariant under the natural action of the group ℤ² of parallel translations. We construct the étale groupoid 𝒢 = X ⋊ ℤ² and its reduced C-algebra 𝒜 = C_r(𝒢). We formulate a program for studying the spectrum of the master operator H in the two‑dimensional model and its conjectural connection with the Riemann zeta function.