The Möbius Function in △‑Ontology: Geometric, Energetic, and Categorical Interpretation

The Möbius function μ(n) is one of the central arithmetic functions, playing a key role in number theory. Classically it is defined through the factorisation of a number into prime factors. In this work we propose three new interpretations of μ(n) within the framework of △‑ontology: geometric (via the axes of symmetry of a △‑mosaic), energetic (via the binding energy of hypotenuses), and categorical (via the spectral representation functor). It is shown that μ(n) = 0 corresponds to the presence of multiple √2‑relations in the △‑mosaic of the number n, and the sign of μ(n) is determined by the parity of the number of distinct prime divisors. A continuous generalisation μ_E(n) — the energetic Möbius function — is introduced. The Möbius inversion formula is interpreted as a natural transformation in the category of △‑mosaics. A connection with the zeta function is established through the spectrum of the master operator ℋ.