Part I: An Operator Model Based on Self‑Similar Triangular Mosaics: Arithmetic, Shift Operator, and Spectrum

Within the framework of Δ‑ontology, where the basic primitive instead of the structureless point is the right isosceles triangle △₁ₓ₁ with legs 1 and hypotenuse √2, an operator model is constructed that unifies arithmetic and spectral analysis. Natural numbers are represented by isomorphism classes of △‑chains, addition by concatenation, and multiplication by rectangular placement. It is proved that this model satisfies the Peano axioms and is isomorphic to standard arithmetic. The shift operator U on the Hilbert space ℓ²(ℕ₀) and its adjoint U* are introduced. The algebraic identities UU = q²(I − P₀) and UU = q²I are rigorously established, where q = 1/√2 and P₀ is the projection onto the vacuum. It is proved that the norm ‖U‖ = q and the spectrum σ(U) is the closed disk of radius q. The spectrum of the master operator H = I − U − U* + UU* in the linear model is completely computed: it coincides with the closed interval [(1−q)², (1+q)²] = [3/2 − √2, 3/2 + √2] and is purely continuous, without isolated points or discrete series. In addition, explicit formulas are obtained for the spectral measure, the heat kernel, the K‑theory of C*(U), and a zeta‑like function associated with the field ℚ(√2). It is shown that the linear model, being a rigorous and complete result, does not contain the discrete spectral structure needed for a connection with the Riemann zeta function; such a structure is expected in the two‑dimensional model of △‑mosaics, which is formulated as a direction for further research. All algebraic results are formalized in Lean 4 without sorry and axiom.