△-Ontology: Structural Basicity of the C*-Algebra of Substitution Mosaics on the Right Isosceles Triangle

This work completes the series of six parts (Parts I–VI) of the △-ontology program. The key idea of the work is to construct a C*-algebra that connects the discrete with the continuous through a single geometric object — the right isosceles triangle △₁ₓ₁ (legs 1, hypotenuse √2). In △₁ₓ₁, the unit (the length of a leg) generates irrationality √2 (the length of the hypotenuse). This makes △₁ₓ₁ the minimal carrier of the connection between the discrete (legs = 1) and the continuous (hypotenuse = √2). Its C*-algebra 𝒜_△ = C_r(X ⋊ ℤ²), associated with the primitive substitution Φ, is the algebra of this connection. Main result. It is proved that the C*-algebra 𝒜_△ is structurally basic — that is, the algebra of the connection between the discrete and the continuous — in the following precise sense: 1. (S1) Minimality. 𝒜_△ is the minimal C*-algebra containing a unit, the irrationality √2, and nontrivial K-theory with torsion ℤ/2ℤ. 2. (S2) Generation. 𝒜_△ generates the class of C*-algebras of substitution mosaics of the right isosceles class via Φ, crossed product, and K-theoretic extensions. 3. (S3) Bridging. 𝒜_△ is a bridge between the discrete (inductive system of finite-dimensional subalgebras) and the continuous (commutative subalgebra C(ℛ𝒞_∞)). (S3) is fully proved. (S1) and (S2) are proved conditionally under standard hypotheses on K-theory and the subdivision of mosaics. Formalization. The combinatorial core has been formalized in Lean 4 without sorry.