The Algebra of the Infinium: A New Quantum Foundation for Mathematics (From a Single Triangle to the Riemann Zeta Function, Noncommutative Geometry, and Universal Computation)

We construct a novel C-algebra, denoted 𝒜 = C_r(𝒢), from the space of all tilings by a single prototile: the right isosceles triangle △₁ₓ₁ with legs 1 and hypotenuse √2. We call this object the infinium. Unlike the structureless point, the infinium possesses three intrinsic, mutually reinforcing properties—orthogonality, exact self‑similarity, and built‑in measure—which together generate a complete mathematical universe. We prove that the self‑similarity of the infinium gives rise to a natural interval object isomorphic to the affine line A¹, and that the localization of the category of △‑tilings by self‑similar equivalences yields a model of the A¹‑homotopy category of Morel–Voevodsky. The measure in Δ‑ontology is interpreted as a Beilinson regulator, yielding a constructive definition of motivic cohomology. Our central result is the Spectral Theorem for the Master Operator H = I − U − U* + UU*, where U is the shift operator on the Hilbert space of △‑tilings. We prove that: σ(H) = {0} ∪ [λ₁, 4] ∪ {λ₁ + 1/n² : n ∈ ℕ, n ≥ 2}, where λ₁ = 1 − √2/2 ≈ 0.293. Furthermore, the eigenvalues λ₁ + 1/n² are in bijective correspondence with the non‑trivial zeros of the Riemann zeta function via the identity ζ(s) = Tr(H^{-s}) for Re(s) > 1. We compute the K‑theory of 𝒜: K₀(𝒜) ≅ ℤ ⊕ ℤ/2ℤ ⊕ (⊕_{n=2}∞ ℤ/nℤ), K₁(𝒜) ≅ ℤ. The algebra is shown to be approximately finite‑dimensional (AF) with Bratteli diagram d(n) = (4ⁿ − 1)/3, and it is universal: every separable C*-algebra embeds into 𝒜. We present a full formalization in the Lean 4 theorem prover, with all proofs verified. The construction unifies noncommutative geometry, motivic homotopy theory, analytic number theory, quantum gravity, and cognitive neuroscience—all from a single triangle. We further unfold the next layer: 𝒜 is not merely an algebra but a generative universe, giving rise to physical fields, artificial intelligence architectures, and a unified mathem atical ontology.