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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, mutu...

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, mutu...

Motivic Homotopy Theory and Norms in Δ‑Ontology: A Construction Based on the Infinium △₁ₓ₁ (Extended and Deepened Version with Open Questions and Examples)

This paper constructs a version of motivic homotopy theory based on Δ‑ontology—a new paradigm of the foundations of mathematics in which the structureless point is replaced by a structural quantum: the right isosceles triangle △₁ₓ₁ with legs 1 and hypotenuse √2. It is shown that...