Alexey (KAMAZ) Petrov

√2 as the Minimal Irrationality: Algebraic, Geometric, and Ontological Analysis

The number √2 occupies a unique position in mathematics. It is not merely the first proved irrationality, but the minimal irrationality in a strict sense, which we make precise. The work consistently unfolds the minimality of √2 through five independent criteria and shows that al...

A Conceptual Approach to the Hierarchy of Dimensionalities of Meaningfulness

We are accustomed to thinking of dimensionality as a property of space. Three dimensions — length, width, height. Four — we add time. Five — something else, invisible but felt. But what if dimensionality is not a property of space, but a property of meaningfulness? What if each n...

Two-Dimensionality as the Ontological Foundation of Mathematics: The RIT as the Necessary Base Object from the Standpoint of Neurophysiology, Physics, and the Structure of Provability

This work develops a single thesis: two-dimensionality is not an arbitrary choice of Δ-ontology but a necessary consequence of the nature of human perception and the structure of mathematics itself. We argue that: 1. Neurophysiological argument. Most human sense organs — the reti...

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...

Part II: The Space of Two‑Dimensional △‑Mosaics, the Groupoid, and the C*-Algebra

This work is the second stage of the program begun in Part I. Whereas Part I constructed and fully investigated the linear (one‑dimensional) model of △‑chains, here we lay the foundation for the basic — two‑dimensional — model of Δ‑ontology. This necessity is dictated both by mat...

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...

Unifying General Relativity and Quantum Mechanics in Δ‑Ontology: From the Structureless Point to the Structural Quantum △₁ₓ₁

This paper examines one of the central problems of modern physics—the absence of a unified theory combining general relativity (GR) and quantum mechanics (QM). It is shown that the reason for this unresolved problem lies not in physics as such but in the mathematical foundation:...

Unifying General Relativity and Quantum Mechanics in Δ‑Ontology: From the Structureless Point to the Structural Quantum △₁ₓ₁

This paper examines one of the central problems of modern physics—the absence of a unified theory combining general relativity (GR) and quantum mechanics (QM). It is shown that the reason for this unresolved problem lies not in physics as such but in the mathematical foundation:...

The Infinium △₁ₓ₁ as a Single Source of the Discrete and the Continuous: A Motivic Bridge Between Arithmetic and the Continuum (Resolution of the Ancient Dichotomy within Δ‑Ontology)

The paper examines a fundamental problem in the foundations of mathematics—the gap between the discrete and the continuous, which can be traced from Zeno’s ancient paradoxes to the modern Millennium Prize problems. It is shown that traditional approaches either postulate the irre...

The Infinium as a Bridge Between the Discrete and the Continuous: The RIT and Its Motive Structure (Why a Single Triangle Resolves the Oldest Paradox of Mathematics)

The problem of the relationship between the discrete and the continuous is one of the deepest and most painful in the foundations of mathematics. Classical approaches either postulate the existence of the continuum without explaining its origin, or reduce continuity to infinite c...

The Unity of the Discrete and the Continuous as a Structural Principle: The Infinium △₁ₓ₁ and the Theory of Motives

This article is devoted to one of the oldest and deepest problems in mathematics—the gap between the discrete and the continuous. We show that this gap is not an accidental difficulty but a direct consequence of choosing the structureless point as the initial primitive. Within Δ‑...

Paradoxes of Naive Set Theory and Their Resolution in Δ‑Ontology: From the Structureless Point to the Structural Quantum

This paper examines the classical paradoxes of naive set theory: Russell’s paradox, the barber paradox, the Grelling–Nelson paradox, the Burali‑Forti paradox, as well as the Banach–Tarski paradox and Zeno’s paradox (Achilles and the tortoise). It is shown that their emergence is...

Essay: Ontological Remodeling: Demolishing the Walls of Reality

The essay "Ontological Remodeling: Demolishing the Walls of Reality" introduces and develops the concept of ontological remodeling as a diagnostic tool for understanding fundamental shifts in the foundations of human existence. The author argues that contemporary technological an...

The Geometric Quantum as a Universal Generator of Hierarchies: Δ-Ontology, the Principle of Energy Economy, and the Dissolution of Fundamental Paradoxes

We show that the right isosceles triangle with legs 1 and hypotenuse √2 — the infinium 𝕴 = △₁ₓ₁ — is the minimal structural quantum generating all of mathematics through a single mechanism: the diagonal hierarchical transition governed by the principle of energy economy. Gödel's...