Infinitum and the New Paradigm of Mathematics: From Logical Necessity to the Complete Theory of the Geometric Quantum

The work consists of three interrelated parts. The first part provides a rigorous logical analysis of the concept of a structureless point as the foundation of mathematics; it is shown that point-based ontology contains an irremovable logical defect, and any geometric figure can serve as an initial foundation. The choice of the right isosceles triangle △₁ₓ₁ (the Infinitum) as the optimal first principle is justified. The second part unfolds the complete theory of the Infinitum: it demonstrates how Lebesgue measure, smoothness, a metric, all known types of numbers and spaces grow from this single geometric quantum, and also gives a new understanding of the nature of prime numbers based on the structural asymmetry of the Infinitum. Special attention is devoted to the reconciliation of △‑ontology with ZFC set theory and category theory; it is shown that the Infinitum serves as a common foundation that unifies the strengths of both approaches and eliminates their limitations. The third part presents a unified architectural proof of ten fundamental problems of mathematics (including the seven Millennium Prize problems, Fermat's Last Theorem, the Goldbach conjecture, and the Collatz conjecture) within the framework of △‑ontology. The central result is the establishment that all these problems are different projections of a single geometric fact — the balance of symmetry and asymmetry encoded in the Infinitum. The logical status of the statements is fixed by forcing: ℑ ⊩ (all ten problems are true in △‑ontology).