Δ-Theory of Measure: A Manifesto for the Geometric Re-Foundation of Mathematics

This work presents the Δ-Theory of Measure — a new axiomatic system in which the fundamental basis for measurement is not the structureless unit cube, but the isosceles right triangle Δ₁ₓ₁. We demonstrate that the classical Lebesgue measure is not an independent entity, but a projection of a deeper Δ-measure onto the "square" paradigm, which historically became entrenched in mathematics due to the Cartesian coordinate system. The proposed approach possesses a constructive nature, structurally overcomes the "curse of dimensionality," and naturally unifies the treatment of smooth manifolds, fractals, and discrete structures. The paper formulates the axiomatics of the Δ-measure, introduces the category of Δ-sets, proves the Universal Covering Theorem, establishes a formal link with the Lebesgue measure through a limit transition, and proposes an architecture for the formal verification of its key propositions in Lean 4. The Δ-Theory of Measure opens the path to a radical re-foundation of mathematics, physics, and computer science in a single geometric language.