Towards a Non-Abelian Arithmetic of Emergent Systems: Formalizing the Identity Deficit in Composite Manifolds

This paper introduces a radical shift in foundational mathematics, proposing a process-oriented arithmetic designed to model complex, emergent systems. Moving beyond the linear addition of Peano axioms (1+1=2), we formalize the concept of the "Identity Deficit" (σ)—the quantitative sacrifice of individual properties required for systemic cohesion. Drawing from the principles of the KAIROS 1.0 logic, this framework reinterprets numbers as "sedimented histories" and provides a rigorous language for the transition from dynamic processes to stabilized structures. It offers a universal tool for quantifying emergence in physics, biology, and information theory.