Temporalization of Euclid Toward a Living Geometry via Temporal and Informational Tensor Fields

Euclidean geometry is traditionally formulated as a static system of spatial relations, implic- itly detached from any notion of internal time or informational dynamics. This paper proposes a foundationalre-interpretationofEuclideanspacebyintroducinganintrinsictemporalparameter τ into its geometric structure. Under this temporalization, Euclid’s axioms are reformulated as generative differential relations, transforming geometric entities from fixed objects into evolving trajectories. We introduce a complex tensor framework, termed Tenson Geometry, in which the geometric field is represented as Tij= Eij + iIij, where Eij encodes energetic (structural) components and Iij encodes informational (organi- zational) components. A generalized set of Cauchy–Riemann-type conditions is proposed to ensure variational consistency between E and I. Under these conditions, a conservation-like equilibrium relation, ∇·(E+ iI) = 0, naturally arises as a structural consistency requirement, referred to as the Tenson Equation. Within this framework, temporal curvature, informational entropy, and Fisher information geometry emerge as interconnected aspects of geometric evolution, suggesting that geometry, once temporalized, may be understood as a self-organizing system balancing stability and trans- formation. The present work does not propose a new axiom system or a physical field theory, but offers a foundational reconstruction of classical geometry suitable for conceptual extensions toward physics, information theory, and the philosophy of space.