Zero free Lorentzian number system

This work introduces and formally develops a novel algebraic and analytic structure called the zero-free Lorentzian number system, denoted L×L×. Unlike conventional number systems such as the real or complex numbers, this system excludes the additive identity (zero) and instead defines all elements as intrinsically Lorentzian states. Each element is constructed as a pair of real components with a generator satisfying ε2=1ε2=1, giving rise to an indefinite Lorentzian norm: N(z)=−t2+x2N(z)=−t2+x2 This norm defines a causal structure analogous to spacetime intervals in Lorentzian geometry. The system supports a complete algebraic structure including: Addition (excluding the additive identity) Multiplication with multiplicative identity Division for non-null elements Conjugation and Lorentzian norm Lorentz-invariant transformations In addition, a full analytic framework is developed over this system, including: Exponential and logarithmic functions Differentiation defined without reference to a zero increment Integration Analytic function theory via Lorentzian Cauchy–Riemann equations Solutions to the Lorentzian wave equation The exponential map naturally generates Lorentz transformations, demonstrating that multiplication corresponds to hyperbolic rotations (Lorentz boosts). This provides a direct algebraic representation of relativistic symmetry. Unlike conventional number systems where zero represents absence or null magnitude, the zero-free Lorentzian system is purely relational: every element represents a state with intrinsic geometric and causal structure. This framework may have applications in: Lorentzian geometry Mathematical physics Relativistic algebraic structures Hyperbolic number theory Alternative algebraic foundations of spacetime Non-Euclidean analytic systems The work provides a complete axiomatic definition, algebraic properties, analytic extensions, and geometric interpretation of the system.