The Geometry of Time

Abstract

The Ze framework proposes a fundamental ontology of discrete counters whose updates constitute primitive events, from which the geometric structure of spacetime emerges relationally. This paper develops the correspondence between Ze dynamics, twistor theory, and spin networks, demonstrating a natural unification of continuous and discrete descriptions of physics. Each event in Ze is associated with a complex counter C = C_temporal + i C_spatial, which is shown to admit a direct interpretation as a point in projective twistor space, following Penrose (1967). The temporal component encodes the event's location along a causal chain, while the spatial component encodes its relational coupling to other events, together forming a twistor coordinate that satisfies the Minkowski norm |C|²_η. Causal sequences of events form a directed graph, which maps naturally onto a spin network wherein edges carry quantized spin labels derived from accumulated counter increments, satisfying j(j+1) ∝ n². This mapping preserves the causal structure while providing a discrete, background-independent representation of quantum geometry in the sense of Penrose (1971) and Rovelli & Smolin (1995). Proper time emerges as the sum of spin labels along causal chains, yielding a combinatorial definition τ = Σ f(j_edge) that reproduces relativistic time dilation and the twin paradox without additional postulates. The discrete Ze Lagrangian L_Ze = Σ [(dC_spatial/dt)² - γ(dC_temporal/dt)²] converges in the continuum limit to the twistor Lagrangian L_twistor = Σ (Ż)ηŻ, confirming that relativistic kinematics are encoded in the counter dynamics. The Ze framework thus achieves a synthesis wherein twistor space provides the continuous, complex-geometric representation of events, spin networks provide the discrete, combinatorial projection, and relativistic effects arise from the internal structure of counters rather than from postulated symmetries. This unification offers a promising foundation for quantum gravity by deriving both quantum discreteness and relativistic spacetime from a single causal ontology.