Temporal Topology and Geometric Symmetry Breaking in a Dual-Time Gauge Framework: The Emergence of the Arrow of Time Through the Observation Mechanism

We formulate a geometric framework in which observable spatial geometry and temporal directionality emerge from the intersection of two orthogonal Lorentzian temporal domains, identified as objective (physical) and subjective (informational). Each domain carries a dual-time structure consisting of a generative temporal coordinate and a manifest temporal coordinate, and is modeled using split-complex geometry that encodes conjugate Lorentzian temporal orientations. Observation is described as an intersection process in which the two Lorentzian domains meet at a Euclidean interface: oppositely oriented manifest temporal components cancel, while generative components combine into an effective temporal magnitude. This intersection yields a three-dimensional Euclidean spatial geometry accompanied by a scalar temporal parameter. The interaction between the domains is formulated using a bi-fibered temporal bundle equipped with independent temporal gauge connections. The associated gauge curvatures encode generative desynchronization, geometric phases, and topological sectors. A discrete temporal interchange symmetry exchanging the two domains is spontaneously broken by a composite temporal order parameter, resulting in an emergent arrow of time. Variation of the action yields effective gravitational field equations in which spacetime curvature receives contributions from the temporal gauge and phase fields. This construction provides a consistent geometric setting in which Euclidean space arises as an observational intersection of conjugate Lorentzian temporal structures, while temporal asymmetry, gauge curvature, and topological quantization emerge from the underlying bi-temporal geometry.