Compact Temporal Geometry and the $T^2$ Framework for Quantum Gravity

We introduce a two-dimensional temporal framework in which time is represented by a compact manifold $T^2=(t_1,t_2)$, with $t_1$ encoding classical causal structure and $t_2$ representing quantum coherence. This construction unifies unitary evolution, decoherence, measurement collapse, and gravitational dynamics within a consistent geometric and algebraic formalism. Compactification of the coherence time $t_2$ yields a minimal temporal resolution $\mathrm{\Delta }{t}_2\sim \sqrt{{\alpha }^{\mathrm{\prime }}}$, leading to a discretized spectrum of temporal modes and regularized ultraviolet behavior in quantum field theory and string-theoretic gravity. We formulate an extended Schrödinger equation and generalized Lindblad dynamics on $T^2$, and demonstrate the compatibility of this structure with local gauge symmetry through a complexified BRST quantization procedure. Using para-Hermitian geometry and generalized complex structures, we derive a covariant formulation of temporal T-duality that accommodates both Lorentzian and Euclidean signatures. The $T^2$ framework provides new insights into modular thermodynamics, black hole entropy, and the emergence of classical time from quantum coherence, offering a compact and quantized model of temporal geometry rooted in string theory and quantum gravity.