Effective Semiclassical Dynamics, Big Bounce, and Time Reflection in Picard-Loop Quantum Cosmology

We present an effective semiclassical cosmological framework combining hyperbolic Picard Hornspatial topology (H3/PSL(2, Z)) with holonomy-corrected Loop Quantum Cosmology (LQC). Byparameterizing the spatial metric deformation via a C2-smooth cutoff function f(a), we resolvethe classical Big Bang singularity without introducing physical stress-tensor discontinuities. Wephysically justify f(a) as a two-scale mechanism: a high-redshift UV geometric percolation cutoffand a late-time IR topological crossover. We rigorously derive the effective second-order dynamicalequations and demonstrate via numerical integration a deterministic Big Bounce at amin ≈ 0.5318.The resulting phase space portrait (a, a ˙ ) is C1-smooth, guaranteeing the continuous evolution of theenergy-momentum tensor Tµν. Under strict unitary quantum evolution, we show that the Planckianvolume truncation at the cusp limits the coarse-grained horizon entropy to a minimum boundaryShorizon ∝ Amin/4l2P , providing a geometric boundary condition for the thermodynamic arrow oftime within a 4D Block Universe without violating quantum unitarity.