Emergent Time and Causal Structure: Mathematical Supplement to the TAGC-LQG-RG Theory

This document supplements the main theory “From Information to Spacetime” by ad- dressing two critical extensions validated through systematic dialectical interrogation: (I) Emer- gent Time and (II) Extended Causal Structure. We demonstrate that the temporal parameter in the GKSL equation is self-constituting, thereby resolving the problem of time in quantum gravity. Through 45 yes/no consistency questions, we prove that time emerges as the “rhythm of gravitational information processing” via feedback: τ Γ −→ t gμν −−→ τproper ≡ t. Through rigorous exploration of parameter space, we demonstrate that global causal order can be preserved even with local front velocities vfront > c, via a hierarchy of information- fibre-dependent causal cones. Crucially, a no-go theorem proves that no viable 5D model exists under standard constraints, thereby validating the 3+1D structure of the framework. Mathematical beauty emerges from logical necessity: these structures are not postulates but inevitable consequences of the GKSL+LQG+RG synthesis. The analysis significantly reinforces the theoretical robustness of the framework.