Atemporal Topological Information Dynamics: Emergence of Time via Categorical Amnesia and Non-Hermitian Friction (Part 1/Draft)

To resolve the 'Problem of Time'—a foundational challenge in quantum gravity and open quantum systems—this paper proposes Atemporal Topological Information Dynamics (ATID), a novel information-theoretic framework that operates without a background spacetime manifold. We formulate an effective non-Hermitian Hamiltonian H_{\text{eff}} describing open-system dissipation alongside a topological action S_{\text{topo}} defined over an atemporal information manifold \mathcal{M}_{\text{info}}. Applying the variational principle \delta S_{\text{topo}} = 0, we geometrically derive a non-commutative friction tensor \mathcal{F}_{ab}^{\text{fric}} that diverges near Exceptional Points (EPs) due to the non-analytic algebraic structure of the Puiseux series. Furthermore, by introducing a Forgetting Functor \mathcal{F}_{\text{orget}} mapping from a high-dimensional microscopic topological category \mathbf{TopoSys} to a 1D causal order category \mathbf{Time}, we prove that thermodynamic entropy and the arrow of time emerge naturally as the 'kernel residue' of lossy topological information compression. Finally, we establish a non-linear algebraic phase delay scaling law (\Delta \tau \propto \Vert\mathcal{F}_{ab}^{\text{fric}}\Vert \ln(I_c / (I_c - I))) near the critical information injection threshold I_c, numerically verifying the model via a Parity-Time (PT) symmetric coupled RLC electronic circuit system and providing an accessible experimental protocol.