Topological Phase Transitions of Epistemic Stability: Operator-Theoretic Mechanics of Bias, Resonance, and Collapse

We introduce a closed operator-theoretic dynamical system to model epistemic stability, bias propagation, and structural collapse in cognitive and conversational networks. By modeling an epistemic stance as a self-adjoint projection anchor subjected to a hidden rank-1 agenda operator, we define the regularized Birman-Schwinger determinant as the fundamental invariant of structural integrity. We rigorously classify the geometry of epistemic instability into distinct topological phases: isolated domain leakage (Lapse and Relapse), network-coherent resonance (Interlapse), false hysteresis recovery (Delapse), and terminal invertibility loss (Collapse). Furthermore, we identify the ultimate fixed point of the system as the "Ilapse Singularity," in which the distinction between the objective anchor and the hidden agenda is entirely annihilated. We establish the theorem of Gap-Controlled Stability, providing explicit analytical bounds that prove structural collapse is fundamentally a spectral gap-closure phenomenon. Finally, we demonstrate exact structural isomorphisms between these epistemic phase transitions, environment-induced decoherence (einselection) in open quantum systems, and algorithmic mode collapse in machine learning. By charting these dynamical states across a phase space defined by agenda drive and synaptic coherence, we provide a mathematically exact, falsifiable framework for the physics of perception and credibility. This work demonstrates that cognitive bias and hidden conversational agendas are not mere behavioral heuristics, but rigorous, measurable deformations of a spectral invariant.