This manuscript constructs a closed operator-theoretic framework that formalizes narcissistic collapse and recovery by embedding Jungian structural decomposition into a rigorous Hilbert space architecture. The system models the psyche across distinct orthogonal sectors (ego/persona, shadow, object-representation, validation-supply, etc.). The dynamic state is governed by a non-self-adjoint distortion operator—comprising false-self projection, confabulation, splitting, and defense mechanisms—acting in opposition to a self-adjoint integration operator. A healthy, individuated state is defined by an Omega-Sigma constraint structure, which enforces structural admissibility and internal integration. Within this mathematical geometry, "narcissistic collapse" is explicitly derived as a spectral rupture. It occurs when the distortion operator overcomes the integration operator, mathematically characterized by a Birman-Schwinger criterion and the zero-set failure of the regularized Fredholm determinant. Furthermore, the non-self-adjoint (skew) nature of the psychological distortion is quantified using an odd Zeta function hierarchy, where the 3rd, 5th, and 7th order coefficients act as exact spectral penalties for logical contradiction, narrative torsion, and archetypal rupture. Conversely, "recovery" is formalized through the Unavailability Protocol. Modeled as a strict mathematical projection that severs external object-validation and reinforcement channels, this protocol restricts the domain of the distortion operator. Absolute recovery is achieved when the norm of this restricted distortion falls below the spectral gap of the integration operator, driving the admissibility defect and odd skew torsion back to zero. Ultimately, this framework is not intended as a clinical diagnostic instrument, but rather as a strict formal language for modeling psychological collapse, defense mechanisms, falsification, and recovery as exact operator-theoretic phenomena.
Andrew Kim (Sat,) studied this question.