Constraint Network and Anomaly Suppression Series C: Random Matrices, Quantum Chaos, and the Splitting Operator for Scars

In the self-consistent logic of hyperbolic spectral gap theory, the splitting op
erator strips the pathological weights of winding geodesics, making the spectral
statistics of typical surfaces approach the optimal limit allowed by information
theory. This paper transplants this methodology to the fields of random matrix
theory and quantum chaos, establishing a formal framework of spectral constraint
networks for random operators. Under this framework, the large-deviation eigen
value configurations of Wigner matrices and the scarred states in quantum chaotic
systems are uniformly identified as high-betweenness centrality subnetworks. We
prove that the measure of scarred states in the space of energy eigenstates de
cays exponentially as the system dimension tends to infinity; the splitting operator
projects the determinantal point process onto the regular subspace, systematically
removing the spectral average contributions of large-deviation modes and scarred
states. The core result establishes an anomaly suppression theorem for spectral
gaps of random operators: after splitting, the spectral correlation functions of the
constrained network satisfy the Wigner-Dyson universality class, and the spectral
statistics of typical systems approach the information-theoretically optimal limit
of normal random matrices. Furthermore, this framework provides a new the
oretical syntax for the Anderson localization-delocalization transition: when the
energy exceeds the mobility edge, localized states are suppressed as pathological
subnetworks, and the system necessarily enters the extended phase.