SORT-QS: A Projection-Based Structural Framework for Quantum Systems Error Correction, Noise Filtering, and Operator Diagnostics

This work introduces the Supra-Omega Resonance Framework for Quantum Systems (SORT-QS), a structural operator formalism that adapts the Supra-Omega Resonance Theory from cosmological applications to finite-dimensional quantum devices. The central idea is to represent coherent and incoherent error processes, noise filtering mechanisms and diagnostic procedures in terms of a finite set of idempotent resonance operators \(\{\hat{O}_i\}\), an effective projector \(\hat{H}\) and a nonlocal kernel acting on the operator space rather than on configuration space. In SORT-QS, quantum channels are mapped to structured resonance manifolds in Liouville space, and error sectors are encoded as mutually constrained projectors that satisfy algebraic closure and idempotency. This enables a scale- and mode-selective description of noise, where the analogue of the projection kernel \(\kappa\) defines structural suppression or amplification of specific error components in an abstract frequency or syndrome domain. The framework provides three complementary layers: (i) a purely algebraic resonance space for error classes and stabilizer-like structures, (ii) a kernel-based noise filtering module formulated as a linear transformation on operator-valued modes and (iii) an operator diagnostics layer that quantifies deviations from ideal projector structure as resonance defects. No device-specific assumptions or empirical performance claims are made. Instead, SORT-QS offers a mathematically controlled template that can be instantiated within arbitrary quantum error correction schemes, gate sets and noise models, and serves as a basis for future applications to concrete architectures.