Operators, Spectral Structure and Effective Physical Regimes in a Correlational Framework

This paper completes the operator-level formulation of a non-ontological correlational framework for fundamental physics. Starting from a closed correlational ontology without primitive space, time, dynamics, or laws, the work analyzes how effective physical structures necessarily emerge once projection to observable subspaces is performed. The central object of the analysis is the Hessian operator associated with a consistency functional on correlational configuration space. By studying its spectral properties under restriction and projection, the paper shows how distinct effective regimes arise, including geometric, quantum-coherent, and open-system (dissipative) descriptions. Elliptic operators, propagators, and GKSL-type generators are recovered as regime-dependent normal forms of the same underlying correlational structure. The role of projection and information loss is emphasized as the origin of irreversibility, decoherence, and effective dynamics. The work clarifies why operators — rather than objects or fundamental equations of motion — carry the relevant physical content in the framework, and why all dynamics remains effective rather than ontological. No new ontological assumptions are introduced, and no specific physical model is postulated. Instead, the paper provides a general classification of admissible effective operators, identifies the limits of operator interpretation, and delineates failure modes where spectral or geometric descriptions break down. This contribution closes the operator-theoretic layer of the correlational program and establishes a rigorous foundation for subsequent numerical, phenomenological, and applied studies based on explicit correlational configurations.