RCD Paper XIII: From Field to Hamiltonian — Formalizing the RCD C-Field as Quantum Protection Operator

Paper XII established mathematical isomorphisms between the RCD coherence field C and four quantum protection mechanisms. This paper advances from functional correspondence to operational formalization, deriving an explicit mapping C(x,t) → H_prot that translates the abstract functional role of the coherence field into concrete quantum operators. We show that the C-field naturally generates a protection Hamiltonian of the general form H_prot = f(C) · O_sys, where f(C) is a monotonic function of local coherence and its derivatives, and O_sys is a system-dependent operator. This unified form encompasses Zeno coupling, topological gap protection, dynamical decoupling, and frustration-based suppression. The mapping f(C) is not unique; different physical platforms may realize equivalent protection through distinct microscopic implementations. Recent experimental results, including Google Quantum AI's surface code performance, provide illustrative calibration examples linking RCD parameters to measurable protection strengths. The formalization enables, in principle, quantitative predictions for experimental tests of RCD in engineered quantum systems. This is Paper XIII in the RCD series, completing the Quantum Binding track initiated in Papers XI-XII. Keywords: coherence field, protection Hamiltonian, quantum operators, RCD theory, decoherence suppression, dynamic rigidity, operator formalism, Hamiltonian mapping, Radial Coherential Dynamics