Logarithmic-Potential Operator Frameworks and the Riemann Hypothesis: Structural Equivalences at the Critical Line

We introduce a conditional operator-theoretic framework based on logarithmic potentials derived from the Riemann ξ-function at the critical line. Two operator realizations are constructed:1. Friedrichs extension L_F (via quadratic forms)2. Natural realization L_N (via Weyl m-function) Under four framework hypotheses, we prove the equivalence:RH ⟺ (L_F = L_N) ⟺ (λ_n ≥ 0) ⟺ (Spectral-Nodal Correspondence) SCOPE: This paper establishes rigorous equivalences within the conditional framework. It does NOT prove the Riemann Hypothesis, nor does it prove the natural realization L_N exists unconditionally. MSC 2020: 11M26 (primary), 47E05, 47B25, 35P05 (secondary) Includes an elementary-level explanation guide for accessibility.