Algebraizable logic and contextual translation for positive-cone semantics

Construct an algebraizable logic whose equivalent algebraic semantics is FA, together with finite contextual translations between the distributive Frobenius–Galois logic FL^{FGC,d}_{ew} and that logic that preserve and reflect derivability in both directions and whose induced algebraic correspondence agrees, up to natural isomorphism, with the Kalman and positive-cone functors K and C of Theorem 5.15.

Background

The paper proves that the distributive Frobenius–Galois logic FL^{FGC,d}_{ew} has equivalent algebraic semantics FARDL and that its logical consequence relation can be characterized using positive cones of algebras in FA. It also proves that FARDL and FA are categorically equivalent through the Kalman and positive-cone functors.

What remains unresolved is a direct logical counterpart on the FA side: an algebraizable logic with FA as its equivalent algebraic semantics, together with finite contextual translations connecting it to FL^{FGC,d}_{ew}. Such translations would strengthen the categorical correspondence by showing that derivability, rather than only semantic consequence and related algebraic properties, is transported and reflected in both directions.