Joint Modal Validities: Explicit Schemas and Partial Decidability Results

An explicit infinite equational schema is given for the formulas valid both in a specified basic modal orthomodular class and in intuitionistic modal logic IK. The same construction treats Wijesekera's propositional constructive modal logic, denoted WK in current terminology, and treats modern CK separately. The orthomodular class has Sasaki implication, a box preserving finite meets and the top element, and the orthocomplement-dual diamond. The schema is formed from finite defining equation lists by placing their pairs in arbitrary terms of the original modal signature and combining them with the lattice majority term. An interval-clamping argument proves that the two generated full-signature congruences have diagonal intersection. Consequently every model of the schema embeds into a product of one algebra from each branch. Common formula validity is equivalent to the existence of a finite equational proof from the schema. Its formation requires neither a branch-validity test nor an enumeration of pairs of branch proofs. The construction applies more generally to any two finitely equationally specified classes of bounded lattice expansions. Version 2 adds the absence of a finite equational basis for the original-signature join varieties and a complete calculus with fixed assumptions for common local consequence, including infinite premise sets. Finite preservation and a decision procedure are proved for the modern constructive and Wijesekera-style branches. Full modal quantum validity reduces to each common-validity problem. Finite prescription results are conditional on the stated residual-finiteness hypothesis, and explicit method-boundary countermodels are given. Decidability and the formula finite-model property of the unrestricted common logics remain open; general residual-finiteness and quotient-prescription obligations are not claimed resolved. The modal quantum branch is an explicitly selected conventional reading of basic, without claiming unique author-confirmed intent. Zenodo preprint.