An Explicit Equational Schema for Joint Modal Validities

An explicit, computably enumerable infinite equational schema characterizes formulas valid both in the stated normal modal orthomodular class with Sasaki implication and in intuitionistic modal logic IK. The quantum class has a box preserving finite meets and the top element, with orthocomplement-dual diamond. The same construction treats Wijesekera's propositional constructive modal logic (WK in current terminology) and treats modern CK separately. The schema is formed syntactically from finite defining equation lists and arbitrary unary term contexts of the original modal signature, using the lattice majority term. An interval-clamping argument proves that the two generated full-signature congruences have diagonal intersection, yielding an embedding into a product of branch quotients. Common validity is equivalent to a finite equational derivation from the schema. The construction also applies to any two finitely equationally specified classes of bounded lattice expansions of a common finite finitary signature. The modal quantum branch is explicitly specified as a conventional reading of "basic", without claiming unique author-confirmed intent. The result is an infinite schema: finite axiomatizability, decidability and priority for the general universal-algebraic construction are not asserted.