Which Base? Modal Closure and Symmetry in the Finite S5 Supervenience Lattice

Once supervenience is represented as invariance relative to a Boolean subalgebra, the question "which base?" becomes a question about the subalgebra lattice of the modal proposition algebra. This paper studies that lattice for finite propositional S5. A base is modally closed when it is stable under necessity. We give an exact partition criterion for modal closure and compute the automorphism group of the finite S5 profile algebra. Because the profile frame is a disjoint union of complete clusters, its bare symmetries form a product of wreath products: clusters of the same size may be exchanged, and their points may be permuted independently. Applying the closure criterion, we show that the cluster base is fixed pointwise by necessity, whereas, for a nonempty vocabulary, the categorical base is not closed and generates the entire proposition algebra once necessity is admitted. We exhibit a sufficient enrichment that restores categorical coordination and compute the resulting contraction of symmetry exactly. The bare frame already recovers an anonymous set of categorical types through its singleton clusters, but not a unique coordination assigning occurrences in larger clusters to those types. Adding the relation of sharing one categorical valuation reduces the automorphism group precisely to permutations acting uniformly across profiles. Restoring the internal Boolean-lattice order reduces the symmetries further to permutations of the propositional dimensions, while naming those dimensions eliminates the symmetries altogether. We also classify all characteristic bases—equivalently, in each fixed finite algebra, all parameter-free first-order definable bases—and prove that each is modally closed. In the one-atom case, exactly seven of the fifteen Boolean bases are modally closed, while not every modally closed base is characteristic. The results separate determination by a base, descent of necessity to its quotient, canonical selection of the base, and generation once necessity is supplied.