Isospectral quantum billiards as generators of the distributive logic algebras (continuation)

A quantum billiard is a compact domain with a free particle localized within the billiard by means of infinitely large outer potential (1D billiard = "a potential pit"). The billiard's spectrum is the set of all energy levels of the particle which are computed from Schroedinger's equation and are invariant under any isometries. But in 1990s-2000s there were found a whole gallery of non-isometric billiard pairs with the same spectra. This challenging effect is mainly induced by so-called cellular billiards as we recently proved [TMF, 224:2 (2025)] using a new method of 2-unitary operator. Such operator intertwines Sobolev's spaces over the cells by means of reflection matrices (= symmetric orthogonal matrices). Two such isospectral m-cellular billiards are constructed by (m-1) mirror reflections of the same subdomain (= the cell) and are connected with a multi-valued cellular isometry. This mapping gives ordinary pairwise cell-to-cell isometries but globally it is an everywhere discontinuous non-homeomorphism inducing an extremely disconnected topology. Acc. to Lindenbaum-Tarski-Stone approach such mapping generates visual representations of logical theories by means of algebras of open-closed subsets. In particular we derive generalized double negation formula which corresponds to the logical foundations of quantum theory. Presentation of this talk is accompanied with numerous visual pictures.