A countable three-generated orthomodular lattice with no σ-complete embedding

We construct a countably infinite orthomodular lattice generated by three specified elements that has no bounded ortholattice embedding into any set-sized σ-complete orthomodular lattice. In particular, it has no embedding into any set-sized complete orthomodular lattice. The example is a generated subalgebra of a cofinite reduced product of finite-dimensional real subspace lattices. Its obstruction requires only one target supremum: every homomorphism for which the image of a specified countable orbit has a supremum annihilates a specified nonzero element. The orbit consists of finite words in the Sasaki projections of the generators. Sasaki adjunction makes its target supremum compatible with the generator images, and compatibility extends to their generated subalgebra. We establish this obstruction in arbitrary orthomodular lattices and then realize its hypotheses by an explicit construction. The target argument requires no representation assumptions or preservation of infinitary operations of the source. AI disclosure The author had obtained the conclusion of this paper before the present use of AI. Codex was used to check the theory and to prepare and proofread the manuscript. Python was used for computational checks over a finite range. Verification scope The manuscript contains an informal mathematical proof. No proof assistant was used. The supplementary exact-arithmetic Python computations cover a finite range and do not prove universal statements. See Appendix B for the verification scope.