This companion note elaborates the object at the center of the Emerging World Models (EWM) framework: the HLLSet lattice and the worlds it carries. The important word in EWM is emerging. The lattice is never built whole; it is grown as measurements arrive. An HLLSet is a compact bit-vector “sketch” of a token collection, and the HLLSet algebra (one universe: every sketch is a node of one Boolean lattice; the morphism is the contract : a deterministic, immutable, content-addressed map from tokens to nodes) is what makes the growth safe. Many users—many worlds—can write into the same lattice simultaneously, and the algebra makes it impossible for them to build anything wrong: the join is mono-tone and convergent, so independent contributions coexist rather than conflict. The hash function (its type and its seed ) is a parameter of the measurement, not part of the lattice, so different users hash into different worlds that are virtually isolated even when they share HLLSets: a shared sketch is the same node on the representation plane, but each world reads it through its own LUT. This built-in isolation is a free consequence of the algebra, not a bolt-on policy; we call it the emergence security of HLLSet Algebra. We formalize worlds as BSS-thresholded node sets with interpretations, prove the closure and isolation properties, describe the internal sub-pipeline by which two worlds communicate (material-ize in world 1, transcribe, ingest in world 2), and close by identifying the bridge with the Grothendieck-topology target of the main paper. The lattice’s size—2^32K nodes at the de-fault configuration—is treated in a single paragraph as the variety a compact presentation affords.