Nearest–Neighbour Cohomology meets Daisy Self–Similarity: a Unified Operator–Homotopy Framework Creators

This revised paper integrates and corrects the combined content of two earlier manuscripts: (1) Pipeline A (Wheel / nearest–neighbour coherence). A “hexagon wheel” is the minimal local data of endomorphisms U1, . . . , U6 of an object X together with near- est–neighbour commutator 2–cells αi,i+1 satisfying braid-type relations. The goal is to make precise what this data does determine, how it rectifies via W–constructions, and how it acts on concrete operator-algebraic models. (2) Pipeline B (Daisy / self-similarity / AF–UHF / KK). A q–daisy is a rooted q–regular tree; its self-similarity yields canonical AF structures and UHF algebras, plus natural renormalisation operators and Cantor-spectrum elements in the diagonal Cartan. (3) Gluing. We glue the local wheel into the global daisy by realising wheel generators as “local” kernel operators supported on petals, then replicating them across depth by self- similarity. The Hochschild/factorisation layer is glued in by treating local kernel algebras as a (Weiss) cosheaf (in the sense of [25]), and reading the resulting E2-structures via braces.