Vanishing of the cellularization map for non-p-nilpotent component groups

Abstract

For every prime p, every field k of characteristic p, and every compact Lie group G whose component group is not p-nilpotent, the canonical k-cellularization map of C_*(Omega((BG)^p-completed);k) induces zero on every homotopy group. This gives an affirmative answer to Greenlees's Question 2.2 in Oberwolfach Report 9/2026. Completion is taken before looping; no adjoint-orientability, nilpotent-action or spectral-sequence-collapse hypothesis is imposed. The proof combines classical proxy-smallness and finite-ideal results for Hopf algebras through a class-by-class compact approximation. Four-page research note with TeX source, independent mathematical and priority audits, exact bounded algebra convention checks and reproduction instructions. AI tools were used extensively in development, verification and writing. Unrefereed; no external human peer review or formal proof-assistant certification is claimed. The conclusion is homotopy vanishing, not nullity in the derived category. Constituent classical results and known restricted cases are credited; the priority audit is bounded.