The §2.9 conjecture of Lester, Haddadin and Gripaios (arXiv:2008.05206v6): three parity-odd variables are necessary

We prove the conjecture in §2.9 of Lester, Haddadin and Gripaios, Lorentz and permutation invariants of particles III (arXiv:2008.05206v6), that their three-variable sets S and S-hat are minimal for their respective four-particle event classes. Any two continuous, Lorentz- and pair-permutation-invariant, parity-odd real variables vanish together on a chiral collision event. The proof constructs a sphere consisting entirely of chiral events, on which parity acts antipodally up to the allowed symmetries, and applies the two-dimensional Borsuk-Ulam theorem. The same obstruction holds for strictly timelike, momentum-completable collisions with any number of additional interchangeable momenta, and for fully interchangeable collections of at least four momenta.