Distinct-root counterexamples for every particle number

Construct, for every integer N>3, a counterexample to Atiyah’s Minkowski-space linear-independence conjecture in which all ordered celestial roots are pairwise distinct.

Background

The paper constructs counterexamples for every particle number N≥3. The three-worldline counterexample has six pairwise distinct ordered celestial roots, but the null-translation stabilization used for N>3 introduces a repeated common factor into the first three binary forms and therefore produces coincident celestial directions. Although such coincidences are permitted by Atiyah’s original formulation, configurations with all ordered roots pairwise distinct would satisfy a stronger genericity requirement. The authors explicitly state that the existence of such counterexamples for every N has not been resolved.