Pure Spinor Factorization of Belinski-Zakharov Solitons and Regular Gravitational Instantons

Abstract

We show that the Belinski-Zakharov soliton construction factorizes into signed Pfaffian tau-functions, revealing a pure-spinor description of the metric and its regularity conditions. We derive analytic restrictions on the integral rod data, prove uniqueness for the asymptotically flat balance problem at fixed rod structure and periods, and give a conditional criterion for global existence. Combining these results with exhaustive enumeration of the admissible data and rigorous existence certificates, we construct a catalogue of regular asymptotically flat solutions representing 356 rod-word classes through twelve turning points, and 52 isolated regular asymptotically locally flat solutions through eight turning points. Analytic positivity and smooth-completion results establish regularity beyond the finite numerical solve. Based on this catalogue, we investigate two organizations of the solution space: generalized rank-two root systems and compositions of simpler endpoint geometries. The endpoint composition rules reproduce exactly all retained asymptotically flat words through twelve turning points.