Onset of a Fold Cascade in a Hopf Texture Driven by a Navier--Stokes Blow-Up Analog

A two-component Bose–Einstein condensate can host a pseudospin Hopf texture while its thermal cloud obeys viscous Navier–Stokes dynamics. Motivated by OpenAI’s reported finite-time singular solution, we drive such a texture with a surrogate of the collapsing core, using an effective Madelung–Schrödinger theory with a Faxén-like advective correction. The texture undergoes fold-mediated creation and annihilation of preimage pairs at conserved Hopf charge with the first one-to-three transition confirmed to obey the square-root opening of a Thom $A_2$