Inverse reconstruction at higher-ramification loci

Develop an inverse reconstruction theorem for the scalar resolvent and quantum fields when ramification points collide or branch-value velocities vanish, using a reformulation of the admissible primitive class compatible with higher-ramification recursion.

Background

The main uniqueness proof requires simple ramification and nonzero motion of the branch values. When these conditions fail, the multiplication map used to determine the primitive becomes singular, although the scalar differential equation and its concomitant may remain well defined.

The higher-Airy examples demonstrate that the resolvent can still be computed on a maximally degenerate curve, but they do not provide an inverse theorem that reconstructs unknown quantum fields there. A higher-ramification version of the admissible primitive conditions is therefore unresolved.