Categoricity in L[U] for small‑cardinality recursive complete second‑order theories

Determine whether, assuming V = L[U] with κ the unique measurable cardinal and U its normal measure, every complete recursively axiomatized second‑order theory T that has a model of cardinality λ < κ where λ is second‑order characterizable is categorical.

Background

In L[U], the authors show that every finitely axiomatized complete second‑order theory is categorical (Theorem 1) and construct recursively axiomatized complete second‑order theories that are non‑categorical at very large cardinals λ > κ (Theorem 2). This raises the question of whether non‑categoricity persists at smaller cardinals below the unique measurable κ.

The problem targets the boundary between the categoricity guaranteed for finite axiomatizations in L[U] and the existence of non‑categorical recursive theories at large cardinals, asking specifically about models of cardinality λ < κ that are second‑order characterizable.