<A HREF="https://en.wikipedia.org/wiki/Weil_conjectures">The Weil conjectures</A> could qualify as a set of heuristics developed into a rigorous proof by Deligne and others:

> What was really eye-catching, from the point of view of other
> mathematical areas, was the proposed connection with algebraic
> topology. Given that finite fields are discrete in nature, and
> topology speaks only about the continuous, the detailed formulation of
> Weil (a heuristic based on working out some examples) was striking and novel. It
> suggested that geometry over finite fields should fit into well-known
> patterns relating to Betti numbers, the Lefschetz fixed-point theorem
> and so on. The analogy with topology suggested that a new homological
> theory be set up applying within algebraic geometry. This took two
> decades (it was a central aim of the work and school of Alexander
> Grothendieck) building up on initial suggestions from Serre.