What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence?

In the theory of percolation and other statistical physics models such as self-avoiding walks, it is common to encounter so-called critical exponents, which are rational numbers that describe a fractional exponent related to the rate of decay of a particular function, often a probability, as a parameter tends to some value.

One example of this is the exponent 5/48, tied to the asymptotic relation $$ \pi(n) = n^{-5/48+o(1)} \qquad (n\to\infty), $$ where $\pi(n)$ denotes the probability that the connected component that contains the origin in critical percolation over a two-dimensional lattice has diameter at least $n$.

Another example is the exponent 5/36, which appears in the formula $$ \theta(p) = (p-p_c)^{5/36+o(1)} \qquad (p\searrow p_c). $$ Here, $p_c$ denotes the critical percolation probability, and $\theta(p)$ is the probability that the origin belongs to the infinite percolation cluster. See Wendelin Werner’s notes, pages 3-4 for these two examples. See this Wikipedia page for many others.

The interesting thing about the critical exponents is that although their values are in many cases known precisely, they have been computed using mostly nonrigorous methods. However, as far as existence goes, they are conjectured to exist (in the sense that the relevant asymptotic relations such as the ones written above hold) in fairly large generality for any “reasonable” lattice, and the actual existence has either not been proved at all (I believe that’s the case for most or all exponents related to self-avoiding random walks), or has been proved only for very specific lattices (using SLE techniques pioneered by Schramm, Lawler, Werner, Smirnov and others). The above two formulas were proved in the case of the triangular lattice, as Werner explains in his notes.