In principle, simulated annealing is able to reach arbitrary accuracy at the expense of computer time. We should, however, remark on a few points. Unlike other minimisation problems, we are not really interested in the solutions that put E=0 exactly. Most likely, these are the data set itself and a few simple transformations of it that preserve the cost function (e.g. a time reversed copy). On the other hand, combinatorics makes it very unlikely that we ever reach one of these few of the N! permutations, unless N is really small or the constraints grossly over-specify the problem. This can be the case, for example, if we include all possible lags of the autocorrelation function, which gives as many (nonlinear) equations as unknowns, I=N. These may close for small N in the space of permutations. In such extreme situations, it is possible to include extra cost terms penalising closeness to one of the trivial transformations of the data. Let us note that if the surrogates are ``too similar'' to the data, this does not in itself affect the validity of the test. Only the discrimination power may be severely reduced.
Now, if we don't want to reach E=0, how can we be sure that there are enough
independent realisations with 
? The theoretical answer depends on
the form of the constraints in a complicated way and cannot be given in
general. We can, however, offer a heuristic argument that the number of
configurations with E smaller than some 
grows fast for large N.
Suppose that for large N the probability distribution of E converges to an
asymptotic form p(E). Assume further that 
is nonzero but maybe
very small. This is evidently true for autocorrelations, for example. While
thus the probability to find 
in a random draw from the
distribution of the data may be extremely small, say 
at 10 sigmas from the mean energy, the total number of
permutations, figuring as the number of draws, grows as 
, that is, much faster than exponentially. Thus, we expect
the number of permutations with to be 
. For example, 
.
In any case, we can always monitor the convergence of the cost function to avoid spurious results due to residual inaccuracy in the surrogates. As we will discuss below, it can also be a good idea to test the surrogates with a linear test statistic before performing the actual nonlinearity test.