When is a Bayes estimator linear? Exact credibility and Goel's conjecture

When is the Bayes estimator of the next observation an affine function of the sample mean at every sample size, the property actuaries call exact credibility? We answer this for all exchangeable models of integrable real observations: apart from degenerate cases, their structure functions are exactly the mixtures of the conjugate priors of regular natural exponential families, all sharing one prior mean and one prior sample size, and the mixing measure is unique. Exactness therefore forces light tails: every conditional law has exponential moments on both sides, so lognormal observations, and Pareto observations with a finite mean, admit no such prior unless the mean is deterministic. Exactness is also stable under mixing, which is why exponentiality needs a regularity assumption: a one-parameter family with fixed support, continuously differentiable log-densities and a prior of full support is a whole regular exponential family with its conjugate prior, which settles a conjecture of Goel. Examples show that neither hypothesis can be dropped and that the sample mean need not be sufficient. The classification extends to several statistics, and shows that a linear Bayes specification fixes the predictive mean but not the Bayes risk, which it bounds sharply for bounded claims.