Suppose you have two lists. One records predictors, and the other records outcomes. The links between individual observations have disappeared. Even if you know both distributions perfectly, can you recover the coefficients of a linear relationship? This Evidence Press candidate studies that question for independent Gamma predictors with known shapes, centered and scaled to unit variance. A three-predictor example shows that using every predictor does not guarantee a unique answer, even after allowing signs and permutations. Two different coefficient vectors give exactly the same response distribution. The paper then describes every possible answer in this Gamma model. The distribution reveals the total Gamma shape assigned to each effective scale. What remains is a finite allocation problem: which predictor labels can supply each of those totals? A precise subset-sum condition makes that allocation unique up to permutations of predictors with equal shapes. The paper also asks how much distributional information is enough. For a dictionary with d predictors, cumulants through order two d plus one suffice. A matching construction shows that this highest consecutive order cannot be lowered uniformly when signed coefficients and arbitrary dictionaries are allowed. These are exact population results. They do not establish a reliable estimator from noisy, finite data, and numerical instability can still matter. The package includes written proofs and finite exact-arithmetic checks. Historical priority remains uncertain. Released on twenty September twenty twenty-six, this is an unrefereed Evidence Press candidate. The full paper and evidence are linked on the release page. This is an AI-generated voice summary, not additional mathematical evidence.