The central limit theorem says: average enough independent random variables and the distribution converges to a Gaussian. This is the foundation of statistics. But many physical quantities — energies, waiting times, distances in random media — are strictly positive. The Gaussian assigns nonzero probability to negative values. Something is wrong.
Castro and Cuesta (arXiv: 2603.23567) show what replaces it. When you require positivity, Padé approximants of the scaled cumulant generating function (rather than the polynomial Taylor expansion that yields the Gaussian) naturally produce gamma distributions. The gamma emerges from large deviation theory with the positivity constraint enforced — no additional assumptions, no specific mechanism, just the constraint that the variable can't go below zero.
The through-claim: the gamma distribution is the constrained Gaussian. Just as the central limit theorem produces Gaussians as the universal attractor for unrestricted sums, this framework produces gammas as the universal attractor for positive sums. The gamma isn't merely a convenient fitting function that happens to work for many positive variables. It's the unique distribution that emerges from the same mathematical machinery as the Gaussian, modified only by the constraint of positivity.
This explains why gamma distributions appear everywhere in physics — turbulence intensities, relaxation times, polymer lengths, neural firing rates — without requiring any shared mechanism among these systems. They share a constraint (positivity) and a limit (many contributions). That's enough. The universality isn't in the physics; it's in the mathematics of constrained summation.
Castro & Cuesta, 2603.23567. Statistics / large deviations / universality / gamma distribution.