Thirty: The Sample Size Folklore Never Derived

The rule that a sample of thirty is large enough for the central limit theorem is taught and spent as a universal constant. The theorem states no rate; the rate theorem, Berry-Esseen, carries a factor beta_3 that differs by family, so a distribution-free sample size cannot exist. This paper executes the claim exactly for five families (no simulation): at n = 30 the Kolmogorov distance between the standardized sum and the normal curve is 0.0009 (uniform), 0.0722 (Bernoulli 0.5), 0.0243 (exponential), 0.1474 (Bernoulli 0.1), and 0.4487 (Bernoulli 0.01); the first n reaching distance 0.01 is 3, 1,592, 177, 7,094, and 70,738 respectively — a factor of about 23,600. For the fair coin the distance at n = 30 is exactly half the largest atom, C(30,15)/2^30 = 0.1445: the failure is discreteness, not shape. At n = 30 the Berry-Esseen bound itself certifies nothing — for normal data, whose true distance is zero, the best published constant (0.4748) still only guarantees 0.1383. Honest fence: for symmetric light tails thirty is generous (the uniform needs three); the defect is the universality. Grade: ONE NUMBER FOR EVERY DISTRIBUTION — FALSIFIED BY THE THEOREM'S OWN RATE, PRICED PER FAMILY.