Four Percent at Full Strength: The Scale-Free Census

For twenty years, network science taught one sentence as if it were a law of nature: real networks are typically scale free, their degree distributions power-law distributed. The sentence was believed on the strength of small samples and straight lines eyeballed on log-log paper. In 2019 Broido and Clauset ran the census the claim had always needed: 928 network data sets from the Index of Complex Networks (927 in the preprint record), every degree distribution fitted with the Clauset-Shalizi-Newman machinery — maximum-likelihood power-law fit, plausibility test, likelihood-ratio comparison against non-scale-free alternatives including the log-normal — and sorted into five evidence tiers from Super-Weak to Strongest. The count: 36 data sets, 4%, reach Strongest; 49% show no evidence of scale-free structure at all; social networks place zero data sets in Strong or Strongest; a handful of technological (3 data sets, 1% at Strongest) and biological (30 data sets, 6% at Strongest) networks genuinely earn the top tiers. The Barabasi-Albert model's own theorem — preferential attachment yields a power law with exponent exactly 3, inside the model — stands untouched; the defect is the leap from the model's theorem to the world's networks, and the one word that carried it: typically. A notebook exhibit shows what the claim would even mean: for a pure power law with exponent in (2, 3) the mean is finite and the variance infinite, verified by truncated-moment integration at exponents 2.5 and 3. Grade: **THE UNIVERSAL CLAIM FAILS ITS OWN CENSUS — FOUR PERCENT AT FULL STRENGTH, AND THE HANDFUL NAMED AS A HANDFUL.** This paper is part of a breadth program; it touches no claim of the physics core.