Synthetic tabular data are valued for preserving inter-column dependency, yet a routine fidelity score says neither where it is lost nor why. Equipping a classifier two-sample test (C2ST) with a gradient-boosted discriminator, we decompose it by controlled permutation into marginal, dependency and numerical--categorical cross components, read against a fully-factorized reference that destroys all dependency while keeping every marginal, and a real-data oracle. The linear detection score in common use rates that reference as nearly real, a known weakness we replicate on four benchmarks. On a flow-matching (TabbyFlow) and a diffusion (TabDiff) generator it exposes a deficit of the same order in both, whose source we narrow by elimination: not a structural limit of the mean-field objective, not mainly sampling discretization, and not closed by a $16\times$ capacity increase where that test has power, while the same measurement responds sharply when capacity is cut eightfold, so the plateau is measured rather than a blind spot. Destroying inter-column dependence while keeping every marginal and every feature--label relation costs between $0$ and $0.22$ minority-class $F_1$ by dataset, and the generators' own shortfalls are not ordered by their deficits. An explicit cross-coupling module and a post-hoc copula leave the deficit in place. Neither generator's objective places a term on the joint at all; that this absence is what sustains the deficit is an explanation we reach by elimination rather than test. A controlled validation calibrates the diagnostic and identifies when it reports a false positive, which none of these generators triggers.

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