The Measurement Limit of Environmental Dominance: Why the Strong Thesis Can Be Neither Confirmed nor Refuted, and What Holds Instead

This is the third and concluding paper of a series. The first [1] postulates that environment dominates an entity's capabilities whenever the variance of the effective density of favourable possibilities exceeds the variance of capability, Var(ln ρ_eff) ≫ Var(ln k); the second [2] measures that inequality at four levels of aggregation. Here we ask whether the conclusion those papers draw from it can be sustained. We distinguish a weak reading (resource or positional dispersion exceeds capability dispersion, a necessary condition) from a strong reading (environment or position causally dominates capability). Using four independent probes we show that the weak reading holds but the strong reading is, on available data, structurally unmeasurable, not merely unproven. A quasi-experimental probe confirms the environment causally displaces outcomes from a held-fixed starting line, yet even there the capability term cannot be recovered, so no dominance ratio forms. The published income headline is partly an artifact: twelve sentinel atoms, 0.65 % of the population, move the within-country variance component by about a quarter, invisibly to the bootstrap. A generative bridge from environment to intergenerational mobility cannot be resolved at all (a pinned-slope test returns r² = 0.008 and is powered only against slopes above ≈ 2, far from the predicted one). Re-operationalising dominance as network position separates a flat visible layer (board interlocks, decisively not a power law: log-normal preferred at p < 10⁻⁹) from a concentrated, heavy-tailed hidden layer (offshore officer network, α ≈ 2.3, power law indistinguishable from log-normal but decisively non-exponential), confirming that the concentration mechanism lives in deliberately opaque structures. Yet the ultimate beneficial owner is recorded in only 2.55 % of relationships even inside the largest leaks. Beneficial-ownership recording is bimodal (near zero in public registries, near unity in leaked internal files), so opacity is a measured structural property, not an assumption, and a clean property of the recording regime rather than of the jurisdiction (the recording source explains 82.5 % of the variance, the jurisdiction 1 % once source is controlled). The public ledger omits the principal by construction. "The powerful hide" is thus vindicated as an empirical finding at the same moment it dissolves the prospect of a clean test of dominance, which is the central methodological point.