This paper proves that Simpson's paradox — the reversal of a statistical association upon aggregation across subgroups — is aliasing in the precise sense of the adversarial aggregation channel (AAC) framework. The aggregation map combining subgroups into a marginal distribution is formalized as a scalar channel compressing a high-dimensional state (the joint distribution of treatment, outcome, and group membership) into a one-dimensional signal (the marginal association). The parameter space has dimension 4n−1 for n subgroups, while the channel output is a scalar, making the channel always sub-Nyquist and the fiber dimension always at least 4n−2. A Simpson–Nyquist theorem establishes a sharp phase transition at three subgroups. For n = 2, reversal is possible but requires specific parameter configurations. For n ≥ 3, reversal is generic: the reversal set is an open subset of the parameter space with positive Lebesgue measure, and the confounding space acquires a second independent dimension that the scalar channel cannot represent without aliasing. The phase transition mirrors the n = 3 threshold in the Capital Nyquist Theorem, Arrow's impossibility theorem, and the scalar impossibility for majorization — in each case, the impossibility is avoidable with two elements and structurally unavoidable with three. A DIP verification theorem establishes that the "Nyquist" label is not a metaphor but a theorem-level instantiation of the Diagonal Intersection Principle from the AAC framework. The Simpson aggregation channel is constructed as a formal AAC, its reachable sets are computed as open intervals in the output space, and the Fiber Dimension Excess condition is verified: the adversarial dimension 2 (n−1) (baselines and treatment rates) exceeds the channel's output dimension 1, with a dimension excess of 2n−3 ≥ 3 for n ≥ 3. The Simpson channel adds a fourth row to the AAC's Fiber Dimension Table alongside Arrow's theorem, the Gibbard–Satterthwaite theorem, and the social-choice Nyquist boundary, completing the structural unification. A quantitative prevalence theorem proves that for n ≥ 3 subgroups under any absolutely continuous distribution on parameters, Simpson reversal occurs with strictly positive probability, and the reversal set contains an open neighborhood robust to parameter perturbations. Explicit lower bounds on the probability of reversal are provided as a function of the baseline heterogeneity scale M, the effect magnitude β, and the number of subgroups n, with the reversal probability approaching 1/2 in the regime M/β → ∞. A Simpson conservation identity, proved via the Enrichment–Corruption Duality of the AAC framework using a rigorous product-measure total-variation argument, shows that no marginal enrichment — additional covariates, reweighting schemes, or post-stratification adjustments — resolves Simpson aliasing without expanding the channel to include the full confounding structure. The proof proceeds in three steps: an upper bound from TV subadditivity with explicit Fubini computation, a lower bound from the data processing inequality for projections, and combining to obtain the exact decomposition Δ = AV + DʳobV (discriminability gained = meta-adversarial capacity + residual security). Under self-referential closure, conservation is tight: AV = Δ. A single additional covariate raises channel dimension to two but leaves aliasing intact whenever n ≥ 4. Full resolution requires observing the complete group-level structure, expanding channel dimension to n−1 to match the confounding dimension. A structural identity theorem proves formally that Simpson's paradox and capital reversal in the Cambridge Capital Controversy are the same aliasing mechanism in different coordinate systems. Both decompose as a within-stratum effect of definite sign plus a between-stratum confounding term that can dominate and reverse it. Both become generic at n = 3 strata. Both are governed by the same conservation identity. The mathematical structure is the wagon-wheel effect: sub-Nyquist projection of a high-dimensional signal producing an apparent sign reversal. An ecological inference isomorphism proves that King's ecological inference problem — estimating individual-level behavior from aggregate data — is a third instance of the same aliasing mechanism. Precincts map to subgroups, demographic shares to treatment rates, voting gaps to within-group effects, and baseline voting rates to stratum levels. King's tomography lines are proved to be the n = 2 fibers of the Simpson channel, and the generic impossibility of ecological inference for n ≥ 3 heterogeneous geographic units is proved as the sub-Nyquist phase transition. The conservation identity recovers King's bounds as the fiber computation and proves that no enrichment with fewer than n−1 covariates can eliminate aliasing. All three isomorphisms — Simpson, capital reversal, and ecological inference — share a common algebraic decomposition: a centered bilinear form ⟨λ, ξ⟩ with ⟨ω, ξ⟩ = 0, where reversal occurs when |⟨λ, ξ⟩| > |⟨ω, α⟩|. A Simpson impossibility theorem completes the static characterization: no continuous scalar functional of the marginal treatment-outcome distribution can consistently recover the sign of within-group effects when Simpson reversal is possible, and the only escape is stratification — expanding the channel dimensionality to match the intrinsic confounding structure. A continuous-confounder extension embeds the discrete results in a spectral aliasing theorem for arbitrary confounding distributions, with the confounding bias as an inner product of baseline and confounding spectra in L² (fG). A dynamic Simpson fragility theory embeds the heterogeneity-to-effect ratio in a stochastic process. The Simpson gap process Z (t) = β̄ (t) − |Δ (t) | is modeled as an Itô diffusion whose sign governs the instantaneous aliasing/recovery regime. A fragility theorem provides: regime-transition times with explicit Laplace transforms, an institutional half-life formula, survival probabilities satisfying a Kolmogorov backward equation with closed-form solution via the reflection principle, and a capacity volatility amplification theorem proving that datasets "safe on average" (EZ > 0) can spend a substantial fraction of time in the reversal regime when the gap volatility is high — converging to 1/2 as T → ∞ under zero drift regardless of the initial gap. A computable Simpson fragility index transforms the static M/β diagnostic into a dynamic institutional fragility measure. The framework is applied to five domains: the 1973 Berkeley admissions dataset (Simpson ratio ρ ≈ 4. 4 correctly predicts the observed reversal), COVID-19 vaccine efficacy estimation from Israeli data (extreme M/β ratio explains apparent negative aggregate efficacy), algorithmic fairness auditing (the impossibility theorem proves aggregate outcome audits cannot detect department-level discrimination and quantifies the minimum audit depth via the conservation identity), meta-analysis of pooled clinical trials (the M/β diagnostic identifies which pooled estimates are vulnerable to sign reversal), and ecological inference in political science (a racial bloc voting example with 8 precincts demonstrates reversal with ρ ≈ 2. 3, and King's bounds are recovered as the fiber computation of the conservation identity).
Kevin Fathi (Tue,) studied this question.