The Dimension-3 Nyquist Theorem: Why Three Is the Universal Threshold for Aggregation Paradoxes

We prove that dimension three is the minimal dimension at which lossy aggregation can produce adversarial exploitation, unifying three classical phenomena under a single algebraic invariant. The Gradient Opposition Lemma establishes that no partition of the binary cube 0, 1ⁿ with n = 3 do. The Angular Channel Capacity Theorem identifies the mechanism: the shrinkage Jacobian Iₙ - 2xxT is a Householder reflector whose trace gives net exploitable capacity n - 2. We generalize to Lᵖ geometry, where the shrinkage divergence is (n - p) /||x||ᵖ, yielding an exploitation threshold of n >= p + 1. The Central Limit Theorem selects p = 2 as the universal attractor for systems governed by aggregation of independent components with finite variance, locking the threshold at n = 3. We confirm the predictive power of the framework by proving the L¹ Stein Paradox: under Laplace noise with L¹ loss, the coordinate-wise maximum likelihood estimator is inadmissible for n >= 2, exactly as the general threshold n >= p + 1 predicts at p = 1. The paper includes a self-contained proof of the Enrichment-Corruption Duality conservation identity from the adversarial aggregation channel framework, and connects the Simpson corollary to the causal inference literature, showing that the n = 3 threshold governs the minimal confounding dimension for collider bias and backdoor non-identification. We establish a nine-way equivalence among conditions that all transition at dimension three, spanning gradient opposition, the discrete Laplacian, continuous divergence positivity, pivotal integral finiteness, James-Stein domination, MLE inadmissibility, Brownian transience, Green's function finiteness, and Simpson reversal. The Banach-Tarski paradox, which shares the n = 3 threshold, is shown to be structurally distinct (a measure-theoretic pathology lacking Decision-Channel-Corruption structure) and is treated as a parallel rather than an equivalence.