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.
Kevin Fathi (Thu,) studied this question.