Settling the Cambridge Capital Controversy via the Adversarial Aggregation Channel

The Cambridge Capital Controversy established that scalar value aggregation of heteroge-neous capital depends on the profit rate, producing reswitching and capital reversal, but left theimpossibility in an informal state that permitted continued use of aggregate production functions.This paper settles the controversy by embedding capital aggregation within the adversarial aggre-gation channel (AAC) framework. The scalar aggregate is an aggregation channel whose input isthe high-dimensional heterogeneous capital configuration and whose output is a real number. Areswitching-as-aliasing theorem proves that reswitching is channel-theoretic aliasing: distinctcapital configurations mapped to identical scalar values. A capital Nyquist theorem establishes asharp phase transition at three techniques: below this threshold, the neoclassical parables hold asadequate approximations; above it, reswitching and capital reversal are guaranteed consequencesof sub-Nyquist aggregation, not anomalous counterexamples. A Goodhart subsumption theoremformalizes Sraffa’s circularity objection as an instance of the AAC conservation identity. Theseresults place the Cambridge impossibility alongside Arrow, Gibbard–Satterthwaite, Myerson–Satterthwaite, and the quantum no-cloning theorem as instances of a single conservation lawgoverning adversarial aggregation. The order-theoretic complement proves that no scalar indexis an order-isomorphism for the majorization preorder on capital states when the technique spacehas dimension at least three, and a comonotonicity impossibility theorem—derived as an AACcorollary—proves that no single-aggregate model can generate the empirically observed crisistaxonomy. The constructive exit is identified: replace the scalar by a distributional capital stateon the probability simplex.