Settling the Cambridge Capital Controversy via the Adversarial Aggregation Channel

This paper settles the Cambridge Capital Controversy by embedding capital aggregation within the adversarial aggregation channel (AAC) framework. The scalar capital aggregate is formalized as a channel mapping high-dimensional heterogeneous capital configurations to a real number, with the endogeneity of prices to the profit rate acting as the adversary. A reswitching-as-aliasing theorem proves that every reswitching event is channel-theoretic aliasing. A capital Nyquist theorem establishes a sharp phase transition at three techniques: below this threshold scalar aggregation is generically adequate; above it, aliasing is a structural guarantee, not an anomaly. A generic necessity theorem proves that for generically chosen Sraffian primitives with three or more techniques, reswitching occurs with probability one. This version adds three major results targeting the empirical workhorse tools of macroeconomics. A Cobb–Douglas vacuity theorem proves that the apparent empirical success of the Cobb–Douglas production function is a tautological consequence of the national income accounting identity and carries zero information about the underlying technology. A Solow residual aliasing theorem proves that total factor productivity generically conflates true productivity growth with composition effects and relative-price revaluation. A growth-accounting impossibility theorem proves that the Solow decomposition is sub-Nyquist inference: its attribution of growth to TFP versus capital deepening is structurally non-unique for every growth episode when three or more techniques are present. Additional results include a Goodhart subsumption theorem formalizing Sraffa's circularity objection, a conservation identity proving that no institutional or econometric patch can resolve aliased configurations without genuinely new information, a scalar impossibility theorem for the majorization preorder, and a comonotonicity impossibility theorem proving no single-aggregate model can generate the empirically observed four-quadrant crisis taxonomy. The constructive exit replaces the scalar aggregate with a distributional capital state on the probability simplex.