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, proved via the Thom transversality theorem applied to the 1-jet extension of the Sraffian cost-difference family, establishes that the single-crossing stratum has codimension at least one in the parameter space of Sraffian primitives, so reswitching occurs with probability one for three or more techniques. This version adds results targeting the empirical workhorse tools of macroeconomics and provides direct empirical validation. 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. An out-of-sample prediction error decomposition theorem proves that predictive success is equally tautological: forecast errors are bounded by factor-share stability and residual smoothness alone, with zero dependence on any technological parameter, closing the "predicts well out of sample" defense. 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. An empirical validation section confronts these predictions with U.S. national accounts data from the Bureau of Labor Statistics Major Sector Total Factor Productivity program (1987–2024, private nonfarm business sector). The BLS aggregates 90 heterogeneous asset types into a single scalar capital services index, placing the U.S. economy 30 times above the Nyquist threshold. The Cobb–Douglas regression achieves R² = 0.999987 with estimated exponent matching the mean capital share to the fourth decimal place, confirming the accounting-identity tautology. The Solow decomposition of the 1995–2005 productivity acceleration (TFP = +1.43%/yr) is shown to be structurally non-unique: a pure-reallocation explanation with zero true TFP produces identical aggregate observables. The aliased fraction of the capital composition simplex is positive in every year. 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 friction irrelevance theorem proving that no cost-based econometric correction can shift the aliasing boundary, a Landauer principle with minimum entropy production bounds, a Second Law of Institutional Entropy proving that successive econometric processing strictly increases structural uncertainty, 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.