Capital as a Profile Distribution: A Constructive Post-Cambridge Framework

This deposit contains two components of the constructive sequel to the rank-obstruction capital aggregation program: an extended theory-and-empirics paper and a companion Jupyter notebook implementing all computations on WIOD data. The paper develops the constructive replacement for scalar capital following the impossibility result proved in the companion theory paper. Five axioms for a post-Cambridge capital object are formulated: price-independence requiring that the capital state be definable without equilibrium prices as weights; label invariance requiring permutation invariance over technique categories; Wicksell-neutrality requiring that the capital measure not change when only prices move with technique assignments fixed; structural comparability requiring a rigorous label-free partial order; and reswitching compatibility requiring that no monotonic capital-deepening condition be imposed in the profit rate. The need for Wicksell-neutrality is motivated by proving that the value of any physical capital bundle is strictly increasing in the profit rate for any irreducible input-output matrix, so any price-based measure fails this axiom by construction. The replacement object is the profile distribution over techniques: a vector of industry gross-output shares lying in the probability simplex, built from counts independent of equilibrium prices. The profile distribution is proved to satisfy all five axioms. The geometry of the capital simplex is developed via majorization, the Hardy-Littlewood-Polya theorem characterizing majorization as existence of a doubly stochastic mixing matrix, the equivalence of majorization with both upper and standard Lorenz order, convex order via relative cell sizes and Karamata's inequality, and Shannon and Renyi entropies as Schur-concave Lorenz-consistent scalar summaries with equality characterizing permutation equivalence. Ordinal majorization entropy is defined and normalized. The Gini coefficient and Theil index are derived and related to Shannon entropy. An incomparability example demonstrates that the partial order is a correct reflection of the Cambridge lesson rather than a defect. A multi-commodity production section separates physical capacity from valuation: aggregate net physical output is a linear function of the profile and labor with no prices, while scalar value additionally requires a valuation vector, with a Wicksell separation proposition showing explicitly how value changes when only the valuation rule changes. A McFadden random-utility microfoundation derives multinomial-logit technique shares from first principles via an explicit Gumbel-integral computation, proves a KL-regularized variational dual representation showing the logit vector as the unique maximizer of expected payoff minus KL divergence from the prior, and derives the full Jacobian of logit shares with respect to payoffs as a scaled covariance matrix proved bijective on the tangent space. A softmax reswitching theorem shows that sign changes in payoff differences produce exactly as many crossings of the technique-share ratio as the payoff difference has zeros. A profile-based growth accounting decomposition separates output growth into labor growth, a reallocation term measuring compositional change, and a within-technique term, with the within-technique term further split into a real-productivity component and a Price Wicksell valuation component via a Sraffian decomposition. A Solow absorption theorem shows that the scalar Solow residual absorbs both components without separating them. Replicator-logit dynamics provide an endogenous law of motion on the simplex via an endogenous-reference update rule in which each technique's share is multiplied by its softmax weight relative to the current distribution. Under constant payoffs the dynamics admit no generic interior steady state, they converge to the payoff-maximizing face, the convergence is proved via a KL Lyapunov theorem showing KL distance from any point on the maximizing face is non-increasing with strict decrease unless already on that face, and an explicit closed-form trajectory is derived. A rate-shock transmission theorem derives the exact first-order linearization of the one-step update with respect to the profit rate, with a current-state covariance approximation theorem quantifying the error from replacing next-period shares by current-period shares. A closed profile-distribution model and empirical design establish calibration protocols, forecasting horse-race designs, and policy experiment designs. Executed on the World Input-Output Database 2016 Release covering 43 countries, 56 industries, and years 2000 through 2014, the panel regression of next-year output growth on the reallocation term, the scalar Solow residual, and labor growth with country fixed effects yields a reallocation coefficient of 1.50 significant at the 1 percent level with country-clustered standard errors, and an incremental R-squared contribution of approximately 1.3 percentage points. Column decomposition shows the predictive content is concentrated in the real-productivity reallocation component rather than the Price Wicksell component. The result survives winsorization, OECD-only restriction, and all 43 leave-one-country-out regressions each significant at 5 percent, with the sole nonrejection coming from the G7-only subsample where power is low but the sign remains positive. The companion Google Colab notebook implements all computational content in 39 cells. It reuses the NIOT and SEA parsers from the falsification notebook with caching optimizations that reduce file reads from 645 to 43. New cells implement the profile distribution and net product computation, the three-term growth decomposition with discrete Tornqvist weights, the Wicksell decomposition of the within-technique term, the replicator-logit one-step update, tau estimation by KL-loss minimization over 2000-2009, the rate-shock transmission linearization and current-state approximation, the KL Lyapunov numerical verification, full-panel decomposition over all 43 countries with CSV export, panel regression using statsmodels with country fixed effects and clustered standard errors, out-of-sample forecasting horse race, industry-level rate-shock heatmaps, and stacked-bar growth decomposition visualizations. The notebook runs on Google Colab using publicly available WIOD data with no proprietary dependencies.