Scalar Capital Aggregation Is Structurally Impossible: Theory, U.S. Asset-Class and 43-Country Evidence, and Replication Code

This deposit contains the theory paper, empirical companion paper, and two replication notebooks for a rank-theoretic falsification of exact scalar capital aggregation. The theory paper proves that exact scalar aggregation under competitive pricing requires the Jacobian of the map from the hidden heterogeneous-capital state to the observable triple (output, profit rate, wage) to have rank at most one, and establishes the converse on regular neighborhoods via the constant-rank theorem. The result extends to Banach-valued capital states and to nonlinear capital accumulation dynamics via the observability codistribution. Explicit no-reswitching Sraffian families — including families with pairwise non-comparable techniques and heterogeneous labor — generate rank-two observable maps, proving reswitching is sufficient but not necessary for aggregation failure. Accounting corollaries show that Cobb–Douglas fit is an algebraic identity once factor shares are fixed, and that one-dimensional measurement corrections (chain-weighting, hedonic adjustment, rebasing) cannot repair a rank-two failure. A noise-robust falsification theorem provides a certified lower bound on the misspecification of every admissible scalar correction. The empirical paper computes the obstruction index from two independent data sources. The primary test uses U. S. Bureau of Economic Analysis Fixed Asset Tables (65 detailed asset types, 1947–2024) and Bureau of Labor Statistics capital-services data to measure σ₂ directly in asset-composition space. At k=5 principal components, σ₂/σ₁ = 0. 46, with a Weyl-certified lower bound positive under 2% noise. The scalar approximation error has a median of 31 basis points in the return on capital, corresponding to 155B in misattributed capital income at 2019 levels. The BLS chain-weighted Törnqvist capital-services index — the official scalar measure — explains only 15. 4% of the joint variation in (Y, r, w), and the residual retains σ₂ > 0: the rank-two obstruction survives the best available one-dimensional correction. A cross-sectional test using BEA–BLS KLEMS Integrated Industry-Level Production Accounts (63 NAICS industries, 1997–2023) confirms σ₂ > 0 in 26 of 27 years. A secondary test using the World Input-Output Database (WIOD 2016 Release, 43 countries, 56 industries, 2000–2014) finds σ₂ > 0 in 645/645 country-year-rate tests, measuring within-technology capital-intensity heterogeneity across industries. High Cobb–Douglas fit (R² > 0. 90 in 74. 9% of country-years) coexists with large obstruction ratios (σ₂/σ₁ > 0. 10 in 98. 8% of those cases), confirming that fit is not identification. Robustness exercises with heterogeneous industry profit-rate proxies preserve the obstruction in 43/43 countries. A dynamic observability test on the G7 panel yields σ₂ > 0 in 7/7 economies. Implications for climate-economy integrated assessment models are discussed. Two replication notebooks are included. The primary notebook (BEACapitalFalsification. ipynb) implements the asset-composition rank test, dollar-valued misattribution calculation, and chain-weighting test using BEA Fixed Asset Tables and BLS TFP data. The secondary notebook (WIODWithinTechnologyFalsification. ipynb) implements the full WIOD pipeline: parsing national input-output tables, Leontief resolvent construction, SVD extraction, 645-combination stress tests, bootstrap under multiplicative noise, heterogeneous-rate robustness, dynamic observability, and fit-vs-obstruction diagnostics. Both notebooks run in Google Colab with no proprietary dependencies. The BEA notebook requires publicly available BEA and BLS data files; the WIOD notebook requires freely available WIOD 2016 Release data.