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 replication notebook 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 at each fixed labor input, 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. A duality theorem identifies the rank condition as the observable-side dual of the Fisher–Gorman–Blackorby–Schworm technology-side separability program. The dynamic obstruction connects to Hankel singular values: a calibrated two-capital DICE computation shows no one-state model tracks investment-composition shocks within 42% relative Hankel error, and a standard decarbonization path is misforecast by 0. 79°C at century scale despite exact baseline fit. In a production-network economy the obstruction reduces to a Domar-share-weighted spread of per-unit returns across capital types. An almost-everywhere extension via Rademacher's theorem brings locally Lipschitz aggregates — kinked, piecewise-linear, and chain-weighted indices — within the scope of the impossibility. Accounting corollaries show that Cobb–Douglas fit is an algebraic identity once factor shares are fixed, that one-dimensional measurement corrections cannot repair a rank-two failure, and that the Solow residual is non-identifying whenever true productivity varies along an observational fiber. The empirical paper computes the obstruction index from U. S. Bureau of Economic Analysis and Bureau of Labor Statistics data under the precise specification the theorem requires. The hidden state is the 65-dimensional vector of real capital stocks across BEA Fixed Asset Table 2. 2 leaf asset types (1947–2024). The observable triple — real output per hour worked, the composition-weighted net-of-depreciation return constructed from BEA Table 2. 4 asset-specific depreciation rates, and the net capital share — is constructed to match the theoretical object exactly. At k=3 principal components, σ₂/σ₁ = 0. 664, with a Weyl-certified lower bound of 0. 0414 under 2% noise and σ₂ > 0 in 200/200 block-bootstrap resamples. A 15-year rolling-window analysis over 1988–2024 shows σ₂/σ₁ bounded away from zero in every window (minimum 0. 433, median 0. 601), ruling out the interpretation that the obstruction is confined to crisis episodes. The invariant second canonical correlation is ρ̂₂ = 0. 59 (block-bootstrap 5th percentile 0. 40) ; the Bartlett-corrected Anderson rank test rejects rank one at p = 0. 007. The Eckart–Young-certified structural floor is 1 − R²∗ = 0. 274; the BLS Törnqvist chain-weighted capital-services index — the official scalar measure used in U. S. productivity measurement — lies a further 62 percentage points below that floor. The rolling-window evidence, as an open-region statement with σ₂ bounded from zero on every window, extends the falsification to the full locally Lipschitz class by the Rademacher theorem. In 2019 economic units the optimal rank-one approximation misattributes the return on capital by a median of 9 basis points and a maximum of 55 basis points, with median capital-income misattribution of 45B. The replication notebook (CambridgeV9PipelineColabfixed. ipynb) computes every number in the empirical paper from a single sequential run on Google Colab. Stage 0 is a mandatory gate that self-validates against archived V7 outputs before any downstream result is trusted. Subsequent stages compute the Specs A–D singular-value tables, k-sensitivity analysis, invariant statistics (ρ̂₂, Bartlett-Anderson LR, HAC-robust Kleibergen–Paap with restricted-bootstrap p-value), block bootstrap with Weyl-certified bounds, 15-year rolling windows with figure export, chain-weighting R² decomposition (BLS Törnqvist vs. Eckart–Young optimal rank-one), dollar-valued misattribution tables on both the headline and full samples, and the production-network closed-form evaluator. Statistical machinery is simulation-validated at the paper's sample dimensions (T=37, m=k=3): canonical correlation is invariant to machine precision under GL×GL reparameterization; Anderson–Bartlett size 0. 053, power 0. 69; Kleibergen–Paap oversized at this T, hence the restricted-bootstrap p-value (size 0. 027, power 0. 60) is the reported inferential quantity. The notebook requires eight vintage-checked data files (BEA Fixed Asset Tables 2. 1, 2. 2, 2. 4; NIPA Tables 1. 1. 5, 1. 1. 6, 1. 12; BLS TFP major sectors; BLS-BEA KLEMS capital accounts) ; all are publicly available. Re-saving BEA workbooks through Excel corrupts the asset-hierarchy indentation that the leaf-extraction logic depends on.