Beyond Reswitching: A Rank-Theoretic Obstruction to Scalar Capital Aggregation and Its Accounting Consequences

This paper isolates and proves the correct necessary and locally sufficient condition for exact scalar capital aggregation in heterogeneous-capital economies, and delivers four upgrades beyond the previous version. The central rank-one theorem states that if a scalar capital aggregate supports a smooth aggregate production relation with competitive pricing, then for each fixed labor input the Jacobian of the map from the hidden capital state to the observable triple of output, profit rate, and wage must have rank at most one. On regular neighborhoods, the converse also holds via the constant-rank theorem. Exact scalar aggregation is therefore locally characterized on regular sets by one-dimensionality of the observable manifold. The first upgrade extends the result to Banach-valued capital states via the Frechet derivative, covering asset distributions, age-efficiency profiles, and capital-service densities. A worked continuum-of-vintages example makes this concrete: for a vintage capital model on an L¹ probability simplex manifold with efficiency kernel phi (a) and rental-rate kernel psi (a), the observable map has Frechet rank two whenever the age-indexed curve in the efficiency-rental plane is not contained in a single affine line. For exponential kernels with distinct decay rates, this condition holds generically, ruling out any local scalar capital statistic. The knife-edge is explicit: exact scalar compression requires efficiency and rental rates to weight every vintage in exactly the same relative proportions. The second upgrade strengthens the Sraffian falsification beyond dominance classes. A proof-level theorem constructs a no-reswitching, rank-two family with pairwise entrywise non-comparable techniques and heterogeneous labor vectors, closing the escape route that the obstruction might owe its force to entrywise dominance or common labor. The third upgrade is a bridge theorem that closes the gap between the audit-bundle obstruction and the aggregate-production obstruction directly. By adjoining a real wage coordinate to an audited output bundle at a fixed profit rate, a production-relevant observable is constructed for which non-collinearity of three technique profiles immediately implies rank two of the full output-wage Jacobian. The explicit no-reswitching family is verified to satisfy this condition numerically, showing that the same economy violates the aggregate-production rank condition directly rather than only through an audit surrogate. The fourth upgrade adds a fiber-dimension count that makes the non-identification theorem quantitative. The observational fiber of the aggregate observable map is a submanifold of dimension equal to the hidden state dimension minus the observable rank, and the subset on which true productivity varies has local manifold dimension one less. Any finite collection of fit or forecast criteria computed from aggregate observables is constant on this fiber, while productivity can differ freely throughout it. A proposition makes the resulting conflict between scalar fit and structural identification fully explicit: two distinct productivity states can generate identical aggregate data, identical Solow residuals, and identical out-of-sample forecast errors, while the richer observable map simultaneously satisfies a positive obstruction index. All accounting corollaries from the earlier version are retained in exact form, and a noise-robust falsification theorem via Weyl's perturbation inequality rounds out the empirical toolkit.