This paper identifies and proves the correct necessary condition for exact scalar capital aggregation in heterogeneous-capital economies, replacing the incomplete framing of the Cambridge capital controversy that centers on reswitching. The central theorem states that if a scalar capital aggregate exists and supports a smooth aggregate production relation with competitive pricing, then for each fixed labor input the Jacobian of the map from the hidden heterogeneous-capital state to the observable triple of output, profit rate, and wage rate must have rank at most one. Whenever two independent directions of heterogeneity move observable outcomes differently, the Jacobian has rank at least two and no exact scalar capital aggregate exists on any neighborhood, regardless of whether any pair of techniques reswitches. The paper first proves that open families of no-reswitching economies exist on every compact feasible profit-rate interval, refuting any claim that reswitching is generic. The key Sraffian result then shows that if three techniques generate non-collinear audited value profiles at any profit rate, the map from technique distributions to audited values has rank two in composition space, ruling out any exact scalar sufficient statistic for the price system locally. An explicit three-technique input-output family is constructed satisfying both properties simultaneously: no reswitching on a wide profit-rate interval and a strictly positive rank-obstruction index. Genericity of the rank-two obstruction over bundle choice is also established once a single witness pair exists. The paper then derives accounting corollaries in exact rather than rhetorical form. A Cobb-Douglas relation with time-varying residual is shown to be an accounting identity once output, capital, and labor are fixed. The out-of-sample prediction error of a fixed-share Cobb-Douglas decomposes exactly into a share-drift term and a residual-drift term, so predictive success generated by accounting stability carries no structural content. A post-processing irrelevance theorem proves that any one-dimensional measurement correction — chain-weighting, rebasing, or hedonic relabeling that still outputs a single scalar — cannot repair a rank-two obstruction. Finally, a fiber non-identification theorem shows that the Solow residual fails to identify true productivity whenever productivity varies along an observational fiber of the aggregate data. The obstruction index, defined as the second singular value of the Jacobian, is shown to equal the minimum first-order error of any local scalar approximation at a given state, and the best rank-one derivative achieves a local pointwise approximation error bounded linearly by this index. The constructive implication is to report a distributional capital state together with a local obstruction index rather than searching for a better scalar aggregate.