This paper isolates and proves the correct necessary and locally sufficient condition for exact scalar capital aggregation in heterogeneous-capital economies, substantially upgrading the earlier version in four directions. 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. The paper now also proves the converse on regular neighborhoods: if the observable map has constant rank one, the constant-rank theorem yields a local scalar factorization. Exact scalar aggregation is therefore locally characterized on regular sets by one-dimensionality of the observable manifold, not merely obstructed by its failure. The first major upgrade extends the result from finite-dimensional hidden states to Banach-valued capital descriptions via the Frechet derivative. If the hidden state is an asset distribution, an age-efficiency profile, or a capital-service density, the same logic applies: exact scalar aggregation implies one-dimensional image of the Frechet derivative, and constant Frechet rank one again yields a local scalar factorization. This transforms the result from a finite-technique curiosity into a general statement about when scalar compression is locally coherent for any capital-state space. The second upgrade strengthens the Sraffian falsification beyond dominance classes. The earlier explicit no-reswitching, rank-two family used entrywise-ordered techniques with a common labor vector. A new proof-level theorem constructs a no-reswitching, rank-two family with pairwise entrywise non-comparable techniques and strictly heterogeneous labor vectors, closing the escape route that the obstruction might owe its force to entrywise dominance or labor homogeneity. The non-collinearity of audited profiles is verified analytically at epsilon equal to zero and extended by continuity to a positive parameter interval, with explicit numerical witnesses provided. The third upgrade connects the aggregation problem to static nonlinear observability theory. The heterogeneous capital composition is the hidden state, audited values are outputs, and scalar capital is a one-state model reduction. The obstruction index, defined as the second singular value of the observable Jacobian, is simultaneously the minimum first-order scalar misspecification error and the local detectability margin of hidden heterogeneity. Audit-bundle design is then posed as maximizing this second singular value over admissible bundle choices, turning the aggregation debate into an explicit measurement-design problem. The fourth upgrade provides a noise-robust falsification theorem using Weyl's perturbation inequality: if the estimated Jacobian is within a perturbation distance of the true Jacobian, and a lower confidence bound on the second singular value stays positive across bundle choices and high-shift regimes, then every admissible one-dimensional measurement correction, including chain-weighting, rebasing, or hedonic relabeling that still outputs a single scalar, inherits a strictly positive first-order misspecification floor. The paper also proves that scalar fit and structural identification can move in opposite directions, making explicit the sense in which a smoother Cobb-Douglas forecast may identify less technology. All accounting corollaries from the earlier version are retained in exact form.