How Many Are There, on a Continuous Spectrum? Three kinds of "separability" judgment, a threshold-free substitute, and where it stops working

Many arguments depend, at one step, on counting a continuous spectrum, or a set of mutually overlapping objects, into an integer: how many layers a cross-scale control system needs; how many mutually independent monitoring lines a field has; into how many components a weakest-link function should be cut. All three usually end at an uncalibrated threshold, and so all three conclusions stall at a provisional grade. This paper argues that the three look like one problem but are three, and that their feasibility differs sharply, in a way simulation reads off directly. First, mutually overlapping but individually identifiable objects can be handled by the effective number of the eigenvalues of their correlation matrix, with no threshold. Recovery is good: twelve objects in two, four, six and twelve independent blocks give order-two effective numbers of 2.54, 4.64, 6.36 and 10.12. Second, the number of separable bands on a continuous spectrum currently has no reliable threshold-free solution. Computing the effective number on the bins of the spectrum measures the width of the spectrum, not the number of bands: for a 256-bin spectrum, the reading lies between 50 and 175 whether the true peak count is two or five. Reducing the problem to the eigen construction improves matters, but the bias reverses with the true value (true 2 reads 6.52; true 6 reads 4.97) and the reading rises systematically with the noise floor, which shows it is mainly counting the degrees of freedom of the noise. Third, the partition problem under min-aggregation is not the same problem at all. The minimum decreases monotonically with the number of components and never plateaus, so no "effective number of components" exists; what is needed is not a threshold but an operational definition of failure independence, and the removal experiment is that definition. A three-tier robustness verdict follows: a conclusion holding across the whole profile is Grade A; one holding on a plateau of stated width is Grade B, and both ends must be reported; one holding only at a point is Grade C and is not accepted. All simulation code is included in Section 9, runs as printed, with fixed seeds and deterministic output.