Non-identifiability of a high-dimensional critical-transition indicator: the absolute-correlation index I_C is blind to the sign and concentration of covariance structure

Early-warning signals (EWS) for critical transitions were formulated for low-dimensional systems and later extended to high-dimensional ensembles sampled as static snapshots. A widely used instance is the index I_C, the ratio of the mean absolute gene-gene correlation to the mean cell-cell correlation of a single-cell ensemble. We study the identifiability of this indicator: given that I_C peaks, what property of the underlying ensemble does the peak certify? We prove an exact algebraic identity showing the I_C numerator is invariant under sign reflection of any subset of variables, so coherent and incoherent reorganizations of equal magnitude are indistinguishable to it; and we show on an independent differentiation ensemble (GSE75748) that I_C assigns nearly equal values to a concentrated transition state and a diffuse terminal state, which the spectral participation ratio separates with non-overlapping bootstrap confidence intervals. We propose the participation ratio as a minimal spectral complement to I_C.