Four Origins of a Characteristic Scale Wavelength Selection, Screened Decay, Instability Threshold and Truncation Constraint, and How to Tell Them Apart

Abstract Cross-domain accounts often read the spacing of animal coat patterns, the radius of an urban market hinterland, the fragmentation scale of self-gravitating gas and the bend of a size distribution as one quantity, the square root of a ratio of transport to decay. An analogy of that kind forbids no observation. This paper argues that a characteristic scale has at least four origins — wavelength selection, screened decay, a long-wave instability threshold and a truncation constraint — and that they can be told apart. The first three are read from the growth rate σ(k) of a linear field operator: whether its maximum is positive or negative, and whether it lies at a finite wavenumber or at k → 0. The fourth lives on the coordinate of a size distribution rather than on a spatial one. Three criteria follow — a spectral peak at finite wavenumber, an identifiable source, and the coordinate of the scale — combined by a sequential rule that admits no vote. The peak criterion reads only the location of the maximum. The screened and long-wave instability types both have monotone spectra and are separated by the source criterion or by the stability of the pattern, never by the spectrum; where only a spectrum is available the report is “not separated”. Term-by-term dimensions separate realisations, not columns: the Debye and Jeans lengths correspond term by term with the sign of the coupling reversed, while the reaction–diffusion screening length differs from both by a factor of T⁻¹ in each term. Four entry conditions for new fields are applied to six objects on the destructive-capacity side; only conventional force projection is admitted at the first tier, and its opposing forms are separated by ratio against difference and by the near field. In hop coordinates the pair-weighted mean of the hop-distance autocorrelation of a standardised field is exactly −1/(N − 1). The trough statistic therefore fails on screened fields: against registered point sources its detection rate is at most 0.021 at every structure fraction, and a threshold set on a short-correlation background gives a false-positive rate of 0.992. A graph-spectral excess over a monotone envelope holds its level between 0.033 and 0.065 across four background families. The paper separates “undecidable”, a statement about data, from “falsified”, a statement about the table, and gives three tests that can overturn the table itself.