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

Abstract A length obtained as the square root of a ratio of transport to decay recurs across unrelated fields: animal coat patterns, urban market hinterlands, galactic clustering, and the bend in size distributions. The common practice is to treat these as applications of one equation, and analogies of that kind cannot be refuted by any observation. This paper argues that the characteristic scale has at least four origins: Turing-type wavelength selection, screened decay, a gravitational instability threshold, and a truncation scale set by finiteness constraints. The first three come from field equations in space; the fourth does not, and lives on the coordinate of a size distribution. The four are observationally separable. The paper gives a nine-row discrimination table and three operational criteria, which ask whether the spatial power spectrum peaks at finite wavenumber, whether an identifiable source exists, and whether the scale lives on a spatial coordinate or on a size-distribution coordinate. It traces the four lineages and on that basis corrects a common attribution: Turing's 1952 paper does not belong to the lineage of the screened-Poisson Green function. It shows that the Jeans length corresponds to the screening length at the level of the ratio and not term by term, and on that basis gives the instability type a column of its own. It gives four entry conditions for admitting a new field to the table and runs them item by item over six objects on the destructive-capacity side, of which only conventional force projection is admitted at the first tier. It gives a complete operational definition of the second tier in hop coordinates and calibrates it on a small-world graph, with fields composed from independent Gaussian coefficients. With a short-correlation background the detection rate is 0.747 and 0.995 at structure fractions of 0.3 and 0.5. With a background whose correlation is comparable to the scale sought, it is only 0.368 and 0.673 at a structure fraction of 0.5. A threshold calibrated on a short-correlation background and transferred to longer-correlation backgrounds gives actual false-positive rates of 0.589 and 0.518 against a nominal 0.05, so the hop-distance autocorrelation trough can be used only with the background family reported and the threshold set under a null of that family. The paper distinguishes “undecidable” from “falsified”: the former means that the present observations carry too little information to separate the four classes, its remedy is more measurement, and it is no evidence about this paper in either direction; the latter means that an observation directly overturns a discrimination rule, and its forms are given in Section 7.