The Principle of Spatial Gradient Excess Screening Length, the Jeans Instance, and a Cross-Scale Applicability Map

Abstract A few nodes carrying most of the flux while vast regions remain quiescent is a spatial pattern that recurs from cities and wealth to stars and galaxies. We propose the principle of spatial gradient excess, which unifies such concentration under a single screened-field picture and strictly separates two sources of inequality. Endogenous inequality arises within a system without any long-range transport, from random multiplicative amplification. Exogenous gradient excess arises in physical space from sustained input together with finite propagation speed, and is governed by the screening length ℓ = √(D/λ). The principal results are these. Under finite screening the point-source profile is not a pure exponential: it has an inner algebraic region with exponent d − 2, logarithmic at d = 2, followed by an outer exponential cutoff. The Jeans length of a self-gravitating isothermal gas is the instance of the screening-length formula in a gravitational field, differing from electrostatic Debye screening only by a sign flip of the λ term, which is the field-equation expression of negative heat capacity; the two share a ratio and play opposite roles. Coupling the cosmological constant into the same field yields a composite screening length whose sign flips over cosmic time, giving a sequence of structure-forming, critical and frozen eras and a locally locked mixed end-state. Read as a spectrum of the environment, the screening length sets a lower bound on the number of layers a system needs; the bound is falsified by a deficit and not by a surplus, and counting separable scales is executable only under a stated window condition in two dimensions. The two sources of inequality act on one multiplicative channel and are sub-additive: the sub-additivity is a distribution-free theorem on the transport channel and is exactly signed on the two-point parameterisation of the mixed channel. The one irreversibility of the dynamical version acquires a fold condition free of parameterisation, a closed-form critical ceiling separating a smooth contraction from a hysteresis loop, and a parameter-free identity entered as a prediction. The three tiers of application acquire operational boundaries, and the first tier is further restricted by two tests, stationarity and the shape of the kernel, so that a one-off blast yields an attenuation length and not a screening length, fallout needs a plume kernel, and planetary climatic mixing yields only a binary reading. A cross-scale applicability map states where the framework holds, where it is only a zero point, where it fails and where it does not apply.