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

Abstract

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). Four principal results. Under finite screening the point-source spatial profile is not a pure exponential but carries an inner algebraic region followed by an outer exponential cutoff, whose inner exponent is dimension-dependent. 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. Coupling the cosmological constant into the same screened field yields a composite screening length whose sign flips over cosmic time, giving a structure-forming, critical and frozen sequence of three eras and implying a locally-locked mixed end-state. And a cross-scale applicability map states explicitly where the framework holds, where it is merely a zero point, where it fails and where it does not apply. V4 makes five completions and no corrections. The five repairs V3 made to V2 stand unchanged and are described after them. Two of the completions were added before deposit and are described last, because each takes a statement this paper already made and replaces it with a closed form: one turns the paper's single irreversibility claim from a qualitative asymmetry into a quantitative prediction, and the other turns its weakest numerical entry into an exactly signed result. Completion one: the screening length, read as a spectrum of the environment, fixes how many layers a system needs (new §4A). Requisite variety together with the fact that one characteristic screening length absorbs one band of the environmental scale spectrum gives the number of layers with no fitted parameter. The correspondence is derived by the adaptation-fidelity companion; it is stated here because ℓ is this paper’s own central object and the count is a consequence of it, and because §4’s dimensional caveat attaches directly to it — a planar profile fitted with a power-law inner segment misreads ℓ, hence the band structure, hence the count. One limitation travels with it and is not discharged: the judgment separable requires a threshold on the ratio of adjacent scales that has no independent calibration. Completion two: the sub-additivity of §5 acquires its theorem and, more usefully, its boundary. The result V3 established numerically on a two-point multiplier is now a distribution-free theorem — Theorem E and its Corollary E1, proved in the cross-scale companion and carried elsewhere in this set under the alias T3B; the two names denote one theorem and mixed forms combining them are not used — but only for two pushers acting through the same allocation exponent. This paper’s two pushers act through different parameters, so V3’s own caution that the sign of the mixed derivative is not universal is exactly right and is retained. What is new is that the boundary can now be drawn precisely rather than left as a general warning: the exogenous pusher is the allocation exponent in the companion’s language and lies inside the theorem’s domain; the endogenous one is the spread and lies outside it. Completion three: the tier boundaries of §10 acquire operational definitions. V3 listed three tiers without a criterion separating the first from the second. The boundary is not whether a network is involved but whether the network correlation length exceeds the physical distance that can still be screened — the Black Death propagated on a contact network yet behaved as tier one because transport limited that length. And tier three acquires a two-sided test: the spatial side asks whether ℓ has a finite value at all; the temporal side asks whether the turning point is independent of one’s own action. Both answered no is what makes a threat tier three. What V3 repaired, retained. The inner exponent of the screened profile is d − 2 and not the far-field exponent (d−1)/2, so in two dimensions there is no inner power law at all. The two sources are sub-additive rather than compounding. The Solar-System verification is demoted from evidence to illustration, because a rank regression cannot distinguish eight planets from eight sorted random numbers and because the plot used signals a constant-ratio hierarchy rather than an exponential distribution. The ideal-gas row of the applicability map is corrected, the exponential being the occupation probability of states and not the marginal distribution of energy. And two numerical statements are tightened, the acceleration-onset redshift and the cluster turnaround radius, with one omitted moment condition restored to the policy corollary of §3. Completion four: the one irreversibility of §6 acquires a fold condition, and the line between a smooth contraction and a hysteretic one acquires a closed form (§6A). V3 and V4 stated that once a single centre has locked in, lowering the transport capacity need not restore the multi-centre pattern, and that this asymmetry is falsifiable. The statement was a shape without a skeleton: nothing in it located the two turning points, and §6's own third discipline — that a crossover is not a phase transition, and that claiming one requires identifying a condensation mechanism — could not be applied to it, because no mechanism had been written. The mechanism is the one §5A already identifies. Exogenous transport raises the allocation exponent, and by the same chain a more concentrated configuration draws from farther still, so the exponent is a function of the concentration rather than a parameter beside it. Written into the concentration equation that is a gain factor, the equation is no longer linear, and the folds satisfy a tangency condition containing no parameterization. Two consequences follow. The critical transport capacity separating a smooth contraction from a hysteretic one is a closed form, so §6's third discipline becomes a test rather than a caution. And the loop carries an identity free of every parameter, which is entered as SG-2. Completion five: the sub-additivity of §5 is exactly signed on this paper's own parameterization, and the 1.2 percent margin is explained rather than defended (§5B). The numerical entry behind that claim compares a joint effect of −0.0903 with a sum of separate effects of −0.0914, a margin of 0.0011, which is 1.2 percent of the sum and 20 percent of the smaller of the two effects. Reported as a single point it is a thin result, and the reporting guardrail this paper submits to requires that a thin result be shown to be robust rather than asserted. It is not thin. On the two-point parameterization the tail index has a closed form, κ★ = ln((1−p)/p)/ln s, from which the mixed second derivative is +[1/(1−p) + 1/p]/(s·(ln s)²), strictly positive on the whole domain p ∈ (0, ½), s > 1. The margin is small only because it is a second-order quantity evaluated at a step of 0.01: the closed form predicts 0.0011431 against the observed 0.0011226. The claim is therefore exact in sign and merely small in size at that step, and §5B says so with a sweep.