The Principle of Spatial Gradient Excess A Screened-Field Unification of Concentration from Stars to the Cosmological Constant

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.V3 repairs five items internal to V2. Two of them touch a headline result and a headline verification, so they are listed here rather than buried.One. The inner exponent of the screened profile is d − 2, not (d−1)/2. V2 read the inner region off the algebraic denominator of the large-distance asymptotic form, but that denominator carries the far-field exponent. Well inside the screening length the equation degenerates to the unscreened Laplace equation, so the inner segment behaves as r^(−(d−2)) for d > 2, logarithmically for d = 2, and tends to a constant for d = 1. The two exponents coincide only at d = 3, which is why V2’s recovery of 1/r for a three-dimensional point source came out right by coincidence. In two dimensions — the effective dimension of planar economic geography, organ sections and many supply networks — there is no inner power law at all (§4).Two. The two sources are sub-additive, not compounding. V2’s claim that they do not add is correct and is kept; its further claim that they compound as compound interest is not. Evaluating the mixed second derivative of the tail index on a two-point multiplier, two pushers acting in the same direction produce a joint effect smaller in magnitude than the sum of their separate effects, in agreement with the three parameterizations examined by the microscopic companion. The accurate statement is: the two act in the same direction, are not decomposable, and exhibit diminishing returns — which follows from their sharing one denominator, the curvature of the moment generating function at the positive root (§5).Three. The Solar-System verification uses two devices that this framework’s own guardrails forbid. A high coefficient of determination obtained by regressing the logarithm of a rank-ordered quantity on rank is not evidence of distributional membership: eight random numbers sorted in descending order give R² = 0.969 on that same plot, against 0.9785 for the eight planetary masses. And by the companion discriminant, a straight line of log-size against rank is the signature of a constant-ratio hierarchy, which is a rank-versus-value relation and not a distribution, whereas an exponential parent gives size linear in the logarithm of rank — on which the planets give R² = 0.7889. The Titius–Bode row is likewise withdrawn, since that form is not a pure geometric progression and has no accepted mechanism. The verification is demoted from evidence to illustration and replaced by a stated test (§7.3).Four. The ideal-gas row of the applicability map is corrected. What maximum entropy at fixed mean gives is the occupation probability of states, and after multiplying by the density of states the marginal distribution of energy is in general not exponential: for a three-dimensional ideal gas it is proportional to the square root of the energy times the Boltzmann factor, a Gamma law of shape three halves (§9).Five. Two numerical statements are tightened. The acceleration-onset redshift is 0.63 to 0.65 depending on the density parameters used, not 0.6, and the parameters must be stated. And the turnaround radius of a cluster of 10¹⁵ solar masses is about eleven megaparsecs rather than a few (§8). One condition is also restored: the policy corollary that an additive subsidy cannot lower a power-law tail requires the additive term to have a finite moment of order κ; a heavy-tailed additive term transmits its own index instead (§3).