Areas add, entropies do not: an additivity obstruction to region-wise Barrow-entropic cosmology

I show that the surface degree-of-freedom count used in emergent-gravity and holographic-equipartition cosmology cannot be assigned separately to sub-regions unless Barrow's deformation parameter vanishes, and I quantify the resulting error. Barrow entropy S ∝ A^(1+Δ/2) is non-additive for Δ ≠ 0, but inherits a composition rule from the Tsallis–Cirto family, the δ-addition rule with δ = 1+Δ/2. Since S^(1/δ) = A/A_p holds identically, that rule is equivalent to the additivity of area. The additive quantity is therefore the horizon area, never the entropy. Consequently N_sur = 4S is not additive over disjoint surfaces unless Δ = 0. This obstructs a step taken throughout the entropic structure-formation literature, in which a modified Friedmann equation whose coefficients depend on the horizon entropy is applied to a collapsing sub-region evaluated at that region's own expansion rate. Splitting a horizon into N equal parts and summing N_sur over the parts misestimates the total by N^(−Δ/2). For |Δ| ~ 5×10⁻³ and N ~ 10⁹ bound structures inside a Hubble volume this is a 5% effect, and it grows only logarithmically with N. The argument is purely geometric — it uses only that disjoint surfaces have additive areas — and therefore does not depend on whether Barrow entropy admits a microstate interpretation, a point on which the literature is divided. Existing multi-horizon constructions that compose √S are shown to be already consistent with the obstruction, since for δ = 2 the square root is the area. This record contains the English and German versions of the same manuscript.