On Limit Constants of Density for Two Classes of Coherent Structures

The paper introduces a formal language for describing the collective behavior of objects in metric spaces with measure, simultaneously accounting for spatial structure, scale spectrum, and temporal evolution. Using the terms medium, existents, and cohorts, we define coherent structures (subsets consisting of pairwise disjoint closed balls) and their saturated versions (maximal by inclusion under an upper-bounded radius spectrum). We consider the local density of a coherent structure — the fraction of the volume of a window occupied by the cells of the structure. Two classes of coherent structures are introduced: class A (with a lower bound on radii) and class B (without a lower bound). For the planar case with Lebesgue measure, two limit results are established. First, for class A it is proved that the maximum local density tends to the classical constant π/√12 (Fejes Tóth theorem); the supremum is attained on the hexagonal packing. Second, for class B it is proved that the supremum of the local density equals one for any fixed parameter: allowing arbitrarily small radii permits hierarchical filling of the gaps between large balls. It is noted that maximality by inclusion is not equivalent to maximal density (the square lattice provides an example of a saturated structure). For intermediate values, a correction factor is introduced, and finding its explicit form is formulated as an open problem.