Interpolation of Local Density via the Gap Landscape in a Dynamic Medium of Coherent Cohorts of Existents

The paper is devoted to the interpolation of the local density of filling of a Dynamic Medium between two limitingregimes: quasi-monodisperse (Fejes Tóth limit 𝜋/√12) and polydisperse (supremum 1). A correction factor Φ(𝜆)is introduced, which shows how many times the density increases in the transition from one regime to the other.The connection of Φ(𝜆) with the geometry of residual gaps (triangular and hexagonal) arising between the cells ofthe hexagonal packing is established. Explicit areas of the gaps are obtained, and it is shown that the hexagonalgap is approximately 25 times larger than the triangular one, which determines the order of their filling. Todescribe the filling, a hardness function and a deformation operator are introduced, allowing cells to adapt to theshape of the gaps. An integral formula for Φ(𝜆) is obtained, and a logarithmic model of filling is considered, forwhich Φ(𝜆) has an explicit form and satisfies the boundaries Φ(0) = 𝜋/√12, Φ(1) = 1. A theorem on thepreservation of coherence under cell deformation is proved: if the hardness function is sufficiently large, thedeformed cells do not intersect pairwise.