Universal Isoperimetric Phase Diagrams for Digit Cayley Graphs

This preprint determines the macroscopic edge- and vertex-isoperimetric phase diagrams of digit Cayley graphs on the integers generated by the separated scales 1,b,…,b^k. For fixed k and large base b, both normalized isoperimetric profiles converge locally uniformly, with error O(1/b) on compact density ranges, to the universal envelopeP_k(t)=min1≤ m≤ k+12m\,t(m-1)/m.Its active branch changes at the explicit critical densitiesc_m=(m-1/m)ᵐ⁽ᵐ⁻¹⁾, 2≤ m≤ k+1,producing k+1 successive effective-dimensional regimes. The proof compares the digit graphs with the cylinders( Z/b Z)^k× Z.For the edge boundary, an exact finite-b lower bound is obtained through a reduction of the Bollobás–Leader torus inequality to a one-variable variational problem. For the vertex boundary, the paper develops a phase-lifting mechanism based on strong-product compression, an exact cyclic coarea identity, and mixed-product isoperimetric inequalities. In both cases, the base-b carry structure is removed through quantitative mapping-torus seam comparisons. The results establish the complete limiting energetic phase diagram. They do not claim a complete classification of finite-b optimizers or prove that optimizer families undergo exactly k combinatorial transitions.