Structural Isoperimetric Phase Transitions in Base-b Cayley Graphs on Z

We answer a question of Briggs and Wells by proving that, for every fixed depth k and all sufficiently large bases b, the edge- and vertex-isoperimetric problems on the associated base-b Cayley graphs of ℤ exhibit k distinct structural phase transitions. Uniformly on compact density ranges, both profiles satisfy Iˣ_{b,k}(⌊t·bᵏ⁺¹⌋) = bᵏ·Pₖ(t) + O(bᵏ⁻¹), Pₖ(t) = min_{1≤m≤k+1} 2m·t^((m−1)/m), for X ∈ {E,V}. The k+1 branches encode successive effective dimensions and meet at k explicit critical densities. A quantitative multi-scale untwisting theorem shows that base-b carries affect the model-cylinder profiles only at lower order. Phase-interior rigidity then concentrates every optimizer in an integer interval of the predicted scale, and the incompatible core scales on opposite sides of a crossing yield a macroscopic defect of inclusion for every pair of optimizers. We also identify the universal critical width O(b⁻¹). Finally, for edge boundary we derive exact discrete cell problems for the competing critical geometries, solve the first cell completely, and formulate the higher micro-transition problem as a separate second-order program.