The Constructal Law says that flow systems evolve their architecture to provide progressively easier access to the currents flowing through them. Rivers branch. Lungs branch. Lightning branches. The classical derivation treats this as static optimization: given finite size and a flow objective, minimize resistance. The optimal architecture is computed — found by solving a constrained minimization problem. The constraint (finite size) limits the solution space, and within that space, the best architecture is selected.
Stiefenhofer (arXiv:2603.06705, March 2026) reformulates the problem as dynamics rather than optimization. The flow architecture is the state of an autonomous nonsmooth dynamical system — a Filippov differential inclusion where the evolution law is discontinuous across switching manifolds. The switching manifolds are not failure boundaries. They are the loci where irreversible transport constraints force the system to change its adjustment law. The architecture evolves not by computing what is best but by following a differential equation that happens to have the best architecture as its unique attractor.
The proof is global. Under a resistance dissipation inequality (encoding the constructal principle as a Lyapunov condition) and a uniform contraction assumption (bounding the spectral properties of the generalized Jacobians), every admissible trajectory converges to the optimal architecture exponentially. There are no local minima. There are no metastable traps. Every starting configuration reaches the same endpoint.
The classical results reappear as features of the dynamics rather than solutions to a problem. The optimal assembly ratios that Bejan computed by static minimization appear here as the switching manifolds of the Filippov inclusion. The classical scaling relations are sliding invariant sets — surfaces along which the dynamics maintain the system once it arrives. The unique optimal architecture sits at the intersection of these surfaces: the place where all switching manifolds and all scaling relations are simultaneously satisfied.
The inversion is structural. Finite size and irreversibility are usually treated as constraints that limit what can be achieved — the system would do better without them. Here they are what guarantees convergence. Remove the finite-size constraint, and the dynamics lose compactness. Remove irreversibility, and the switching manifolds disappear. The constraints that appear to fight the optimum are the conditions that enforce it. The architecture is not optimal despite its limitations. It is optimal because of them.
Stiefenhofer, "Constructal Evolution as a Nonsmooth Dynamical System: Stability and Selection of Flow Architectures," arXiv:2603.06705 (March 2026).