A designed tree model with a sharp congestion–fragmentation transition

We give an analytically tractable designed-tree example motivated by Aldous's conceptual congestion toy-model request. For a bounded-degree binary physical tree with k leaves on each root side and k squared cross-root demand streams, strictly positive linear capacity prices uniquely give the cheapest fixed-topology enlargement carrying doubled unit demands. After design, each stream grows affinely at an independent fixed-law Exp(1) rate. An exact transportation-cut formula and an all-cuts probability estimate confine the joint marginal-admission and spare-connectivity change to an interval of width 16 sqrt(log(k)/k) around mean doubled demand. In the two outer phases every optimizer respectively leaves the whole tree spare or saturates every edge. The window is an upper bound, not an exact critical scaling theorem. Dense traffic, capacities increasing with graph size, quadratic time variance and normalization by the realized offered-demand slope are essential declared features. No canonical right-model, universal congestion, adaptive-growth, critical-profile or first-transition claim is made. Classical flow/concentration, random transportation, HOT and prior congestion/percolation models are credited, including Aldous's earlier tree-realism reservation. A bounded primary-source priority audit found no authenticated exact earlier construction/theorem in inspected material; the final 1993 Karp-Motwani-Nisan journal body, unindexed/unpublished work and uninspected editions remain limitations. The paper and source/verification ZIP preserve all 24 original research files, independent flow/probability controls and source comparisons. A new narrowly tested replay derivative preserves the original wrapper, requires exact active second-turn output and permits only explicitly bounded older floating diagnostics and documented metadata/projection differences. Python 3.9.6, NumPy 2.0.2 and SciPy 1.13.1 were tested; unpinned historical bitwise replay is not claimed. The preserved optional priority checker separately requires Python 3.10 or later, was tested with Python 3.14.6 and exactly reproduces its 32 retained controls. The analytic proof does not depend on finite computations. AI tools were extensively involved in mathematical research, model and proof development, computational verification, literature search, manuscript preparation and adversarial auditing. This is an unrefereed preprint and has not undergone formal independent human peer review. Internal AI reviews do not constitute human peer review.