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    "tree_boundary_checked": true,
    "stationary_equations_checked": true,
    "scope": "Numerical checks plus analytic tree boundaries; no general-network global optimality proof"
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  "limits": [
    "Dimensionless illustrative cost gamma=.9, not a fitted Poiseuille leaf volume budget",
    "Topology fixed while each load realization reaches a linear steady state",
    "All tree boundaries analytic; interior searches are candidates, not general proofs",
    "Transition threshold refers to the explicitly symmetric design family",
    "Disconnected fixed source/sink balance is infeasible; physical demand could instead decline"
  ]
}