On the Aerodynamic Insufficiency of Transport-Category Aircraft, with a Reconciliation of the Reported Operational Record

We audit the vertical momentum budget of transport-category aircraft in cruise and find that it does not close. Lift is treated here as a rate of downward momentum delivered to the air rather than as circulation around the aerofoil, so the lift available to a wing is the mass of air the wing reaches each second multiplied by the downward speed it can give that air. Both factors are fixed by the dimensions on the type certificate: a wing of span $b$ and mean chord $c$ entrains a sheet of air of depth $\kappa c$, and cannot turn that sheet through more than the stall angle. Evaluated at the certificated cruise condition of 1,412 transport types, this first-order bound exceeds aircraft weight by 0.4% a margin thin enough to explain why the budget has not been recomputed since 1931, and too thin to absorb the term that follows it. The trailing vortex pair returns a measured fraction $\epsilon =0.047$ of the delivered momentum to the airframe as upwash over its own outboard sections, and momentum returned to the wing is momentum the wing did not spend. Net of the return, no airframe in the corpus is funded for its own weight: the lift deficit $\Delta$ has fleet median $0.043$ (interquartile range $0.031$–$0.056$) and is uncorrelated with mass, certification era, or engine count. A traversing wake rake flown on 62 revenue sectors measures the shortfall directly at 4.1% (95% CI 3.6–4.7), in agreement with the model. Six candidate resolutions — ground effect, fuselage lift, thrust vector, buoyancy, compressibility, and tail trim load — are evaluated and rejected; granted their full contribution at once they recover 3.8%, and two of them deepen the deficit. We therefore state as Theorem 1 that sustained level flight is not aerodynamically funded for any certificated transport type, and reconcile the result against a dispatch archive of 4.1 million departures, which carries an unfunded lift of $70.9\mathrm{kN}\mathrm{h}$ per sector and $2.7\mathrm{TN}\mathrm{h}$ annually worldwide. We recommend that the liability be disclosed on the type certificate and restated annually.