Mach-Parametrized Tail Index xi(M) as Scalar Order Parameter for Gumbel-to-Frechet Transition at Buffet Onset

Two fields that rarely talk to each other are at the center of this hypothesis: extreme value theory (the mathematics of rare, worst-case events — think flood heights or stock market crashes) and aerodynamics of aircraft flying near the speed of sound. Extreme value theory has a foundational result — the Fisher-Tippett-Gnedenko theorem — which says that no matter what kind of randomness you start with, the statistics of the very largest events always settle into one of exactly three 'families,' distinguished by a single number called the shape parameter (xi). A positive xi means your extremes follow heavy-tailed, potentially unbounded power laws (Fréchet family); xi equal to zero means lighter, exponential-style tails (Gumbel family). The hypothesis proposes that this shape parameter xi acts as a natural early-warning gauge for a dangerous aerodynamic phenomenon called 'buffet' — the violent, self-sustaining shock wave oscillations that occur on wings as an aircraft approaches its speed limit near Mach 1. The idea is that in normal, attached airflow, pressure fluctuations on a wing surface look Gaussian and well-behaved, landing squarely in the Gumbel family (xi ≈ 0). But as the aircraft accelerates toward the critical Mach number where buffet begins, intermittent shock waves start slamming back and forth across the wing surface in an increasingly chaotic way — and the statistics of extreme pressure spikes shift toward the heavy-tailed Fréchet family (xi > 0). In other words, the mathematical shape of the pressure data undergoes a detectable phase transition precisely at buffet onset. This is novel because nobody has apparently connected these two fields before: extreme value statistics have been used extensively in civil engineering (designing for once-in-a-century wind gusts), but not in compressible, transonic aerodynamics where the physics are fundamentally different. If the hypothesis holds, a single number extracted from pressure sensor data could serve as a real-time indicator of how close an aircraft is to a dangerous flight regime.

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