Rigorous Helmholtz Smoothing and the Three-Inflection Theorem for Navier–Stokes-𝛼 Boundary Layers

We revisit the Navier–Stokes-𝛼 boundary-layer similarity equa-tion, separating formal differential-algebraic solvability fromanalytic regularity and replacing an overstrong separant-basedinference with a sharp spectral Helmholtz estimate. For Cheski-dov’s fifth-order turbulent similarity problem, we derive a stablecurvature representation and a tail-hazard identity that lead tocomplementary inflection-count theorems. A computer-assistedanalytic proof establishes an open turbulent parameter region inwhich every physically acceptable similarity profile has exactlythree logarithmic-coordinate inflection points, thereby resolvingCheskidov’s 2004 three-inflection conjecture for this similaritymodel. Conversely, three-inflection profiles are impossible for0 < 𝑎 ≤ 14.5130165319 . . .. Turbulent uniqueness remainsopen.