Establish robustness of the displacement marker with independent observables

Establish whether the resolved collective displacement is recovered with a front marker based on pressure, Mach number, or an alternative gradient rather than solely on the density half-jump location.

Background

The primary marker is the density half-jump location, and its inferred angular envelope is strongly correlated with the inverse maximum mean density gradient. This creates a residual possibility that part of the apparent angular structure reflects marker sensitivity or crossing-position uncertainty rather than a physical displacement.

Full-field pressure, Mach-number, and translational-temperature template checks provide complementary evidence, but they do not constitute a distinct front-extraction procedure. A marker constructed from an independent observable or gradient is therefore needed to determine whether the collective displacement survives an alternative measurement definition.

References

Because crossing-position uncertainty scales as |∂sρ̄|−1, the angular envelope alone cannot rule out marker sensitivity. The full-field zero-shift localization, multi-moment consistency and completed seed/loading repeats reduce this ambiguity, but invariance to a distinct pressure- or gradient-based front marker is not established by the present archive and remains a stated limitation.

— Noise-separated evidence for a slow collective displacement in a rarefied hypersonic bow-shock layer  (2608.17285 - Shoja-sani et al., 18 Aug 2026) in Section 3.4, “Angular covariance and collective displacement shape”; Section 4.7, “Limitations and scope of the conclusions”