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A density-gradient diagnostic identifies the compression-layer station \\(x_s\\), while a jump-based thickness \\(\\delta_j=\\Delta\\rho/\\max|\\partial\\rho/\\partial x|\\) defines a shock-centered coordinate \\(\\xi_j=(x-x_s)/\\delta_j\\). In physical coordinates, the leading proper orthogonal decomposition (POD) mode of the centerline density profiles captures on"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"In the registered (ξ_j, η) frame, the first density mode captures 94.98% and the first two modes capture 99.05% of the fluctuation energy. 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