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Section\u00a0SI reviews the Wilson-loop phase and hybrid Wannier-center description, including the quantization on mirror-invariant lines in the two-band model and its loss in the three-band case. Section\u00a0SII describes the sample design of the phononic delicate topological insulator, and Sec.\u00a0SIII outlines the fabrication procedure. Section\u00a0SIV summarizes the signal analysis, while Sec.\u00a0SV explains the orbital-resolved measurements and normalization. Section\u00a0SVI presents details on the band-structure measurements, and Sec.\u00a0SVII explains the cross-sample frequency alignment used to compare measurements on different samples. Section\u00a0SVIII provides details on the impurity-spectroscopy measurements and additional reconstructed mode profiles of impurity-induced states. Section\u00a0SIX discusses the boundary obstruction and edge states of the phononic delicate topological insulator and presents corresponding measurements. Sections\u00a0SX and SXI describe the extraction of the effective model and the full tight-binding model with parameters. Section\u00a0SXII presents a detailed discussion of impurity-induced states and the impurity-projected Green\u2019s-function numerics for the two- and three-band descriptions. Section\u00a0SXIII evaluates the effective impurity strengths realized experimentally. Section\u00a0SXIV shows how the three-band model can be adiabatically transformed into an effective two-band model, Sec.\u00a0SXV discusses the adiabatic swap within the mirror-odd subspace, and Sec.\u00a0SXVI presents a real-space Wannier diagnostic of multicellularity and its loss.","type":"other"},{"key":"7ts9-mxcyCc26R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-025-64671-w","type":"journal-article"},{"key":"7ts9-mxcyCc27R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.107.195110","type":"journal-article"},{"key":"7ts9-mxcyCc28R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.42.1698","type":"journal-article"},{"key":"7ts9-mxcyCc29R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41563-017-0003-3","type":"journal-article"},{"key":"7ts9-mxcyCc30R1","doi-asserted-by":"publisher","DOI":"10.1038\/nature25156","type":"journal-article"},{"key":"7ts9-mxcyCc31R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.4.023177","type":"journal-article"},{"key":"7ts9-mxcyCc32R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.101.245132","type":"journal-article"},{"key":"7ts9-mxcyCc33R1","doi-asserted-by":"publisher","DOI":"10.1103\/lysb-n9zp","type":"journal-article"},{"key":"7ts9-mxcyCc34R1","unstructured":"COMSOL AB, comsol multiphysics\u00ae v. 6.4, Stockholm, Sweden, https:\/\/www.comsol.com.","type":"other"},{"key":"7ts9-mxcyCc35R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.80.115216","type":"journal-article"},{"key":"7ts9-mxcyCc36R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.84.035307","type":"journal-article"},{"key":"7ts9-mxcyCc37R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.106.094105","type":"journal-article"},{"key":"7ts9-mxcyCc38R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.109.155116","type":"journal-article"},{"key":"7ts9-mxcyCc39R1","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6633\/ae5bcb","type":"journal-article"},{"key":"7ts9-mxcyCc40R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.3.013239","type":"journal-article"},{"key":"7ts9-mxcyCc42R1","doi-asserted-by":"crossref","unstructured":"C. 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