Online Supplementary Appendix PRL 01 Single-Sector Density Bounds and an Echo-Slope No-Go Test

The paper presents a no-go constraint for horizonless echo models. If a single bounded-density nonclassical sector both regularizes the central singularity and supplies the reflective surface transition while carrying a non-vanishing mass fraction, then its proper radial support cannot remain microscopic. The resulting lower bound dprop≳M1/3d_{\rm prop} \gtrsim M^{1/3}d_{\rm prop} \gtrsim M^{1/3} implies that the normalized echo delay must satisfy ddln⁡M(Δtecho8M)≤23,\frac{d}{d\ln M}\left(\frac{\Delta t_{\rm echo}}{8M}\right) \le \frac{2}{3},\frac{d}{d\ln M}\left(\frac{\Delta t_{\rm echo}}{8M}\right) \le \frac{2}{3}, with equality only for compact saturation of the bound. A fixed independent microscopic wall (slope = 1) is incompatible with this single-sector rule. The finite-matrix construction provides one explicit realization that also caps central curvature. A population-level regression on the echo-delay slope is proposed as the falsification test.