Gaussian Critical-Threshold Instability in Real Phase Retrieval

We establish the natural scale of instability in real Gaussian phase retrieval at the critical injectivity threshold. Let $A$ be a $(2M-1)\times M$ matrix with independent standard Gaussian entries and let $K_M=\binom{2M-1}{M}$. For every $w_M\to\infty$ with $\log w_M=o(M)$, we prove that $\mathbb{P}\{(w_M\sqrt{M}K_M)^{-1}\leω(A)\le w_M/(\sqrt{M}K_M)\}\to1$, where $ω(A)$ is the Balan--Wang stability parameter. Consequently, $-M^{-1}\logω(A)\to\log4$ in probability. For full-spark matrices at this threshold, $ω(A)$ equals both the optimal lower Lipschitz constant of $x\mapsto|Ax|$ and the minimum least singular value over all square row submatrices. The upper bound follows from a second-moment analysis of overlapping minors. A weighted Gaussian inverse-tail asymptotic and inverse-Wishart concentration yield asymptotic independence for central overlaps; rectangular hard-edge bounds control the remaining overlaps. The matching lower bound follows from a union bound and a square Gaussian hard-edge estimate.