Possible new result in random matrix theory, Looking for mistakes, references, or prior work

I’ve been working on the following problem and would appreciate any feedback.

Let (A\_M) be a ((2M-1)\\times M) matrix with iid standard Gaussian entries. Define

**\[**
**\\omega(A\_M)**
\\min\_{S:,\\mathrm{rank}(A\_{S\^c})<M}
\\sigma\_M(A\_S).
\]
I claim to prove that, almost surely,
\[
\-\\frac{1}{M}\\log \\omega(A\_M)\\to \\log 4,
\]
equivalently,
\[
\\omega(A\_M)=4\^{-M+o(M)}.
\]

The proof first reduces the problem to the minimum singular value among all (M\\times M) row submatrices. The lower bound is essentially a union bound over Gaussian row-to-hyperplane distances. The upper bound uses a second-moment argument that explicitly handles dependence between overlapping submatrices via a quotient-space reduction to correlations of random spherical slabs.
What I’m looking for:
Any references proving this already (or something equivalent).
Any obvious gaps or mistakes.
Whether the result is known in frame theory / phase retrieval / random matrix theory under different terminology.

I have attached the proof and results below

[Manuscript](https://drive.google.com/file/d/1VB15UawpehhWr73ksBldrhlHZTFzZO0V/view?usp=drivesdk)