# P12881: Exponential instability at the real phase-retrieval threshold

- ID: `P12881`
- Reference: `randomstrasse-20-exponential-instability-real-phase-retrieval`
- Page: https://theoremdb.org/statements/P12881
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/randomstrasse-20-exponential-instability-real-phase-retrieval?representation=markdown
- Record maturity: Reviewed problem

## The problem

For an integer $M>1$, let $A\in\mathbb R^{(2M-1)\times M}$ be a matrix such that every $M$ rows span $\mathbb R^M$. For $S\subseteq\{1,\ldots,2M-1\}$ let $A_S$ be the submatrix with those rows, and define $\omega(A)=\min_{\operatorname{rank}(A_{S^c})<M}\sigma_M(A_S)$, where $\sigma_M$ is the smallest of its $M$ singular values. Do there exist universal constants $C>0$ and $0<\beta<1$ such that $\omega(A)\le C\beta^M\max_k\|A_k\|_2$ for every such $M,A$?

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `randomstrasse-20-exponential-instability-real-phase-retrieval`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.