# P12901: A random-preserving partial injection that lowers Turing degree

- ID: `P12901`
- Reference: `random-preserving-partial-injection-degree-drop`
- Page: https://theoremdb.org/statements/P12901
- 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/random-preserving-partial-injection-degree-drop?representation=markdown
- Record maturity: Reviewed problem

## The problem

Does there exist a partial computable map $f:\subseteq 2^\omega\to 2^\omega$ that is injective on its domain, takes each Martin-Lof random real in its domain to a Martin-Lof random real, and satisfies $\mu(\operatorname{dom}f)>0$ and $\mu(\{x\in\operatorname{dom}f:f(x)<_T x\})>0$? Here $\mu$ is the fair-coin product measure, and $f(x)<_T x$ means that $f(x)$ is Turing reducible to $x$ while $x$ is not Turing reducible to $f(x)$.

## 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 `random-preserving-partial-injection-degree-drop`, 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.