Remark: Family 021. A quadratic bound for Jacobsthal's function

Answers Jacobsthal's quadratic-bound question: every interval of $C{k}^2$ consecutive integers contains an integer coprime to any prescribed positive integer with at most $k$ distinct prime divisors, for an absolute constant $C$. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.