Remark: Family 137. One-tape time simulation in two-fifths-power space

Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time $T$ using $O(T^{2/5}{\log }^C(T+2))$ space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of $T$, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted.

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.