A Descent-Forced Variant of the Collatz Map: ...

We study a deterministic procedure, denoted $\mathcal{P}$, built from three operations: $3n+1$ on odd values, division by two ($\mathrm{d2}$) on even values, and a correction step $-1$ that is applied whenever the current odd value exceeds the previous odd value. The correction keeps the sequence of odd values below its starting point, so the ``lower part'' of every start is maintained by construction. We prove that for each odd start $n\ge 3$ the procedure reaches an odd value $G(n)