A Descent-Forced Variant of the Collatz Map: ...
Year: 2026
Type: preprint
Abstract: 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 $G(n)<n$ in a number of operations determined by a $2$-adic valuation, that $G$ has a closed form, and that iterating $G$ reaches $1$ in at most $\lfloor \log_{4/3}(n-1)\rfloor+1$ descents. We then show that $\mathcal{P}$ coincides with the Collatz map only until its first correction, and we apply the minimal-counterexample (contradiction) method. The argument closes if and only if a single merging statement holds: for every $y\equiv 2 \pmod 3$ the Collatz orbits of $y$ and $2y+1$ meet. This statement is equivalent to the Collatz conjecture, so $\mathcal{P}$ locates the difficulty precisely but does not remove it. Worked examples are tabulated.
Author: Tantisukarom, Chaiya
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223784329