Coverage Bounds for a Contracting Inverse-Word Sieve in Z₃
This research note grew out of an AI-assisted investigation of exact reduction methods for the Collatz map. Rather than claiming a solution to the conjecture, it addresses a narrower question: how much can one precisely defined family of inverse-step certificates cover—and what remains beyond its reach?
The paper gives a computer-assisted bound on the portion of 3-adic space left uncovered by this particular sieve, even when inverse words of every length are included. Writing u∞u_\infty for the normalized measure of that uncovered set, the result is:
0.5301284713<u∞≤283552531441<0.5335531132.0.5301284713<u_\infty \le\frac{283552}{531441} <0.5335531132.
In other words, more than 53% remains uncovered by this specific sieve. That is a statement about the reach of a defined mathematical construction—not the percentage of integers that fail to converge, and not a limitation on every possible approach to Collatz.
The paper and supporting archive include the proofs, executable computations, exact rational outputs, and a separately implemented reconstruction of the coverage calculation. I’m sharing the work for independent examination and feedback. Some original-source literature comparisons remain incomplete, and no historical-priority claim is made. This is not a proof of the Collatz conjecture.
