1. Negative Cycles

Smallest \(|n|\)	Length	\(k\)	\(m\)	Elements
1	2	1	1	−1, −2
5	5	2	3	−5, −14, −7, −20, −10
17	18	7	11	−17, −50, −25, −74, −37, …

Every starting value in \([-200\,000,-1]\) enters one of these three cycles.

2. Statistical Indistinguishability

Parity vectors, modular structure, stopping classes, drift and tail density are identical for the negative map. Any proof of uniqueness on the positive side must exploit positivity.

3. Divergent Trajectories and the 2-adics

The set of parity walks that remain forever above the critical line has measure zero yet is nonempty in the 2-adic integers. Whether any positive ordinary integer belongs to it remains open.