ON THE COMMENSURATING FULL GROUP

The paper proves the equivalence (abstract/topological isomorphism of [T]com ⇔ flip-conjugacy) via: (i) spatiality from Fremlin’s many-involutions theorem, (ii) transporting T1 into [T2]com and forcing equality of orbit relations, and (iii) the almost positive/negative decomposition plus the Belinskaya–Katznelson conjugacy-on-common-orbits argument (Proposition 4.5), yielding flip-conjugacy. See Theorem 4.9 and Example 4.7/Thm. 4.8 for spatiality; Proposition 3.5 and Proposition 4.5 for the decomposition and the flip-conjugacy step . The candidate solution follows the same structure and is mathematically sound. The only issue is a mis-citation: it attributes [T]com = [T−1]com to Proposition 2.19, which in the paper states [T]p < [T]1 < [T]com; however 3⇒2 remains immediate by spatial conjugation as used in Theorem 4.9.