Example. A rotated sign update fails to commute [ftip-00BG]
Example. A rotated sign update fails to commute [ftip-00BG]
For \(G=(1,1)\) and the 45-degree rotation matrix \(Q\) of Example [ftip-00BF], one has \(GQ=(0,\sqrt 2)\). With \(\operatorname {sign}(0)=0\),
\[ \operatorname {sign}(GQ)=(0,1), \qquad \operatorname {sign}(G)Q=(0,\sqrt 2). \]Thus even this two-coordinate update violates the commuting equation in Definition [ftip-00B7]. The calculation is the explicit Appendix B.4 witness of The loss does not see the basis, but Adam does[singh2026lossbasis].