Remark. Product symmetry and the scope of optimizer conclusions [ftip-00C0]
Remark. Product symmetry and the scope of optimizer conclusions [ftip-00C0]
The relevant optimizer results appear in version 1 of [⧉], submitted 5 August 2026: Sections 3--4 and 6, Appendices B.1--B.6 and B.9, with limitations in Section 11.
The full product symmetry acts on the factors by
\[ (U,V)\longmapsto (UA,VA^{-\mathsf T}). \]It does not use \((UA,VA)\). For row-vector attention weights, the corresponding action uses right multiplication, as in Example [ftip-00BX].
The general rectangular preconditioner representation need not have the square-case uniqueness. A probability-one full-rank claim additionally needs a law for the gradient; a generic full-rank observation does not specify one.
The coordinatewise theorem is a zero-state, memoryless obstruction. The clock theorem is continuous-time and excludes scalar-Adam's momentum. Experimental optimizer and attention results remain observations reported by the source. Equivariance is not identified with low-rank recovery.