Corollary. The square invertible preconditioner representation [ftip-00BI]
✍️source

In the specialization \(k=n\) with \(G\) invertible, let \(P^{1/2}\) denote the unique positive-definite square root of \(P\). The function in Theorem [ftip-00BH] has the unique canonical representative

\[ H(P)=\Phi (P^{1/2})P^{-1/2}, \qquad P=GG^{\mathsf T}>0. \]

Proof.

The polar decomposition writes \(G=P^{1/2}Q\). Equivariance gives \(\Phi (G)=\Phi (P^{1/2})Q=H(P)G\). Evaluating at \(G=P^{1/2}\) proves uniqueness in this square invertible case.

For rectangular \(G\), the action of \(H(P)\) away from the range needed to multiply \(G\) is not determined by \(\Phi \). The proof above establishes uniqueness in the square invertible specialization.

Corollary. The square invertible preconditioner representation [ftip-00BI]
✍️source

Proof.

Equivariant and basis-sensitive update rules [ftip-00B6]

Definition. Memoryless right-equivariant update [ftip-00B7]

proposition. Gradient descent and momentum preserve the orthogonal gauge [ftip-00B8]

Proof.

proposition. A shared scalar second moment preserves the orthogonal gauge [ftip-00B9]

Proof.

proposition. Full-matrix Muon and Shampoo preserve the orthogonal gauge [ftip-00BA]

Proof.

proposition. Coordinatewise equivariance forces linearity [ftip-00BB]

Proof.

Corollary. Fixed nonlinear coordinatewise updates break the gauge [ftip-00BC]

Definition. First-step Adam map and gauge defect [ftip-00BD]

proposition. Exact represented-matrix defect after one Adam step [ftip-00BE]

Proof.

Example. A two-coordinate Adam defect witness [ftip-00BF]

Example. A rotated sign update fails to commute [ftip-00BG]

Theorem. Full-rank memoryless equivariant rules are Gram-determined [ftip-00BH]

Proof.

Corollary. The square invertible preconditioner representation [ftip-00BI]
✍️source

Proof.

Remark. What the structure theorem leaves unresolved at rank deficiency [ftip-00BJ]

Theorem. Full-rank memoryless equivariant rules are Gram-determined [ftip-00BH]

Proof.