Theorem. Full-rank memoryless equivariant rules are Gram-determined [ftip-00BH]
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    Let \(k\leq n\), and let \(\Phi \) be defined on full-column-rank matrices \(G\in \mathbb R^{n\times k}\). Then \(\Phi (GQ)=\Phi (G)Q\) for every \(Q\in \mathrm O(k)\) if and only if there is a matrix-valued function \(H\) of \(GG^{\mathsf T}\) such that

    \[ \Phi (G)=H(GG^{\mathsf T})G. \]

    Proof. 

      The displayed form is immediately equivariant. Let \(G^+\) denote the Moore--Penrose pseudoinverse. Conversely, set \(X(G)=\Phi (G)G^+\). Then \(X(G)G=\Phi (G)\) and \(X(GQ)=X(G)\). Two full-column-rank matrices with the same left Gram matrix have the same range and differ by a right orthogonal factor. Hence \(X(G)\) depends only on \(GG^{\mathsf T}\); take \(H=X\).

      See Theorem 4.5 and its proof in Appendix B.6 of The loss does not see the basis, but Adam does[singh2026lossbasis]. It classifies the full-rank stratum only.