proposition. A shared scalar second moment preserves the orthogonal gauge [ftip-00B9]
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    Replace Adam's entrywise second-moment denominator by one scalar whose updates depend only on gauge-invariant quantities such as the pooled mean of the squared entries of both factor gradients. With a covariant first-moment buffer, the resulting stateful update is gauge-equivariant.

    Proof. 

      The first moment is a linear exponential average and transforms by right multiplication by \(Q\). The pooled squared-entry mean is the joint squared Frobenius norm divided by the entry count, hence is unchanged by \(G\mapsto GQ\). Its bias correction and positive scalar denominator are unchanged, so the complete update transforms covariantly.

      Proposition 4.2(2) and its proof are in Appendix B.3 of The loss does not see the basis, but Adam does[singh2026lossbasis]. The result concerns this declared shared-scalar variant, not stock Adam.