Definition. First-step Adam map and gauge defect [ftip-00BD]
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    Fix \(\epsilon >0\). For zero-state, bias-corrected Adam define the entrywise first-step direction

    \[ D_\epsilon (G)=G\mathbin {\oslash }(|G|+\epsilon ) \]

    and, for \(Q\in \mathrm O(k)\), define its gauge-aligned defect by

    \[ E_Q(G)=D_\epsilon (GQ)Q^{\mathsf T}-D_\epsilon (G). \]

    Here \(|G|\), addition, and division are entrywise. Bias correction makes the first direction independent of Adam's two decay coefficients under the declared convention [singh2026lossbasis, Proposition 4.4].