Remark. Scalar-Adam is outside the common-scalar flow theorem [ftip-00BQ]
Remark. Scalar-Adam is outside the common-scalar flow theorem [ftip-00BQ]
The shared-scalar Adam variant in proposition [ftip-00B9] retains a first-moment exponential moving average. It is stateful and is not the memoryless flow in Definition [ftip-00BL]. Remark 4.7 of The loss does not see the basis, but Adam does[singh2026lossbasis] treats its agreement with gradient flow as empirical rather than as a consequence of Theorem 4.6.
The theorem likewise does not apply to discrete Adam, PPO, RLVR update loops, clipping, weight decay, and mixed optimizers. Gauge equivariance and time reparameterization are separate properties.