arXiv:2607.04535v4 Announce Type: replace-cross Abstract: Orthogonal and Stiefel constraints are widely used in neural network optimization to control weight geometry, improve conditioning, and stabilize signal propagation. Yet standard Stiefel optimization imposes a hidden restriction: while learning the directions of a linear map, it fixes every singular value to one, preventing the model from learning how strongly different directions should be amplified or suppressed. We introduce ManifoldFlow, a spectrum-learnable extension of Stiefel optimization that separates directional geometry from spectral scaling through $W=QS^{1/2}$, where $Q^\top Q=I$ and $S\succ0$. Beyond simply making the spectrum trainable, ManifoldFlow uses the gradient component rejected by the Stiefel constraint as a signal for how the spectrum should change, accumulates this signal across aligned tangent spaces, and jointly updates the orthonormal basis and a bounded SPD spectrum. This construction preserves exact spectral interpretability because $W^\top W=S$, while recovering standard Stiefel optimization when $S=I$. We establish the geometry of the rejected gradient, characterize optimization over the resulting Stiefel--SPD product space, and derive how the learned spectrum affects conditioning, signal propagation, and spectral complexity. Across diverse tasks, ManifoldFlow consistently outperforms matched fixed-spectrum Stiefel baselines, with particularly strong gains in recurrent language modeling. Our code is available at https://github.com/Hik289/manifold_flow.git.