Fixed-Time Riemannian Barrier Optimizer

First integrate the method at the optimizer update: replace the standard minibatch update by this vector field. Begin with ordinary Euclidean weights, G=I, on a small MLP or Transformer block. For explicit parameter bounds \ell_i< heta_i<u_i, use B( heta)=-\sum_i[\log( heta_i-\ell_i)+\log(u_i- heta_i)]; for spectral constraints, use a softplus barrier on \sigma(W)-c. A practical non-Euclidean variant uses a diagonal metric G_{ii}=1+\gamma heta_i^2 or a layerwise positive-definite Shampoo-like approximation. Pseudocode is: compute minibatch loss L; compute B and \Psi=L+\kappa B; backpropagate
abla_ heta\Psi; form g=G^{-1}
abla_ heta\Psi; compute r=\sqrt{g^{\mathsf T}Gg+\varepsilon^2}; update heta\leftarrow\operatorname{Retract}_ heta[-\eta(a r^{p-1}+b r^{q-1})g]. Clip only barrier gradients when overflow occurs. The mechanism supplies the two-power flow, barrier, metric norm, and theoretical time bound. Estimate \mu from the smallest Hessian eigenvalue near the final iterate, and estimate \Psi_\star from the best feasible value. In the first cheap experiment, train a two-layer MLP on MNIST against Adam and normalized gradient descent from five initialization scales, using p=0.5 and q=2. Tune a,b for a predicted 50-step settling bound. Measure constraint violations, potential residuals, and settling time. The prediction is that, after entering the locally strongly convex basin, settling times across initialization scales differ by less than 20%; removing either power term should restore initialization-dependent tails. Varying \varepsilon should produce a final residual scaling approximately linearly with \varepsilon.