Avoiding Premature Collapse: Adaptive Annealing for Entropy-Regularized Structural Inference

Differentiable matching layers and residual connection paradigms, often implemented via entropy-regularized Optimal Transport (OT), serve as critical mechanisms in structural prediction and architectural scaling. However, recovering discrete permutations or maintaining identity mappings via annealing $\epsilon \to 0$ is notoriously unstable. In this work, we identify a fundamental mechanism for this failure: $\mathbf{\text{Premature}}\mathbf{\text{Mode}}\mathbf{\text{Collapse}}$. By analyzing the non-normal dynamics of the Sinkhorn fixed-point map, we reveal a theoretical thermodynamic speed limit: standard exponential cooling outpaces the contraction rate of the inference operator, which degrades as $O(1\mathrm{/}\epsilon )$. To address this, we propose $\mathbf{\text{Efficient}}\mathbf{\text{Piecewise}}\mathbf{\text{Hybrid}}\mathbf{\text{Adaptive}}\mathbf{\text{Stability}}\mathbf{\text{Control}}\mathbf{\text{(EPH-ASC)}}$, an adaptive scheduling algorithm that monitors the stability of the inference process. We demonstrate that EPH-ASC is essential for stabilizing Manifold-Constrained Hyper-Connections (mHC) during large-scale training on the FineWeb-Edu dataset, effectively preventing late-stage gradient explosions by enforcing a linear stability law.