Joint Syntactic & Semantic Graph Alignment via Unbalanced Optimal Transport. Differentiable Low-Rank Gromov–Wasserstein Surrogates, and Log-Sinkhorn Stability; ALIGN100- 1/7

This manuscript presents a mathematically rigorous and practically scalable framework for aligning heterogeneous sentence representations — Universal Dependencies (UD), Abstract Meaning Representation (AMR), and Semantic Dependency Parsing (SDP). The approach integrates three core contributions: (1) an Unbalanced Optimal Transport (UOT) alignment core that formally handles null/extra nodes; (2) a differentiable low-rank Gromov–Wasserstein (GW) surrogate with provable approximation bounds that enables scalable structural regularization; and (3) a finite-precision analysis of log-domain Sinkhorn iterations with practical stability guarantees. 
In addition, the manuscript introduces structured calibration for transport plans, extending calibration theory to structured outputs and linking calibration error to downstream decision risk. Algorithms, theoretical results, proofs, and an experimental plan with ablation matrices are provided. The work is presented in a hybrid style: it is focused and self-contained while forming
the first installment of a planned seven-paper program on robust syntactic–semantic governance and adversarial resilience.