Appendix of VarDim-Transformer: A Unified Framework for Variable-Dimension Time Series with Unknown Missing Patterns

It provides rigorous mathematical proofs (Theorem 4.2 on degenerate equivalence), engineering implementation details (Algorithm B.1 with PiRegistry lookups and soft gating), comprehensive complexity analysis, training hyperparameters, and extensive experimental results.The included materials support the reproducibility of the VarDim-Transformer model, including:Full derivation of the model’s reduction to standard Transformer under fixed-dimension conditions.Optimized pseudocode for dynamic projection and VarDim attention.Time complexity analysis and empirical training time comparison.Detailed statistical reports (pairwise p-values, Mann–Whitney U tests, seed-level performance).Supplementary figures and tables (ranking heatmaps, sensitivity analysis, top‑K score distributions).These data are essential for researchers wishing to understand, implement, or extend the VarDim-Transformer for variable-dimension time series forecasting with unknown missing patterns.