Temporal Structure in Multivariate Forecasting: A Mathematical Bridge from Causal Sequence Models to Time-Field-Informed Prediction

This working paper develops an open mathematical bridge between modern multivariate forecasting systems and explicit temporal structure, using Google Research’s TimesFM-3 architecture as a current engineering case study. The paper proves, within clearly stated operational definitions, that a causal forecasting system retains temporal structure when it preserves ordered observations, an information boundary between known and unknown states, and nontrivial change. Removing timestamps, clocks, or temporal terminology does not remove that structure when the underlying order and causal relationships remain. The paper also shows that comparing multiple variables at the same indexed position contains an implicit synchronization assumption. Real channels may instead contain clock drift, sampling irregularities, measurement delays, and differences in temporal phase. To make these conditions measurable, the paper connects multivariate forecasting with the phase mismatch, coherence window, lock-state, and stability concepts developed in the Chronos Communication framework. Independent synthetic experiments test these mechanisms using transparent forecasting models. Across 40 high-drift trials, explicit phase alignment reduced the mean forecasting error by 72.45 percent compared with assuming that corresponding data rows were synchronized. Coherence gating also protected forecasts when synchronization deteriorated, although actual realignment performed substantially better. The results demonstrate the importance of temporal alignment in controlled forecasting tasks. They are not presented as a benchmark of TimesFM-3 or as empirical confirmation of a universal physical time field. The paper includes a claim ledger that clearly separates mathematical results, computational demonstrations, proposed extensions, and open physical hypotheses. An accompanying reproducibility package provides the complete source code, deterministic synthetic generators, trial-level results, summary tables, figures, and PDF-generation code. The work is released for public inspection, reproduction, criticism, and extension.