TopoPrimer: The Missing Topological Context in Forecasting Models

Many leading time series foundation models (TSFMs) forecast individual series based predominantly on their own temporal histories, relying on implicit cross-attention to capture any broader context. It remains largely unexplored whether explicit, domain-wide population structure can improve these models. Specifically, it is unclear if a series' relative position within a population, or the global shape of the population itself, offers a recoverable signal for forecasting. To investigate this, we introduce TopoPrimer, a framework that injects the global topological structure of a series population as an explicit prior into forecasting backbones. TopoPrimer captures this structure through two frozen encodings, precomputed once per domain: a spectral relational coordinate that locates each series within the population, and a persistent-homology fingerprint that describes the population's global shape. These descriptors are injected either per-token into models trained from scratch, or via a lightweight adapter into frozen, pre-trained foundation models. We find a strict division of labor: the relational coordinate varies across series and drives accuracy on frozen backbones when history is abundant, whereas the topological fingerprint is constant within a population and relies entirely on cross-cohort structural diversity to provide predictive signal. TopoPrimer improves accuracy on three of four public benchmarks, reducing MSE by 7.4% on ECL. Notably, on the ECL dataset, the relational coordinate outperforms full fine-tuning of Chronos while training fewer than 1% of the parameters. On a large-scale internal corpus, the fingerprint reduces peak-demand error by 9 to 14% and improves cold-start MAE by 13% at launch. Because a replication on the public M5 dataset yielded no aggregate cold-start gain, we scope this zero-shot advantage to training distributions spanning structurally distinct populations.