Task-Aware Temporal Coarsening under Finite Budgets: Exact Risk, Tractable Surrogates, and Downstream Transfer

Compressing an $H$-step history into $K\ll H$ contiguous measurements requires deciding where to allocate temporal resolution. Standard heuristics allocate it uniformly or logarithmically, ignoring both the history covariance $\Sigma$ and the downstream task geometry $M$, while the exact task-weighted compression risk $J(\mathcal{P};\Sigma ,M)$ is not block-additive and cannot be minimized exactly by the classical contiguous-partition dynamic program. We study task-aware temporal coarsening with $\Sigma$ and $M$ estimated on development data only: a dynamic program globally minimizes a block-additive surrogate $\tilde{J}$, and its minimizer is refined under exact $J$. This separates three gaps that are usually read as a single gap: approximation, optimization, and transfer. Over $20,000$ audit partitions per dataset–horizon cell at $K=16$, surrogate rank fidelity ranges from $0.953$ on Weather to $-0.081$ on Electricity. Exact $J$ orders six controlled constructions against a held-out linear reader at mean rank agreement $0.980$, against $0.362$ for the surrogate, yet in a post-hoc validation audit, lower-$J$ partitions reduce neural-reader error in only six of thirteen cases. Under the linear reader, task-aware allocation beats both uniform and logarithmic spacing in all $36$ cells; under a Transformer reader, it beats uniform in all twelve cells, negligibly in one, but does not dominate a fixed logarithmic grid. Evaluating coarsened representations, therefore, requires auditing approximation, search, and transfer separately rather than attributing downstream performance to a single objective.