Exact loss envelopes for cumulative forecasts under endpoint constraints

An increment forecast, its cumulative trajectory, and its final value are different evaluation objects. We give a self-contained finite-dimensional account of the exact relations among their squared errors. At a fixed horizon, prescribed increment energy and terminal error determine a compact feasible family. Its minimum and maximum cumulative squared error are obtained explicitly from the Dirichlet sine basis and two one-dimensional secular equations. We give the extremizing paths, prove global optimality, and distinguish the disconnected two-step case from the interval of attainable losses at longer horizons. The construction yields a necessary and sufficient certificate for an ordering to follow from those summaries alone. Generalized eigenvalues give the exact increment-loss margin needed to preserve a quadratic-loss ordering, including singular weighted losses. For positive trajectories reconstructed from log increments, sign-aware bounds and an endpoint-matched family show why these quadratic results cannot be read as relative-error rankings. The operator spectra and trust-region tools are classical; this note assembles and specializes them for transparent forecast comparison rather than claiming a new general optimization theory. All examples are explicit mathematical constructions, not empirical evidence. Self-contained mathematical technical note. No empirical dataset is used. Editable LaTeX and synthetic arithmetic checks accompany the manuscript. Chien-Tung Lin is the first and corresponding author ([email protected]). Sheng-Ye Yu is the second author. This is a technical-note preprint, not a peer-reviewed journal article.