We study online prediction for a specific finite-alphabet, exogenously driven source with infinite input memory. Independent Rademacher inputs $(U_t)$ are observed sequentially, and the next binary mark has logit $\sum_{j=1}^{t}θ_jU_{t+1-j}$, where $\abs{θ_j}\leq r_j$ and $\sum_jr_j\leq B$. Regret is expected cumulative excess log loss. Lag $j$ can affect prediction by scale $r_j$ and enters only $n_{T,j}=T-j+1$ prediction rounds, leading to the lag-resolved spectrum $Γ_T(r)=\sum_{j=1}^{T}\log\!\left(1+n_{T,j}r_j^2\right)$. For every summable envelope, a localized Bayesian mixture proves $\cR_T(r)\leq CΓ_T(r)$. For exponential and polynomial envelopes, under the stated finite-sample dimension condition, a Toeplitz-design converse proves $\cR_T(r)\geq cΓ_T(r)$, with constants allowed to depend on the fixed decay parameters and the logit bound. Thus $Γ_T(r)$ is the minimax cumulative-regret scale for this source class in these canonical regimes, giving $Θ(α^{-1}\log^2T)$ for $r_j=Ae^{-αj}$ and $Θ(T^{1/(2s)})$ for $r_j=Aj^{-s}$, $s>1$. The converse is specific to the exogenous lagged model and is not a profile-only theorem for arbitrary stationary infinite-memory sources. Retaining only the most recent $h$ inputs costs order $\sum_{j>h}n_{T,j}θ_j^2$, yet the same worst-case truncation profile can correspond to polynomially different regret. A scaled online Newton predictor attains the spectrum upper bound.
Robust forecast aggregation combines the predictions of multiple information sources to perform well in the worst case across all possible information structures. Previous work largely focuses on settings with a known binary state space, where the state is either 0 or 1. We study prior-agnostic robust forecast aggregation in which the aggregator observes only experts' reports, yet is ignorant of both the underlying joint information structure and the full prior, including the underlying state space. Unlike the standard model that fixes the binary state space {0, 1}, we allow the (binary) unknown state values to be arbitrary numbers in [0, 1], so the same reported probability may correspond to very different realized outcome frequencies across environments. Our main contribution is a simple, explicit, closed-form log-odds aggregator that linearly pools forecasts in logit space, together with (nearly-)tight minimax-regret guarantees across three knowledge regimes. We first show that under conditionally independent (CI) signals, robust aggregation with an unknown state space is strictly harder than in the known-state setting by establishing a larger lower bound, and our aggregation rule can achieve a worst-case regret of 0.0255. Along the way, we also characterize tight regret bounds for Blackwell-ordered structures and for general information structures. In the classical setting with known state space {0,1}, our aggregator achieves regret strictly below 0.0226 for CI structures. To the best of our knowledge, this is the first explicit closed-form aggregator that achieves a regret upper bound strictly less than 0.0226. Finally, we extend the model where the aggregator additionally knows each expert's marginal forecast distribution; in this setting, with the CI structures, we show that a generalized log-odds rule achieves regret of 0.0228, complementing with a lower bound of 0.0225.