Geometry-Aware Multi-Armed Bandits for Antenna Beam Selection on Spheres, Tori, $\text{SO}(3)$, and Reconfigurable Intelligent Surfaces

Beam alignment in mmWave phased arrays and RIS-assisted links is a stochastic bandit under both short TTI budgets and Doppler-induced non-stationarity. The arm space is a Riemannian manifold: ${\text{sphere}}^2$ for steering, ${\text{torus}}^n$ for phase combining, $\text{SO}(3)$ for panel orientation, or the discrete torus $({\mathbb{Z}}_B{)}^M$ with up to $K\text{\hspace{0.17em}⁣}\sim \text{\hspace{0.17em}⁣}{10}^{90}$ configurations for $B$-level RIS ($B\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}{2}^b$, $b$ bits/element); the intrinsic Matérn kernel of Borovitskiy et al.\ provides the base GP. We contribute two algorithmic pieces. $\mathbf{\text{(C1)}}$ A Kronecker-factorised intrinsic-product Matérn kernel on $({\mathbb{Z}}_B{)}^M$ evaluating in $O(M)$ table lookups, making GP-UCB tractable at $K\sim {10}^{90}$ where the extrinsic alternative is infeasible. $\mathbf{\text{(C2)}}$ AdaptiveGP-v2, an online sliding-window controller that selects $W$ by per-sample marginal likelihood, with predictive-variance and drift $z$-score reset triggers and a post-reset $\beta$-boost. On a four-speed ($v\text{\hspace{0.17em}⁣}\in \text{\hspace{0.17em}⁣}\{0.02,0.08,0.12,0.20\}$~km/h), $20$-seed paired campaign at $T\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}3000$, AdaptiveGP-v2 is statistically indistinguishable from the hand-tuned fixed-window oracle at every speed (Holm–Bonferroni-corrected paired differences cross zero); the operational benefit is the absence of a deployment-time per-speed calibration step, not a mean-regret improvement. On four static 3GPP-style mmWave benchmarks, intrinsic-kernel GP-UCB reduces cumulative regret by $25$–$45\mathrm{\%}$ vs.\ codebook UCB1/Thompson and by $10$–$33\mathrm{\%}$ vs.\ Euclidean-ambient GP-UCB on the toroidal arm spaces; a wideband OFDM ablation on a $100$~MHz channel confirms the advantage persists under frequency-selective fading ($\sim \text{\hspace{0.17em}⁣}32$~Mbps/UE at initial access vs.\ UCB1). A third-party-simulator sanity check on Sionna CDL is reported in Section~V.