Geometry-Aware Multi-Armed Bandits for Antenna Beam Selection on Spheres, Tori, $\SO(3)$, 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: $\sphere^2$ for steering, $\torus^n$ for phase combining, $\SO(3)$ for panel orientation, or the discrete torus $(\mathbb Z_B)^M$ with up to $K\!\sim\!10^{90}$ configurations for $B$-level RIS ($B\!=\!2^b$, $b$ bits/element); the intrinsic Matérn kernel of Borovitskiy et al.\ provides the base GP. We contribute two algorithmic pieces. \textbf{(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. \textbf{(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 $β$-boost. On a four-speed ($v\!\in\!\{0.02,0.08,0.12,0.20\}$~km/h), $20$-seed paired campaign at $T\!=\!3000$, 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 $25$--$45\%$ vs.\ codebook UCB1/Thompson and by $10$--$33\%$ 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 ($\sim\!32$~Mbps/UE at initial access vs.\ UCB1). A third-party-simulator sanity check on Sionna CDL is reported in Section~V.