Constant-Rank Confidence Amplification for Parameter-Free Heavy-Tailed Graph-Unimodal Bandits

We study stochastic graph-unimodal bandits with arm-wise symmetric rewards and unknown finite centered $(1+a_i)$-th moments. Fixed-sample coverage alone does not control the decision cost of adaptive cache reuse, so we charge each fixed-prefix failure by the maximum number of decisions its cached endpoint can govern. Amplified prefix reuse with localization (\algname) pairs constant-rank resampled-median-of-means (RMM) endpoints with dyadic median-of-means (MoM) leader scores; independent-base amplification makes the confidence dependence logarithmic, and prefix–epoch budgets keep the expected total bad-cache decision charge finite. The charge bounds suboptimal local choices in the leader's own clock, and periodic leader pulls with summable MoM score tails make suboptimal-leader occurrences finite in expectation. Both the certified and practical settings attain expected regret $O({\sum }_{j\in N(i^{\star })}[{\Delta }_j+M_j^{1/a_j}{\Delta }_j^{-1/a_j}]\log T)$ plus a finite term. Under a global clock, \globalalg attains the all-arm $O(\log T)$ guarantee, improving the published RMM-UCB horizon factor from ${\log }^{2}T$. A symmetry-preserving Gaussian-subfamily lower bound matches the horizon order and optimal-neighbor support, with equal moment–gap powers at $a_j=1$. Across 30 paired trials on two infinite-variance path instances, matched comparisons place most of the certified–practical regret gap before cache activation and show the MoM leader raising the optimal-neighborhood pull share.