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Dense Associative Memory (DAM) generalizes Hopfield networks through higher-order interactions and achieves storage capacity that scales as $O(N^{n-1})$ under suitable pattern separation conditions. Existing dynamical analyses primarily study the thermodynamic limit $N\to\infty$ with randomly sampled patterns and therefore do not provide finite-size guarantees or explicit convergence rates. We develop an algorithmic analysis of DAM retrieval dynamics that yields finite-$N$ guarantees under explicit, verifiable pattern conditions. Under a separation assumption and a bounded-interference condition at high loading, we prove geometric convergence of asynchronous retrieval dynamics, which implies $O(\log N)$ convergence time once the trajectory enters the basin of attraction. We further establish adversarial robustness bounds expressed through an explicit margin condition that quantifies the number of corrupted bits tolerable per sweep, and derive capacity guarantees that scale as $\Theta(N^{n-1})$ up to polylogarithmic factors in the worst case, while recovering the classical $\Theta(N^{n-1})$ scaling for random pattern ensembles. Finally, we show that DAM retrieval dynamics admit a potential-game interpretation that ensures convergence to pure Nash equilibria under asynchronous updates. Complete proofs are provided in the appendices, together with preliminary experiments that illustrate the predicted convergence, robustness, and capacity scaling behavior.