Approximate nearest neighbor (ANN) search in high-dimensional spaces is a foundational component of many modern retrieval and recommendation systems. Currently, almost all algorithms follow an $ε$-Recall-Bounded principle when comparing performance: they require the ANN search results to achieve a recall of more than $1-ε$ and then compare query-per-second (QPS) performance. However, this approach only accounts for the recall of true positive results and does not provide guarantees on the deviation of incorrect results. To address this limitation, we focus on an Error-Bounded ANN method, which ensures that the returned results are a $(1/δ)$-approximation of the true values. Our approach adopts a graph-based framework. To enable Error-Bounded ANN search, we propose a $δ$-EMG (Error-bounded Monotonic Graph), which, for the first time, provides a provable approximation for arbitrary queries. By enforcing a $δ$-monotonic geometric constraint during graph construction, $δ$-EMG ensures that any greedy search converges to a $(1/δ)$-approximate neighbor without backtracking. Building on this foundation, we design an error-bounded top-$k$ ANN search algorithm that adaptively controls approximation accuracy during query time. To make the framework practical at scale, we introduce $δ$-EMQG (Error-bounded Monotonic Quantized Graph), a localized and degree-balanced variant with near-linear construction complexity. We further integrate vector quantization to accelerate distance computation while preserving theoretical guarantees. Extensive experiments on the ANN-Benchmarks dataset demonstrate the effectiveness of our approach. Under a recall requirement of 0.99, our algorithm achieves 19,000 QPS on the SIFT1M dataset, outperforming other methods by more than 40\%.