While provably secure steganography provides strong concealment by ensuring stego carriers are indistinguishable from natural samples, such systems remain vulnerable to real-world edit errors (e.g., insertions, deletions, substitutions) because their decoding depends on perfect synchronization and lacks error-correcting capability. To bridge this gap, we propose Alkaid, a provably secure steganographic scheme resilient to edit errors via distance-constrained encoding. The key innovation integrates the minimum distance decoding principle directly into the encoding process by enforcing a strict lower bound on the edit distance between codewords of different messages. Specifically, if two candidate codewords violate this bound, they are merged to represent the same message, thereby guaranteeing reliable recovery. While maintaining provable security, we theoretically prove that Alkaid offers deterministic robustness against bounded errors. To implement this scheme efficiently, we adopt block-wise and batch processing. Extensive experiments demonstrate that Alkaid achieves decoding success rates of 99\% to 100\% across diverse error channels, delivers a payload of 0.2 bits per token for high embedding capacity, and maintains an encoding speed of 6.72 bits per second, significantly surpassing state-of-the-art (SOTA) methods in robustness, capacity, and efficiency.