The widespread use of Large Language Models (LLMs) in text generation has raised increasing concerns about intellectual property disputes. Watermarking techniques, which embed meta information into AI-generated content (AIGC), have the potential to serve as judicial evidence. However, existing methods rely on statistical signals in token distributions, leading to inherently probabilistic detection and reduced reliability, especially in multi-bit encoding (e.g., timestamps). Moreover, such methods introduce detectable statistical patterns, making them vulnerable to forgery attacks and enabling model providers to fabricate arbitrary watermarks. To address these issues, we propose the concept of trustworthy watermark, which achieves reliable recovery with 100% identification accuracy while resisting both user-side statistical attacks and provider-side forgery. We focus on trustworthy time watermarking for use as judicial evidence. Our framework integrates cryptographic techniques and encodes time information into time-dependent secret keys under regulatory supervision, preventing arbitrary timestamp fabrication. The watermark payload is decoupled from time and generated as a random, non-stored bit sequence for each instance, eliminating statistical patterns. To ensure verifiability, we design a two-stage encoding mechanism, which, combined with error-correcting codes, enables reliable recovery of generation time with theoretically perfect accuracy. Both theoretical analysis and experiments demonstrate that our framework satisfies the reliability requirements for judicial evidence and offers a practical solution for future AIGC-related intellectual property disputes.