We demonstrate that widely deployed Large Language Model (LLM) inference stacks harbor a steganographic channel that requires no modification to model weights, sampling code, or output distributions. The channel exploits a structural property of deterministic decoding: pseudo-random number generators (PRNGs) used in inverse-transform sampling produce a seed-dependent sequence of token-level probability intervals that can be reconstructed from the generated text alone. A sender encodes a secret message in the PRNG seed before generation; a receiver reconstructs the intervals and recovers the seed, and thus the hidden payload, by exhaustive search over the seed space. We formalize two operational modes. In the known-prompt setting, sender and receiver share the prompt, enabling exact interval reconstruction and perfect seed recovery via forced alignment. In the unknown-prompt setting, only the generated text is available; approximate interval reconstruction combined with a maximum-hit-count scoring strategy still permits reliable recovery from sufficiently long outputs. Extensive experiments across six model families and five heterogeneous text domains show that, in the known-prompt setting, full 32-bit seed recovery from the complete 2^32 candidate space achieves up to 100% accuracy, depending on model and text domain, within 300 tokens and under 35 seconds on a single GPU. In the unknown-prompt setting, recovery reaches near-perfect accuracy at 600-800 tokens in about 12 seconds. We further analyze the influence of prompting strategies, tokenization ambiguities, and sampling hyperparameters on channel reliability. Moreover, we discuss several applications of our results: First, it allows for the steganographic transmission of 32 bits, but also shows that ignorance of the prompt is not a valid security assumption.