This paper proposes a novel semiotic framework for analyzing Large Language Models (LLMs), conceptualizing them as stochastic semiotic engines whose outputs demand active, asymmetric human interpretation. We formalize the trade-off between expressive richness (semiotic breadth) and interpretive stability (decipherability) using information-theoretic tools. Breadth is quantified as source entropy, and decipherability as the mutual information between messages and human interpretations. We introduce a generative complexity parameter (lambda) that governs this trade-off, as both breadth and decipherability are functions of lambda. The core trade-off is modeled as an emergent property of their distinct responses to $λ$. We define a semiotic channel, parameterized by audience and context, and posit a capacity constraint on meaning transmission, operationally defined as the maximum decipherability by optimizing lambda. This reframing shifts analysis from opaque model internals to observable textual artifacts, enabling empirical measurement of breadth and decipherability. We demonstrate the framework's utility across four key applications: (i) model profiling; (ii) optimizing prompt/context design; (iii) risk analysis based on ambiguity; and (iv) adaptive semiotic systems. We conclude that this capacity-based semiotic approach offers a rigorous, actionable toolkit for understanding, evaluating, and designing LLM-mediated communication.