The prevailing interaction paradigm with Large Language Models (LLMs) as simple query-response tools is inefficient, leading to the squandering of their latent potential. This paper redefines the interaction model with complex logical systems by establishing a functional framework demonstrating that high-value outputs are not trivially requested but are "mined" through deliberate cognitive investment from the user, a process we term "cognitive proof-of-work." Drawing a functional analogy to the economics of cryptocurrency mining, we employ conceptual analysis to synthesize principles from complexity science, particularly the theory of emergence in large-scale systems, and formally deconstruct advanced prompt engineering techniques. The analysis establishes a formal model where user-LLM interaction is a value-driven exchange governed by cognitive merit. The principal findings are: (1) The concept of a "clarity window" is defined, identifying rare, high-value outputs as a scarce cognitive resource. (2) The model's latent potential is framed as a perishable "clarity potency charge," which is irreversibly collapsed by low-complexity prompts. (3) A causal link is identified between the user's intellectual labor, the complexity of the prompting strategy employed, and the computational energy consumed during inference, giving the proof-of-work a physical basis. (4) We distinguish between "explicit" proof-of-work via prompting and "implicit" proof-of-work via gamified interfaces, drawing parallels to biological processing in closed-loop systems like Formula 1. The "cognitive proof-of-work" framework functions as a systemic filter that inherently rewards critical inquiry and intellectual rigor, fostering a new cognitive symbiosis between human and machine that augments and refines human thought rather than replacing it.