Persistence as Recursive History Formation: A Conditional-Information Framework under Diffusive Conditions

Persistence need not be understood as the preservation of order or material identity. Living and other persistent systems are often described as maintaining organization in the face of thermodynamic dissipation. Yet this description obscures a more fundamental question: what, precisely, is retained across change? This paper proposes a minimal conditional-information framework for describing persistence under non-equilibrium, dissipative, and diffusive conditions. The central expression is PₙH: = I (H₍+₁;FₙH E₂, ₍H), where Hₙ denotes a history-relative structure, E₂, ₍H the contextually relevant environment for that structure, FₙH the future possibility space opened under that context, and H₍+₁ the subsequent history formed through update. Rather than identifying persistence with the invariance of material components or the preservation of static order, the framework describes it in terms of the conditional informational contribution by which some future possibilities become incorporated into subsequent history. In short, what persists is the continuity of recursive history formation: history conditions future possibility, and some future possibilities are in turn incorporated into history. The framework is proposed as a structural schema, not as a new physical law. It is therefore not opposed to the second law of thermodynamics. Under diffusive conditions, many differences dissipate, while some are incorporated into local histories and thereby constrain the subsequent future possibility space. This distinction may be represented by separating FₙH into incorporated and dissipated components. The recursive form of the framework suggests that persistence consists in repeated history formation under changing contextual conditions. The paper also discusses a possible route toward empirical connection. Since FₙH is typically not directly observable, one preliminary approach is to examine observable fluctuations or candidate variables Xₙ FₙH, such as fluctuations in intracellular signaling, gene-expression noise, or candidate state transitions, and ask whether they leave conditional information in subsequent history: I (H₍+₁;Xₙ E₂, ₍H) >0. This provides a preliminary way to distinguish fluctuations that contribute to history formation from noise that dissipates without persistence-relevant effect. The paper concludes by situating the framework as a modest but potentially useful tool for describing persistence, history formation, and biological organization under diffusive conditions.