This paper presents a continuous, information-theoretic extension of the Free Energy Principle through the concept of Markov blanket density, i.e., a scalar field that quantifies the degree of conditional independence between internal and external states at each point in space (ranging from 0 for full coupling to 1 for full separation). It demonstrates that active inference dynamics (including the minimization of variational and expected free energy) naturally emerge from spatial gradients in this density, making Markov blanket density a necessary foundation for the definability and coherence of the Free Energy Principle. These ideas are developed through a mathematically framework that links density gradients to precise and testable dynamics, offering a foundation for novel predictions and simulation paradigms.

翻译：本文通过马尔可夫毯密度的概念，提出了自由能原理的一种连续信息论扩展。马尔可夫毯密度是一个标量场，用于量化空间各点处内部状态与外部状态之间的条件独立程度（取值范围从0表示完全耦合到1表示完全分离）。研究表明，主动推理动力学（包括变分自由能与期望自由能的最小化）自然地从该密度的空间梯度中涌现，使得马尔可夫毯密度成为自由能原理可定义性与自洽性的必要基础。这些思想通过一个将密度梯度与精确可检验动力学相联系的数学框架得以发展，为新的预测与仿真范式提供了理论基础。