We introduce conserved active information $I^\oplus$, a symmetric extension of active information that quantifies net information gain/loss across the entire search space, respecting No-Free-Lunch conservation. Through Bernoulli and uniform-baseline examples, we show $I^\oplus$ reveals regimes hidden from KL divergence, such as when strong knowledge reduces global disorder. Such regimes are proven formally under uniform baseline, distinguishing disorder (increasing mild knowledge from order-imposing strong knowledge. We further illustrate these regimes with examples from Markov chains and cosmological fine-tuning. This resolves a longstanding critique of active information while enabling applications in search, optimization, and beyond.

翻译：本文引入守恒主动信息 $I^\oplus$，作为主动信息的一种对称扩展，用于量化整个搜索空间中的净信息增益或损失，并遵循无免费午餐守恒原理。通过伯努利分布与均匀基线示例，我们证明 $I^\oplus$ 能够揭示 KL 散度所无法反映的机制，例如当强知识降低全局无序度时的情况。在均匀基线条件下，这些机制被严格证明，从而区分了无序（由温和知识增加所引起）与有序（由强知识施加所引起）。我们进一步通过马尔可夫链和宇宙学精细调节的实例阐释了这些机制。这一工作解决了长期以来对主动信息的批评，同时使其在搜索、优化及其他领域中的应用成为可能。