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：我们引入保守活跃信息 $I^\oplus$，这是活跃信息的一种对称扩展形式，用于量化整个搜索空间中的净信息增益/损失，并遵循“没有免费午餐”守恒定律。通过伯努利分布和均匀基线示例，我们证明 $I^\oplus$ 能够揭示 KL 散度无法探测的区域，例如强知识减少全局无序的情况。在均匀基线假设下，我们严格证明了这些区域的存在，从而区分了无序（增加微弱知识）与强知识（施加有序结构）。我们还通过马尔可夫链和宇宙学微调示例进一步阐释了这些区域。这解决了对活跃信息的一个长期批评，同时使其能够在搜索、优化等领域中得到应用。