Conserved active information

Abstract

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.