Measuring the quality of public deliberation requires evaluating not only civility or argument structure, but also the informational progress of a conversation. We introduce a framework for Conversational Information Gain (CIG) that evaluates each utterance in terms of how it advances collective understanding of the target topic. To operationalize CIG, we model an evolving semantic memory of the discussion: the system extracts atomic claims from utterances and incrementally consolidates them into a structured memory state. Using this memory, we score each utterance along three interpretable dimensions: Novelty, Relevance, and Implication Scope. We annotate 80 segments from two moderated deliberative settings (TV debates and community discussions) with these dimensions and show that memory-derived dynamics (e.g., the number of claim updates) correlate more strongly with human-perceived CIG than traditional heuristics such as utterance length or TF--IDF. We develop effective LLM-based CIG predictors paving the way for information-focused conversation quality analysis in dialogues and deliberative success.