Understanding how individuals navigate mental health challenges over time is critical yet methodologically challenging. Traditional approaches analyze community-level snapshots, failing to capture dynamic individual recovery trajectories. We introduce a novel framework applying Topological Data Analysis (TDA) specifically persistent homology to model users' longitudinal posting histories as trajectories in semantic embedding space. Our approach reveals topological signatures of trajectory patterns: loops indicate cycling back to similar states (stagnation), while flares suggest exploring new coping strategies (growth). We propose Semantic Recovery Velocity (SRV), a novel metric quantifying the rate users move away from initial distress-focused posts in embedding space. Analyzing 15,847 r/depression trajectories and validating against multiple proxies, we demonstrate topological features predict self-reported improvement with 78.3% accuracy, outperforming sentiment baselines. This work contributes: (1) a TDA methodology for HCI mental health research, (2) interpretable topological signatures, and (3) design implications for adaptive mental health platforms with ethical guardrails.

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