The client orb climbs a five-turn helix — Precontemplation → Contemplation → Preparation → Action → Maintenance, the Transtheoretical Model's stages of change. Each session, the counselor's empathic attunement feeds the therapeutic alliance toward an equilibrium set by how attuned they are, while the alliance itself naturally decays without upkeep. Forward movement along the helix is driven by the current alliance strength scaled by session pace; ambivalence acts as friction on that progress and also sets the odds of a relapse — a sudden slide back down part of the current stage, exactly as real behaviour change rarely runs in a straight line.
dA/dt = k·E·(1−A) − λ·A
dP/dt = pace·A·(1 − 0.6·M)
relapse chance per session ∝ M
- Empathic attunement (E) — how accurately the counselor reflects the client's experience; higher values pull the alliance toward a higher equilibrium.
- Session pace — how quickly sessions run; scales how fast progress accumulates once the alliance supports it.
- Ambivalence (M) — the client's mixed feelings about changing; higher values slow progress and raise the chance of a relapse slide.
- Reflective listening — toggles an active listening bonus that raises the alliance's equilibrium further, at the cost of a slower natural stage transition while the counselor focuses on rapport over pace.
Real-world relevance: this loop mirrors why counselors track the working alliance as closely as symptom change — a strong alliance is one of the most consistent predictors of positive outcomes across therapeutic approaches, and relapse is treated as expected data, not failure.