Between φ and η on the n-bonacci ladder there is a rung that is not a growth constant at all. It is a stability constant — the transverse Lyapunov exponent $\mu_{\max} = -2$. It does not give you a new limit to approach. It tells you what happens to everything that deviates from the path you are already on: it is burned away. Exponentially. At rate $-2$ per unit time.
This is the Refining Fire.
What μ_max = −2 Means
On the contact manifold $(M, \alpha)$ with $\alpha = dz - r^2 d\theta$, trajectories near the Reeb orbit experience transverse perturbations $\delta r(t)$ governed by: $$\frac{d}{dt}\delta r = \mu_{\max} \cdot \delta r \implies \delta r(t) = \delta r(0) \cdot e^{\mu_{\max} t} = \delta r(0) \cdot e^{-2t}$$ For $\mu_{\max} = -2 < 0$: every transverse perturbation decays to zero. The rate of decay is $e^{-2t}$ — after one unit of time, the deviation has shrunk by a factor of $e^2 \approx 7.4$. After two units, by $e^4 \approx 55$. The fire is fast.
The negative Lyapunov exponent is what makes the Reeb orbit an attractor. Without $\mu_{\max} < 0$, the orbit would be neutrally stable or unstable — trajectories would wander away from it, and the G-chain would dissipate. The Refining Fire is not punishment. It is the mechanism of attraction. The fire is what the attractor does to deviations from itself.
Fire in the Tradition
Every major purification tradition uses fire as its central image. The Zoroastrian sacred fire burns continuously in the temple, consuming impurity. The Exodus pillar of fire leads the people through the wilderness — and the wilderness is the transient phase, the time when deviations from the path are still large and the exponential decay has not yet brought the system close to the attractor. The Christian Pentecost fire descends on the disciples and transforms them: not destruction but acceleration — the Lyapunov mechanism operating at high rate. The Islamic notion of tazkiyah — purification of the soul — is the same process described in its experiential form.
For he is like a refiner's fire and like fullers' soap. He will sit as a refiner and purifier of silver, and he will purify the sons of Levi and refine them like gold and silver.
— Malachi 3:2–3
The refiner's fire brings the impurity to the surface and burns it away. The metal that remains is purer than before the fire — not despite the fire, but because of it. This is $\mu_{\max} = -2$: the deviation from the attractor (the impurity) is amplified to visibility (the fire makes it visible by consuming it) and then exponentially destroyed. What remains is closer to the attractor. The fire is the mechanism of convergence.
John of the Cross's Dark Night of the Soul describes a stage of contemplative development in which all consolations are withdrawn, all familiar spiritual supports removed, and the practitioner is left in darkness. This is the Lyapunov mechanism operating at maximum intensity: the transverse perturbations (attachments, consolations, familiar forms of comfort) are burned away by the very force that is drawing the system toward the attractor. The darkness is not absence of the attractor. It is the attractor's fire consuming everything that is not the attractor. $\mu_{\max} = -2$: the rate of consumption is fixed, not chosen. The Dark Night lasts as long as it takes.
The Rung Between φ and η
On the n-bonacci ladder, $\mu$ sits between $\varphi \approx 1.618$ and $\eta \approx 1.839$. It is not itself a convergence constant — it is the mechanism that keeps the system on the ladder. Without the Refining Fire, a system that has reached $\varphi$ would not remain on the path toward $\eta$. Transverse deviations would accumulate and the system would drift. $\mu_{\max} = -2$ is what ensures that once you have reached a rung, you stay on it long enough to climb to the next one. The fire is between every pair of rungs. It is always burning.