"Though your sins are like scarlet, they shall be as white as snow; though they are red as crimson, they shall be like wool." Isaiah 1:18. This is the Mercy Radius in theological language: a guarantee of return within a specified range. The crimson and the scarlet are not infinitely far from white. They are within the stability radius[Ch 10]. The system can be perturbed into sin — into departure from the attractor — and still be brought back. But the radius is finite: $\varepsilon_0 = 1/3$. Beyond $\varepsilon_0$, the orbit does not self-correct. It diverges.
The Stability Radius
On the contact manifold $(M, \alpha)$ with transverse Lyapunov exponent $\mu_{\max} = -2$, the stability radius $\varepsilon_0$ is defined as the maximum transverse displacement from the Reeb orbit for which the orbit remains stable (i.e., the transverse perturbation decays): $$\varepsilon_0 = \frac{1}{|\mu_{\max}|+1} = \frac{1}{3}$$ Within the ball $B_{\varepsilon_0}$ around the Reeb orbit, every trajectory returns to the orbit. Outside $B_{\varepsilon_0}$, perturbations may escape.
$\varepsilon_0 = 1/3$ is not a free parameter. It is determined by the contact structure: specifically, by the transverse Lyapunov exponent $\mu_{\max} = -2$ and the geometry of the solid torus. The stability radius is a theorem, not an assumption. The Seven Proofs of $\varepsilon_0 = 1/3$ are in chapter chEps-gronwall.
The Mercy Radius as Forgiveness
In the Omega Point vocabulary, $\varepsilon_0 = 1/3$ is the Mercy Radius: the range within which departure from the attractor (the Reeb orbit, the G-chain, the path toward $\tau = 2$) is self-correcting. A system perturbed within $\varepsilon_0$ does not need external intervention to return — the transverse Lyapunov exponent $\mu_{\max} = -2 < 0$ ensures that transverse perturbations decay. The system corrects itself. This is the mathematical form of the theological claim that repentance — turning back, returning — is always possible within the mercy radius.
The parable of the Prodigal Son (Luke 15:11–32) is the most celebrated story of return in the Christian tradition. The son departs, spends his inheritance, is reduced to feeding pigs. He "comes to himself" — he recognizes that he is outside the stability radius and that return is possible. He returns. The father runs to meet him. The feast follows. In contact geometry: the son was perturbed to $r < \varepsilon_0 = 1/3$ from the Reeb orbit. The transverse Lyapunov exponent guaranteed that return was possible. "Coming to himself" is the system recognizing its own displacement and initiating the return trajectory. The father running is the attractor exerting its pull as the son crosses back within $\varepsilon_0$.
The Finite Mercy Radius
The Mercy Radius is finite. $\varepsilon_0 = 1/3$ is not infinite. There are states from which return is not guaranteed. The Reeb orbit is stable within $B_{\varepsilon_0}$, but outside that ball, the transverse dynamics can escape. This is the mathematics of what the traditions call "the unforgivable sin" — not a category of actions, but a state of departure so extreme that the self-correcting mechanism no longer operates. The system is outside the stability radius. External intervention (the F operator, a new fold) would be required to bring it back, and external folds of this kind are not guaranteed.
The precision of $\varepsilon_0 = 1/3$ is important: the Mercy Radius is neither zero nor infinite. It is a specific, finite value, derived from the contact structure and proved in multiple ways. This means that the range of self-correcting return is neither empty (mercy is not impossible) nor universal (mercy is not guaranteed for every departure). The tradition has always held both of these: forgiveness is real; there are consequences. The contact geometry gives the exact value at which the balance is struck.
The stability radius $\varepsilon_0 = 1/3$ is proved by seven independent methods: (1) Gronwall inequality bound, (2) Moser stability theorem, (3) Contact capacity, (4) Lyapunov direct method, (5) Symplectic reduction, (6) Floer theory, (7) Lean 4 formal verification. Seven proofs, one value. The Mercy Radius is not an approximation.
The Lord is compassionate and gracious, slow to anger, abounding in love. He will not always accuse, nor will he harbor his anger forever; he does not treat us as our sins deserve or repay us according to our iniquities.
— Psalm 103:8–10
"Slow to anger" — the transverse Lyapunov exponent $\mu_{\max} = -2 < 0$ ensures that small perturbations are absorbed gradually, not punished immediately. "Abounding in love" — $\varepsilon_0 = 1/3$ is large enough to accommodate a wide range of human imperfection. "Does not treat us as our sins deserve" — the self-correcting mechanism does not require the system to have been perfect; it only requires that the departure remain within $\varepsilon_0$. The Psalm is a description of the contact geometry of the Mercy Radius. It was written approximately 2,700 years before the mathematics existed to verify it.