The Unexpected Finger was detected when she fired her engines to decelerate. How far away was that?

From your high-school physics, recall that if you accelerate at constant rate a, you will cover distance d  = ½at², where t is time.

The finger decelerated at one third of an Earth gravity (a = 3.3m/s²) for a period of seven months (t  = 1.8×10⁷s).

So the Finger's deceleration covered a distance of d  = ½at² = ½(3.3m/s²)(1.8×10⁷s)² = 5.3×10¹⁴m.

(Yes, I am using the words accelerate and decelerate interchangeably. This is okay.)

In other words, the Finger fired her engines (and was detected) at a distance of 530 billion km which is 20 light days or 3500 au, near the inner edge of the Oort cloud.

Everything in that last paragraph was wrong, because the Finger's deceleration was not constant. She braked slower at the beginning, when she was heavy with fuel. This is also when her velocity was highest, so she actually covered more distance than is implied by the formula for constant acceleration.

Sorry about that, but the error is only about 5%, and fixing it requires calculus or numerical simulations. I solved it with numerical simulation, and the real answer is 540 billion km, which is 21 light days or 3600 au. But I won't show my work, because putting a numerical simulation into a free math supplement to a popular novel would be dumb.