This paper describes a simple, universal pattern in how a single remnant horizon first appears when two black holes merge. The authors show that, in the fully nonlinear stage of a merger, the newly born common apparent horizon splits into two nearby branches whose separation grows like the square root of time after formation. That scaling and a shared linear motion together make the local shape look like a tilted parabola, independent of special symmetries of the system.

The work studies marginally outer trapped surfaces (MOTSs). A MOTS is a closed surface where outgoing light rays neither expand nor contract, and a connected stack of MOTSs through time is a marginally outer trapped tube (MOTT) — a useful, locally defined black‑hole boundary used in numerical relativity. The key mathematical object is the MOTS stability operator. This operator describes how the outgoing expansion changes if the surface is nudged outward. The authors show that, at the moment the first common horizon appears, this operator loses invertibility: its principal eigenvalue reaches zero and the corresponding eigenfunction can be chosen strictly positive. That spectral change controls what can happen next.

Using a standard reduction method from nonlinear analysis (Lyapunov–Schmidt reduction), the authors reduce the complicated geometric problem to a small set of leading terms. Those terms produce a square‑root branch separation together with a shared linear drift in time. Concretely, nearby spatial slices after formation intersect the smooth MOTT in two branches — an outer and an inner common horizon — whose distance from each other scales as (t − t*)1/2, where t* is the formation time. Other horizon quantities that respond to the same critical mode inherit the same square‑root behaviour and linear term.

The paper tests these predictions in three numerical binary‑black‑hole simulations, including a challenging case that is eccentric, precessing, and has unequal masses. In all three runs the worldtube geometry and quasilocal scalar measures of the horizon follow the predicted scaling. When the authors include the next‑order term in their fits, the fitted power law recovers the 1/2 exponent to within half a percent, and different horizon diagnostics give formation times that agree to about 5×10−5 M (M is the total mass scale used in the simulation). The authors also report a correlation between horizon shear (a measure of how outgoing light rays are distorted at the horizon) and the gravitational‑wave news, which suggests the formation event might leave a short imprint on the waveform.