The surgical criterion detects when/where to surgically remove spacetime regions before singularities form. The covariance is what makes it a genuine geometric trigger for surgery, valid for all observers, in all reference frames.
No horizon finding needed, the cut surface is the emerging horizon.

1. Modified Curvature ∥Rμν∥g

• Measures departure from smooth soliton geometry

• Includes matter + geometric flow terms

• Blows up at singularities

2. Causal Distance dg(p,∂J−(p))

• Maximum proper time from point pp to its past light cone boundary

• Shrinks as region becomes causally isolated

• Zero at singularities (light cone pinches off)

3. Threshold Θ

• Dimensionless constant

• Calibrated so surgery happens just before classical singularity

• ~1 in Planck units

Curvature × Distance² is:

• Dimensionless → universal threshold

• Scale-invariant → detects shape, not size

• Monotonic during collapse → guaranteed trigger

• Finite at singularities → triggers at finite value

∞ (curvature) × 0 (distance^2) → finite

Physical

During gravitational collapse:

1. Curvature increases (star compresses)

2. Causal distance decreases (light cones narrow)

3. Product grows

4. When ≥Θ→ cut along ∂J−(p) → replace with smooth soliton

The boundary ∂J−(p) becomes the surgical surface, automatically the apparent horizon.

One invariant does:

1. Singularity detector (flags bad regions)

2. Horizon locator (finds cut surface)

3. Surgical trigger (tells when to cut)

Eliminates: Horizon-finding elliptic solves, singularity-handling hacks, angular momentum loss during excision.