Layer-by-layer convergence in the Engel group

The corner path γa,b first follows X₁ then turns into X₂. Theorem 3.4 gives convergence at each depth: H₁ (displacement), H₂ (signed area via Z), H₃ (area moment via W).
Stage
j = 3
rj = 1/j
—
u
0.000
H₂[qj]
—
H₂ target
—
H₃[qj]
—
H₃ target
—
a = 1.0
b = 1.0
Speed: 0.18

1. Blow-down qj(u) in V₁

The corner path: first along X₁, then a right-angle turn into X₂. The turn creates signed area (H₂) and area moment (H₃).
compressed previous affine bridge exact corner target γ (dashed)

2. Depth-two: H₂[qj] / Z (signed area)

Converges to −ab/2. The corner sweeps area in the [X₁,X₂] = Z direction.

3. Depth-three: H₃[qj] / W (area moment)

Converges to a²b/12. The X₁ leg precedes the turn, creating a moment in the [X₁,Z] = W direction. This layer is invisible when [X₁,X₂] is central.