Subsequential convergence — Proposition 3.6

Along the subsequence T_j, the blow-down converges to the target arc γ. Error → 0, H₂ → target.

Stage

j = 3

T_j

—

r_j = T_j−1/T_j

—

0.000

Exact arc begins

—

‖q_j−γ‖_∞

—

H₂[q_j] / Z

—

H₂ target

—

Sequence T_j =

Presets:

Radius R

R = 1.0

Turning angle θ / π

θ = 2.00π (full circle)

Animation speed

Speed: 0.18

advance j automatically show target γ

Macroscopic time decomposition at stage j

0 r_j 2r_j 1

compressed previous stage affine bridge exact circular arc target γ (dashed)

The three construction pieces shown in color. As j increases, the compressed and bridge segments shrink toward u = 0.

For sequences with r_j → 0 (e.g. j!), the error decreases to zero — convergence succeeds. For bounded r_j (e.g. 2^j), it plateaus above zero.

Dashed green: target value −(R²/2)(θ − sin θ). The central coordinate locks onto the target.