Same trajectory, two limits

One input, two subsequences. Odd stages see a semicircular arc; even stages see a straight segment. Same horizontal endpoint — different H₂. (Remark 5.3)
Current stage
j = 3
Target
—
rj
—
u
0.000
‖qj−target‖∞
—
H₂[qj] / Z
—
H₂ semicircle target
—
Shared endpoint
—
R = 1.0
Speed: 0.18
Odd stages → semicircular arc

Blow-down at Tj, j odd

Dashed: semicircle target η₂. Both targets share endpoint (2R, 0).
Even stages → straight segment

Blow-down at Tj, j even

Dashed: straight target η₁. Same endpoint — but H₂ = 0.

H₂[qj] / Z vs stage

Two traces from one trajectory. Purple: odd stages converge to πR²/2 (enclosed area of semicircle). Red: even stages converge to 0 (straight line encloses nothing). Same endpoint, different development.
odd stages (semicircle) → πR²/2 even stages (straight) → 0