Same trajectory, two modes of convergence

One input. Left: blow-down at the subsequence T_j — converges (Prop 3.6). Right: blow-down at scales T between T_j and T_j+1 — never settles (Prop 3.5 fails without the full family). Bottom: H₂ tells both stories.

Stage j

λ = T/T_j+1

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r_j

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‖q_j−γ‖ (subseq)

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‖ℓ_T−γ‖ (full)

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H₂ subseq

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H₂ full

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H₂ target

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Radius R

R = 1.0

Turning angle θ / π

θ = 2.00π (full circle)

Speed

Speed: 0.12

Along the subsequence T_j — converges

Blow-down at T_j

The Prop 3.6 construction. At each T_j, the blow-down locks onto γ.

compressed bridge exact arc target γ

Between T_j and T_j+1 — drifts

Blow-down at T between T_j and T_j+1

The same trajectory viewed at scales between the special times. Radius R/λ, extent θλ — the shape changes continuously and never settles.

current ℓ_T target γ

H₂ — both modes on one axis

Green dots: H₂ at the subsequence T_j — converges to the dashed target. Red sawtooth: H₂ at scales between T_j and T_j+1 — oscillates with overshoot, never settles.

subsequential H₂ (dots) full-family H₂ (sawtooth) target (dashed)

Same trajectory, two modes of convergence

Blow-down at Tj

Blow-down at T between Tj and Tj+1

H₂ — both modes on one axis

Blow-down at T_j

Blow-down at T between T_j and T_j+1