Shape gallery — any Lipschitz path can be realized

Proposition 3.6: every Lipschitz path based at the origin occurs as a macroscopic limit. The construction is shape-agnostic — each path carries its own H₂. Explore several and watch all traces converge.

Shape

—

Stage

j = 3

r_j

—

Endpoint

—

H₂[q_j]

—

H₂ target

—

Select a target path — try several; all H₂ traces accumulate on the bottom panel

Sequence T_j =

Presets:

Speed

Speed: 0.22

1. Blow-down q_j(u) in V₁

Each target shape is Lipschitz and based at the origin. The construction realizes it as the macroscopic limit of a bounded input.

compressed previous affine bridge exact target target (dashed)

2. H₂[q_j] / Z vs stage — all shapes

Each shape's trace converges to its own H₂ target — but only when r_j → 0. Try 2^j or j² and watch the traces plateau short of their targets.

Shape gallery — any Lipschitz path can be realized

1. Blow-down qj(u) in V₁

2. H₂[qj] / Z vs stage — all shapes

1. Blow-down q_j(u) in V₁

2. H₂[q_j] / Z vs stage — all shapes