Learning goal: the source and target are the same ring, but changing the endpoint pairing can make an intermediate law _t collapse to lower dimension. Then an ordinary ambient-space density and its score may not exist. The γ_t z term can add full-dimensional interior noise.

Pale blue: source ring p₀; pale orange: target ring p₁; white: current x _t Green: twelve paired conditional paths

For antipodal pairing and γ=0, move t to 0.5; the ring collapses to one point. Both principal spreads vanish, so an ordinary two-dimensional density and ∇log p _t are unavailable there. Positive interior noise instead produces a cloud with thickness: this is what the γ _t z term supplies. Same-point pairing stays on the ring without collapsing to a point, though the ring itself is still singular relative to two-dimensional area. Deterministic pairing need not cause additional collapse; density-based formulas still require the appropriate regularity and reference measure.