Refuted: a homotopy that is immersive at every earlier time is a regular homotopy

For every $t<1$ the slice $H_t(\theta )=((1-t)\cos \theta +t,(1-t)\sin \theta )$ has derivative $H_t^{\mathrm{\prime }}(\theta )=(-(1-t)\sin \theta ,(1-t)\cos \theta )$ of norm $1-t>0$, hence is an immersion with image the circle of radius $1-t$ centred at $(t,0)$; at $t=0$ this is the standard unit circle.

At $t=1$ the slice is the constant map $H_1(\theta )=(1,0)$, whose derivative vanishes identically: the differential has rank $0<1=\dim S^1$ at every point. So the family contains a rank drop at the final time and $H_1$ is not an immersion.

Therefore $H$ is not a regular homotopy in the sense of [F1], since a regular homotopy requires every slice to be immersive, and it cannot certify any regular-homotopy claim: in particular this shrinking family does not show that the circle is regularly homotopic to a point, because the point is not an immersion and the rotation number (here $1$ at the initial slice) would have to remain constant while no rotation number is defined for the final slice. The failed conclusion is exactly the rank-drop phenomenon separated in [F3]: the final slice fails the injective-differential condition.

The family $H$ is smooth in $(\theta ,t)$, but its final slice is constant. Any perturbation that keeps this final slice still has zero final derivative, so it still fails to be a regular homotopy, independently of its size. Self-intersections are compatible with immersive slices; a rank drop is not.