That was close!

Starting with two points P1 and P2 that are 1 unit apart, the following construction (which makes use of only 6 circles) places P10 and P11 1.436x10-7 unit apart!

C1 = circle(P1,P2)
C2 = circle(P2,P1)
P3 = intersect(C2,C1)
P4 = intersect(C1,C2)
C3 = circle(P3,P4)
P5 = intersect(C2,C3)
C4 = circle(P5,P2)
P6 = intersect(C2,C4)
P7 = intersect(C4,C3)
C5 = circle(P6,P7)
P8 = intersect(C5,C1)       // zoom P3 for this one
P9 = intersect(C4,C2)
C6 = circle(P8,P9)
P10 = intersect(C1,C6)      // not quite P2! zoom a lot to see the intersection
P11 = intersect(C6,C4)      // zoom a lot more to differentiate from P10!

Here's a succession of zooms, with P10 & P11 at the center of each image.

4 zooms