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In the coordinates of L30, squared length is $q(a,b)=a^2+ab+b^2$. The exact checker enumerates all 36 pairs. Their squared lengths and multiplicities are

$$\begin{array}{c|rrrrrrrr} q&1&3&4&7&12&13&19&21\\\hline m_q&7&3&2&8&5&6&4&1. \end{array}$$

The only collinear triple is $(1,2,3)$, with indices starting at zero. The only cocircular quadruples are $(0,2,3,4)$ and $(4,6,7,8)$; each has no collinear triple, so each is a genuine circle.

In each distance class take its lexicographically first edge as a base, and subtract its squared length from every other edge's. This gives 28 polynomial equations in 18 coordinates. The derivative of an edge with difference $(a,b)$ is $(2a+b,a+2b)$ at its first endpoint and its negative at the other endpoint.

Fix both coordinates of points 0 and 1. These points are distinct, and there is a unique orientation-preserving similarity taking any nearby ordered pair to this fixed pair. Thus this is a continuous local gauge for the four similarity parameters. In the remaining coordinate order $(a_2,b_2,\ldots,a_8,b_8)$, use the 14 equations with edge/base pairs

$$\begin{split} &(03,01),(24,01),(28,01),(48,01),(56,01),(67,01),\\ &(23,13),(57,13),(38,37),(07,02),(08,02),(17,02),\\ &(26,02),(27,02). \end{split}$$

Their exact square Jacobian has determinant $-1102248$. The complete matrix is retained as an input to the checker, which recomputes it and its determinant from the points. The inverse function theorem therefore isolates the original gauged configuration among solutions of these equations. In particular, a sufficiently close realization of the entire distance partition is similar to it.

Choose disjoint open intervals about the eight original positive lengths. By continuity, in a sufficiently small neighborhood each edge remains in its original interval, and each interval still contains an edge. A configuration there with at most eight distinct distances must have exactly one distance in each interval. No original class can split, so the entire distance partition is preserved. Apply the preceding local uniqueness after the continuous gauge. Similarities preserve collinearity and cocircularity, proving the final assertion.