User blog comment:Hyp cos/SCG(n) and some related/@comment-1605058-20140915163630/@comment-11227630-20140916044019

Goucher's sequence has my level $$\vartheta(\Omega^{\omega2})$$ graph in the 3rd one, so we cannot have any triple fusing cycles later.

In his graph ordering, two triangle fused has level $$\vartheta(\Omega^\omega)$$, triangle fused square has level $$\theta(\Omega^\omega,1)$$, triangle fused k-gon (k>4) has level $$\vartheta(\Omega^\omega+\Omega^{k-5})$$. Then two triangle fused with each one linked to a dot has level $$\vartheta(\Omega^\omega2)$$, two triangle fused with each one linked to a stick has level $$\vartheta(\Omega^\omega3)$$. Then two square fused has level $$\vartheta(\Omega^{\omega+1})$$, square fused pentagon has level $$\vartheta(\Omega^{\omega+2})$$.

Without multiple fusing cycles it won't go far.