S1 Text. The proof of three properties
Proof of Property 1
The definition of the maps 1 and  2 indicates that if x1i  0 , y1i  0 , x2i  0 ,
y2i  0 , then gi  A ; if x1i  0 , y1i  0 , x2i  0 , y2i  0 , then gi  C ; if x1i  0 ,
y1i  0 , x2i  0 , y2i  0 , then gi  G ; if x1i  0 , y1i  0 , x2i  0 , y2i  0 , then
gi  U ; if x1i  0 , y1i  0 , x2i  0 , y2i  0 , then gi  A ; if x1i  0 , y1i  0 , x2i  0 ,
y2i  0 , then gi  C ; if x1i  0 , y1i  0 , x2i  0 , y2i  0 , then gi  G ; if x1i  0 ,
y1i  0 , x2i  0 , y2i  0 , then gi  U  . Therefore, any two of the three curves can
uniquely determine the original RNA secondary structure, which makes the mapping
on plan X-Y non-degenerative.
Proof of Property 2
From Fig. 2 in the main context, it is obvious that the distribution of points in
 A, G, C,U } is dense while that in others is sparse, which indicates that the content
of bases  A, G, C,U } is higher than the content of bases { A, G, C ,U } in the
characteristic sequence of TSV-3. We can also immediately conclude that the AU content is higher than the CG -content, the AC -content is similar to CU -content, and
the AG -content is higher than the CU -content.
Proof of Property 3
According to the definition of maps 1 and  2 , we can obtain z11 
z12 
1 An Cn
 l and
n l 1
1 An Gn
 l , where An , Cn , and Gn are the cumulative occurrence numbers of A , C ,
n l 1
and G , respectively, in a characteristic sequence of the RNA secondary structure.
z11  z12 implies
An  Cn An  Gn

, and thus c  g . Similarly, z11  z31 implies c  u ;
n
n
z12  z31 implies g  a ; z13  z23 implies c  g  ; z13  z33 implies c  u ; z13  z24
implies a  u and others in the same way. Therefore, the variables of z11 , z12 , z13 , z14 ,
z12 , z 22 , z23 , z 24 , z31 , z32 , z33 , and z34 indicate the distribution of any two bases
frequencies of a, g , c, u  or a, g , c, u .