Box Projection UV Mapping on single vertex mesh?

I'm making a procedural generator, for example in the picture doing some caves, and I'm texturing using box projection/cubic projection, which is awesome, but it leaves a line of triangles with wonky texture.

you can see the areas below the yellow dots are UVmapped with a side texture and areas above the red dots are mapped to a top texture.

I want to make a 2nd vertex on all the red dots so that the distance between the yellow dots and the red dot is 0, and there is a seam at that point.

Do you have a clear mind about what the task is? do I have to make a new triangle for every wonky triangle? And a new vertex for each corner of the triangle? how could I measure it, and make the triangles, and do all the comparisons? are there some tricks I could use?

   //Box Projection textures
     var mesh : Mesh = GetComponent(MeshFilter).mesh;
     var vertices : Vector3[]  = mesh.vertices;
     var normals : Vector3[]  = mesh.normals;
     var uvs : Vector2[] = new Vector2[vertices.Length];
 
     for (var i = 0 ; i < uvs.Length; i++){
     if ( Mathf.Abs(normals[i].x) > Mathf.Abs(normals[i].y) && Mathf.Abs(normals[i].x) > Mathf.Abs(normals[i].z) ){ 
         uvs[i] = Vector2 (vertices[i].y, vertices[i].z);}
         
     if ( Mathf.Abs(normals[i].y) > Mathf.Abs(normals[i].x) && Mathf.Abs(normals[i].y) > Mathf.Abs(normals[i].z) ){ 
         uvs[i] = Vector2 (vertices[i].x, vertices[i].z);}
         
     if ( Mathf.Abs(normals[i].z) > Mathf.Abs(normals[i].x) && Mathf.Abs(normals[i].z) > Mathf.Abs(normals[i].y) ){ 
         uvs[i] = Vector2 (vertices[i].x, vertices[i].y);}
         
     Debug.Log(  " vertex  "+vertices[i] +  " uvs " +uvs[i] );                
 
     mesh.uv = uvs;
 }