'Clear Sharp' doesn't seem to be working I'm having a weird issue where 'Clear Sharp' seems to do nothing to edges that Blender half recognizes as sharp. 

I have a model that does not show to have any edges marked as sharp: 



Nothing seems wrong when in object mode either: 



But when I render the model without a subsurf modifier, I get this as a result: 



And when I go to the Select Menu and hit 'Sharp Edges', those same edges become selected:



However neither doing so from this selection nor selecting all will clear them if I do a Mesh > Edge > Clear Sharp

Before I discovered this was the case, I thought something was just wrong with the normals, but smoothing vertex/edge/faces did nothing and neither did recalculating normals or checking auto smooth or adding an edge split modifier. I'm not sure what else to try.

Here's the blend file: https://www.dropbox.com/s/9h6arxu9fn74yrm/SharpEdgeProblem.blend?dl=0

edit: A few extra things to note: this topology was the result of the add-on RetopoFlow 2.0, but I have had this issue before with a different model simply by importing an OBJ file and deleting edge loops.
There are no sharp edges on the mesh. What you are observing there is geometry obstructing itself or forming some unexpected shadows because of the way it is triangulated. When rendered, the mesh is triangulated and some faces that may look smooth when you rotate them in the 3d view get divided into triangles that obstruct each other looking at them at some angles or produce unexpected shadows: 

 

There is nothing to do about it - the geometry is just not detailed enough to be rendered nicely, you do need to use Subsurf modifier or modify the geometry some other way.