Strange black shading. Cannot fully light mesh. (Cycles) Here is the viewport in material view.  The node setup is just a PNG texture into a diffuse shader.



And this is in rendered view.  I've tried using environmental lighting as well as a directional light, and while the lit parts become brighter/washed out, the darker parts stay dark.

Normals are correct and there is no z-fighting.  Ambient Occlusion is not enabled.  Changing to Flat shading makes it worse.

Here's the .blend

Update:  Any modifier that alters the mesh (bevel, subdivide, etc) make the problem go away instantly.
You have Custom Split Normals Data added to the geometry, to fix this, while selecting the object, go to Data Tab in Properties Panel, scroll to find Geometry Data, and click Clear Custom Split Normals Data.

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