Sharpening on cropped or uncropped images

The sharpening controls inside Lightroom Classic work on pixels. For example, you can change the pixel radius over which sharpening is determined. Amd the recommendation is to work at 1:1 zoom (or failing that, to use the small 1:1 preview window inside the Detail panel) so that you can see what is happening with the pixels and the picture content in its own local terms.

How large or small a part this local area of pixels plays within the overall image, including how prominently the sharpening of the picture details will show, is a separate matter. But that is no different than how prominently the pixels themselves will show, carrying with them (as they zoom larger in effect, when you crop smaller) any lens softness or diffraction or demosaicing issues which this same sharpening is attempting to mitigate. 

There's another aspect though: when exporting, output sharpening can be selected off / low / medium / high. For this purpose, if you crop down to a 4MP selection and then choose export settings amounting to 4MP, your export is more or less happening at original pixel resolution. But if you don't crop down from 24MP, and your export settings amount to 4MP, meeting that requires considerable downsizing, Output sharpening will 'understand' the practical difference between different export scenarios, and tries to allow for that towards a similar looking result regardless (when considered in terms of the final output pixels). Loosely: those will look the same amount sharper than they would otherwise have been, no matter whether you are exporting FROM 4MP or 24MP. Naturally that will also look different depending whether you are exporting TO 4MP ro 24MP.

The best way to evaluate all this, and there is a lot of sheer preference involved, is to simply try it.

I guess you could call it Schrodinger's Image Editor, in that any edits, including sharpening and cropping, are simultaneously there (as parametric edits you can preview) and also not there (not affecting the original); the edits are permanently applied only to an exported copy.

I just tried it, so it’s easy to confirm. I cropped an image, applied enough sharpening to be obvious, then open up the crop, and the sharpening is already in the area that was formerly cropped out.

Probably a good general way to think about it is:

If a non-masked edit is applied to a cropped image, it applies to the entire image, regardless of cropping. Because even the crop is permanently applied only to an exported copy.

So if we’re talking spatially, only a mask will persistently restrict the area of an adjustment.

The only exception I can think of is Post-Crop Vignetting, which by definition always stays within the crop area.

The crop, like all edits, is totally nondestructive. All the edits are just instructions for processing upon Export (or Open in Photoshop). If and when you render the raw, it is what it was asked to be: cropped (maybe to 4MP) and sharpened for that crop. The original is untouched.