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Intresting thing. Quickest way for me would be crop with that 1:1,618 (cm) and leave blank resolution. Than will change resolution as you say blow up a lot. Thats total okey.

1. Make all your golden crops.

2. Put them all in one folder.

3. Record Action which change Image Size that way:

[ATTACH]5407[/ATTACH]

set resolution to your origin (assuming all images had same resolution). 72ppi or 300ppi.

That will restore origin resolution and set correct dimension (however for me at 10cm it gaved 9,98 cm so little error but i think you can live with that).

4. Make batch to your croped folder with that action.

All your images should be now golden crop and "human" dimension.