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If I put a capo on 3, retune everything dead on there and check the first frets then: same thing. They are sharp on the first three. Worse on fret one (4 in this case), getting better as you move toward 6...
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This happens because fretting the string near the nut (or the capo) bends the string at a sharper angle, causing the tension to increase in a nonlinear fashion.