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Puzzling intonation issue: first 4-5 frets note sharp

Recently, three different guitars have come to me, the owners complaining of sharp intonation on the first 4-5 frets. All newish electrics, one Telecaster, one Gibson ES-137, one Epiphone. All three have what I consider very tall frets(.053" on the Gibson and Epiphone).

I checked the nut slot, and 12th fret intonation, and adjusted where necessary. On the Tele, I installed a NOS Earvana compensated nut(but the owner was not happy with the result). On the Epiphone, my lowering the nut slots helped slightly. The Gibson needed neither adjustment, but intonated sharp.

The Tele owner uses extra lights(with the .009" first string), the Gibson/Epi owner uses .011"sets. 

I went so far as to order a Gibson fret scale from Stewmac, to check the fret layout on the Gibson: the fret layout on the Gibson does not match any of the three 'short' scales on the scale(but then, neither does the 1951 Gibson LG2 I also have in my shop at the moment).

I am puzzled by this problem, and do not have a good answer. I have tried to research this issue online, and I'm aware this is a common complaint on several guitar forums.  Has anyone here dealt with this complaint, and found a solution?

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Sounds like the frets are misplaced. If I were you I would use Stewmacs fret calculator and try to match the numbers with the fret positions on the guitar. The open string length for the treble e string, or twice the length from the nut to the 12th fret, is a good starting value for the calculator.

If so, remove the faulty frets and glue matching wood in the cuts. Then saw new cuts for the frets in the right position.

I remember someone saying that gibson had a slightly odd scale which changed when they went to Montana , that scale is still used by the old factory (Heritage I think ). That may have been an input from Finger Lakes ? Also I remember about 10 years ago I had a run of Fender electrics (USA) where I had to take 0.012"off the fretboard at the nut to get them intonated , then they were fine , and I run nuts as low as workable .

there is a thing about gibson scales being sort of "off"

not "gibson" 24 3/4 inch scale as found on epiphone and many others, but actual gibson guitars. first off it's closer to (but not exactly) 24.6", and second, luthier david collins has hipped us all to how the actual ratios are a little weird, like they didn't do the math right or there were tolerance errors in the early fret slotting machines and they just kept it.

unbelievable i know but it seems to work fine, including the first few frets not being sharp if it's set up right. (just went through one today, a new 2020 les paul, and after filing the slots down where they belonged and thinking about this question i made a point of checking with my super-accurate turbo-tuner and yeah, first four or five frets were not appreciably sharp.)

Yes it was about calculating the spacing using the rule of 18 where the frets advance by 1/18th of the string length versus the 12th root of 2 method . Imagine working that out long hand !

After, checking & rechecking the fret layout, I removed the nut, and trimmed 1mm off the fingerboard. Made a new nut, and the intonation is very much improved, and the owner is very pleased. I think the very tall Gibson factory frets are not helping(it's absurdly easy to push the strings sharp, when fretting), but that's a separate issue. Thanks for all your replies.

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