The trace only detects the magic of the caster. The reason that Harry got the note for Dobby’s use of magic is because Dobby used inscrutable house elf magic to fake Harry’s magical signature.
Or, alternatively, they knew that tracking every spell everywhere there was a child would result in them being inundated with reports, be a violation of privacy, etc., so they have it set to disregard spells cast in known magical areas—basically allowing carte blanche for wizard-raised students, while completely shafting the muggle-raised. That also seems consistent with their policies.
As yet another possibility (and that I was going to use in my HP fic that died with my old comp) the Trace is twofold, both the spell on their wand and the spell on a location, generally their home. This secondary spell is the one that actually detects the magic, while the first merely serves to identify the caster as underaged. Because Dobby would still have to fake something, it would appear to suffer from a complexity penalty, but it would allow for...ah.
I had thought that Riddle had killed his family to create the ring horcrux while he was in school and so should still have the Trace, which research backed up, but which also revealed that he used the wand of the person he was framing for the deeds. Which is rather unhelpful, even if it is obvious. Though he also False Memory-Charmed the man to think that he had committed the deed, which he couldn’t use that man’s wand for since it was going to be checked, and he was never caught for -that-. That does seem to imply that location is significant.
There is a simpler possibility.
The trace only detects the magic of the caster. The reason that Harry got the note for Dobby’s use of magic is because Dobby used inscrutable house elf magic to fake Harry’s magical signature.
Or, alternatively, they knew that tracking every spell everywhere there was a child would result in them being inundated with reports, be a violation of privacy, etc., so they have it set to disregard spells cast in known magical areas—basically allowing carte blanche for wizard-raised students, while completely shafting the muggle-raised. That also seems consistent with their policies.
As yet another possibility (and that I was going to use in my HP fic that died with my old comp) the Trace is twofold, both the spell on their wand and the spell on a location, generally their home. This secondary spell is the one that actually detects the magic, while the first merely serves to identify the caster as underaged. Because Dobby would still have to fake something, it would appear to suffer from a complexity penalty, but it would allow for...ah.
I had thought that Riddle had killed his family to create the ring horcrux while he was in school and so should still have the Trace, which research backed up, but which also revealed that he used the wand of the person he was framing for the deeds. Which is rather unhelpful, even if it is obvious. Though he also False Memory-Charmed the man to think that he had committed the deed, which he couldn’t use that man’s wand for since it was going to be checked, and he was never caught for -that-. That does seem to imply that location is significant.