My cat scratches the carpet right outside the bedroom door every morning at 5am.

He has trained you very effectively, and he knows it. I say that without judgment because it is one of the clearest examples of how intelligent cats are about finding what works. At some point, probably early on, the scratching produced a result, either you came out, called to him, or he got access to you, and he has been refining the strategy ever since. The spot outside the bedroom door is deliberate. It is the place most likely to reach you.

Ignoring it is the right instinct but it is very hard to sustain at 5am, and cats are extraordinarily persistent when a strategy has worked before. The scratching post next to the spot will not solve this on its own because the scratching is not about the carpet. It is a communication tool aimed at you.

What tends to work is a combination of making that specific spot genuinely unappealing, a heavy mat he cannot get traction on, or a textured surface he dislikes, while also adjusting his feeding schedule so that the early morning wake-up call loses its purpose. If he is waking you for food, an automatic feeder set for just before his usual start time can interrupt the cycle. If he is waking you for company, the solution is different. If you want to map out his routine and what has been tried in detail, the Work With Me assessment is a good place to work through this properly.