It is a good thing that you got a better job for yourself. However, you leaving your previous work and going for the new one without giving a formal notice to your employer is a breach of your employment contract.
As you have stated, you are to submit your resignation letter within 4 weeks to you leaving so that your vacancy will be filled at the workplace within a reasonable time.
Because your boss can decide to escalate the matter and sue you for a breach of contract which might not be a good story for you.
Not only suing you, the company you previously worked with can send a letter to this new company stating that you have a questionable personality as you have left your position unannounced and it’s likely you might do the same to them. This can bring up questions that might lead to terminating this present contract and if care is not taken, subsequent contracts might be affected.
Working etiquette should be a 100% skill.
Since your boss is already aware, I suggest you meet with him/her, apologize for your unannounced absence and formally send a resignation letter to them and leave.
You should always try to leave an environment with a clean slate.
Chuks
University of Bradford Alumni Rep