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To ensure compliance and maintain proper documentation, I recommend gathering your proof of employment with the sponsoring employer when your green card was approved. It's important to request written confirmation from your new employer showing they were prepared to file Supplement J before your transition. Keep any prepared  Form I-485 Supplement J documentation, even if it wasn't filed due to the timing of your green card approval. Additionally, maintain detailed job descriptions from both positions to demonstrate they were "same or similar" positions as required by immigration law. Store all these documents securely for the long term, as you may need them during the naturalization process to verify compliance. This documentation will help protect you and demonstrate that you followed proper procedures during your employment transition, even though the green card approval occurred before Form I-485 Supplement J could be filed.

If you have an EB-1A-based EAD and a pending I-485, starting an unrelated business (such as teaching) is risky. Your EB-1A petition asserts that you intend to work in your field of extraordinary ability. Shifting to an unrelated field could raise questions about your original intent and impact your green card application.

Additionally, traveling on Advance Parole in this situation may pose some risks. It’s advisable to either have a job offer or a clear plan to work in your field (such as consulting with clients) before traveling. Proceed with caution.