What are the rules of simultaneous investment immigration program participation?

That is an interesting question. There are no rules/restrictions regarding investment immigration programs in multiple countries; however, as far as U.S. immigration laws are concerned regarding the concept of permanent residence, (which is what you would be given when you successfully process through the immigrant investor processing), you are declaring your intent to maintain permanent residence in the U.S.(ie., presence in the U.S. (more than six months out of the year), maintenance of residence/home in the U.S., paying taxes in the U.S., etc). Sounds like in your case this would not conflict with your residence requirement in Portugal since it only requires you to be in Portugal seven days a year.

Once you obtain a U.S. green card, you have permanent residency, which may be revoked if you are out of the U.S. for more than one year (and maybe less) or fail to pay U.S. taxes. Having permanent residency in multiple countries may also be an issue, as your green card can be revoked if you move to another country intending to live there permanently.

U.S. lawful permanent residence (green card) is for individuals who wish to reside permanently in the United States. There is no prohibition against a green card holder owning property, making investments, or having residences in other countries, but the green card holder should have permanent residence in the United States. Certain activity, including extended absences from the U.S. of longer than six months, may raise a presumption that a green card holder has abandoned his or her intent to permanently reside in the United States. As long as the person maintains the intent to be a U.S. permanent resident and does not abandon the green card by failing to enter the U.S. frequently enough, there is no prohibition against participating in investment programs in multiple countries. Those other countries might have different rules, so it would be wise to check with a lawyer admitted in those jurisdictions.

There are minimum residency issues which need to be addressed, along with other complicated immigration issues you need to consider carefully prior to making a move.