Officers don't receive their authority from the President, they receive their authority from the Constitution and statutory law, to include DOD directives and the UCMJ.
So long as Obama is in that office, irrespective if he was installed in the office under fraudulent means, he still wields the authority of the office, to include the authority vest him by the USC as Commander-in-Chief. It is that simple. I can't make it any simpler.
If he is not eligible the Constitution says he can’t be in that office.
The US Constitution doesn’t say that no person shall be eligible unless they’ve been sworn in. Being sworn in has nothing to do with whether a person is eligible for the office. You can argue that the person is a “de facto” president for purposes of stability, but a “de facto” president is not the same thing as a CONSTITUTIONAL president, and that is what officers swear to uphold - the Constitution.
If Obama is not the right age, is not a natural born US citizen, and has not lived in the US long enough, then he is not a Constitutional president - the only kind of president the officers’ oath will allow them to obey.