Article II, Section 2 states: "The President shall have Power to fill up all Vacancies that may happen during the Recess of the Senate, by granting Commissions which shall expire at the End of their next Session."
Article I, Section 5 states: "Neither House, during the Session of Congress, shall, without the Consent of the other, adjourn for more than three days."
The House did not consent to a recess. The Senate, therefore, cannot be in recess.
Furthermore, no reasonable person could assert the Senate can pass bills while in recess, which they did (and Obama signed without objection to the status of the Senate which passed it) in the same status they were in when Obama made his "recess" appointments.
Since it is indisputable that A: the House did not consent to a recess, and B: the Senate was capable of conducting business (as evidenced by the passage of a bill the president then signed into law), then it is not possible the Senate was in recess and unable to conduct business.
If the Senate was not in recess and was able to conduct business, the recess appointments clause cannot be invoked and the appointments were unconstitutional.
QED.