What is a Code of Criminal Procedure 46.05 Review?

Texas's Court of Criminal Appeals issued opinions Wednesday, June 27, 2012 holding that Jonathan Marcus Green was competent to be executed Saturday, June 30, 2012. They changed their minds and reversed the stay on the 30th.
Let's start with a chronology.

• Christina Neal  disappeared around June 20, 2000.
• Green was convicted in July 2002 of her murder and was given the death penalty. 
• June 23, 2010 Green filed his first subsequent application for a writ of habeas corpus, claiming to be incompetent and requesting a Texas Code of Criminal Procedure article 46.05 hearing to determine his competence.
• June 28, 2010 the district court held a competency hearing and found him competent. Green appealed this ruling and also filed for a writ of habeas corpus accompanied by a brief arguing that competency-to-be-executed claim are cognizable on habeas corpus. The CCA stayed Green's execution and ordered the trial judge to file with them a written clarification of the standard she used.
• The CCA ruled June 27, 2012.
• They re-stayed the execution on June 30, 2012.

Judge Paul Womack delivered the opinion of the court in which Presiding Judge Keller, and Judges Meyers, Keasler, Hervey and Cochran joined. Judge Tom Price wrote a concurring opinion joined by Judges Johnson and Alcala. Everybody agreed that Green had not shown his incompetence to be executed, that Green did not deserve any relief on his habeas application, and that it was OK in a hearing on a motion to recuse the trial judge that the trial judge not be required to present herself as a witness. The majority believed that the review of the article 46.05 hearing was like a direct appeal, giving great deference to the trial judge. The concurrence argued that the 46.05 review was like a post-conviction habeas corpus review, giving less deference to the trial judge. It would follow from the majority's view that habeas would not lie for competence since a habeas is not appropriate where there is a remedy by direct appeal. The concurrence group agreed that habeas didn't lie in this case, but argued that there might be circumstance when it would, particularly if the incompetence manifested itself for the first time less than 20 days before the execution (46.05 requires that the defendant give at least 20 days' notice before execution to get a reviewable hearing). The majority ruled that the recusal matter was filed too late- you can't file a refusal complaint after you've submitted your appeal. The concurrence noted that on post-conviction habeas, defendants regularly brought up additional matters between the first submission of a writ application up until the habeas had been ruled on. The concurrence said that the recusal motion wasn't late, it was just unmeritorious.
Green v. State, Nos. AP-76,374, AP-76,376 & AP-76,381 (Tex. Crim. App. June 27, 2012).