In fact, he explained, a mortally wounded Zarqawi was the only one of the six people in the house to initially survive the blasts of the two 500lb bombs dropped on the house, a two-storey white structure in a date grove near the town of Baquba.
Police found Zarqawi alive and placed him on a stretcher. At a certain point, when he "attempted to sort of turn away off the stretcher, everybody resecured him back on... He died shortly thereafter from the wounds he received." Maj-Gen Caldwell added that "he mumbled something - whatever it was, it was very short."
It is also not entirely clear why Zarqawi's face, as it appeared afterwards, seemed comparatively intact. "We cleaned the blood off," Maj-Gen Caldwell said, explaining how the released photos showed little more than a few cuts, although the house itself, judging from other photos, was totally flattened.
Whatever the discrepancies over Zarqawi's last moments, his death - coupled with the completion at last of a new government in Baghdad - has provided the Bush administration a fleeting opportunity to show it has a strategy for prevailing in a war which a clear majority of Americans now believe was a mistake.
The successful raid has not only deflected attention from the investigations into alleged atrocities against Iraqi civilians by US Marines. It has also proved that US and its allies now have the ability to penetrate al-Qa'ida in Iraq.
US special forces have been tracking Zarqawi ever since the 2003 invasion and have had several near misses, most recently in April during raids on the southern Iraqi city of Yusufiyah.
They finally got their man thanks to a combination of electronic and human intelligence. The key was the identification and surveillance of Sheik Abd al-Rahman, described as the new "spiritual adviser" of Zarqawi, which ultimately led the US to the Baquba safe house on Wednesday.
But that meeting, it is understood, was only pinpointed by an informant within Zarqawi's organisation, in an operation in which Jordanian intelligence played a major part. This gave the US "definite and unquestionable information about where the terrorist leader was located," Maj-Gen Caldwell said.
US forces then sought to capitalise on the "treasure trove" of information uncovered in Baqubah with raids on 17 sites around Baghdad, in which 25 people were detained and one suspected insurgent killed.
The military push is being accompanied by a major political push. On Monday and Tuesday, Mr Bush will hold strategy meetings on Iraq with his top advisers at the presidential retreat Camp David, culminating in what he called a "joint cabinet meeting" of Iraqis and Americans, involving a video conference with Nouri al-Maliki, the Iraqi premier, and other members of the new government in Baghdad.
The White House hopes that the killing of Zarqawi will give at least a temporary boost to Mr Bush's dismal approval rating, currently standing at an all-time low of barely 30 per cent. However, in keeping with his deliberately low- key reaction when he first spoke in public on Thursday about Zarqawi's death, Mr Bush was again careful not to exult yesterday.
"A war is not won with the death of one person," he said after meeting Anders Rasmussen, the Danish Prime Minister, at Camp David. The death of Zarqawi "helps a lot" and "shows we are making progress". But there would still be "death and destruction" in the period ahead.
Mr Bush also played down talk that Zarqawi's death might accelerate cuts in the 130,000-strong US military force in Iraq later this year, despite demands for precisely that from many senior Democrats and Republicans alike. Only last month, 1,500 extra troops were sent from Kuwait to the insurgent stronghold of Anbar province.
Mr Bush refused to commit to firm dates for a withdrawal of forces. This would happen only when circumstances on the ground permitted, and when Iraqis were able to take over from American troops. That, Mr Bush made clear, depended on the performance of the new Iraqi government, which on Thursday finally named a defence minister and filled other top posts. "I think we'll get a realistic idea of the capacity for standing up Iraqi troops as this new government begins to function," the President added.
Meanwhile, attention here is already turning to who will take over from Zarqawi as al-Qa'ida's de facto commander on the ground in Iraq. Maj-Gen Caldwell said the most likely figure was the Egyptian-born Abu Ayyub al-Masri. In February 2005 he was named by the US military as a close associate of Zarqawi, and a $50,000 (£27,000) bounty was placed on his head.
Masri's links with the al-Qa'ida supreme commanders Osama bin Laden and Ayman al-Zawahiri are unclear. Yesterday, al-Jazeera broadcast a video of the latter praising Zarqawi. But it appears to have been made before the latter's death.
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