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“There were several opportunities where we could have engaged with 40 mm ensuring zero [collateral damage estimate] on any buildings,” the navigator testified. “The opportunity was definitely there for us to engage those two guys or even provide containment fires to try to slow their movement.”

Investigator: “Did you ask to engage them?

Navigator: “Yes, sir.”

Investigator: “And it was denied, right?

Navigator: “Yes, sir.”

AC-130 commander: “I think he spoke with the Ground Force Commander and he said, ‘No. No-go. Just maintain eyes-on.’”

Mr. Zinke, the former SEAL, said he talks to guys coming back home who are frustrated because the rules of engagement “are too restrictive.”

“I’ve always been a champion of, if we are going to fight, fight to win,” said Mr. Zinke, a candidate in the Republican primary for a House seat in Montana. “And you’ve got to give our troops that are in harm’s way every tool and every advantage that is possible.

“And when you start restricting [rules of engagement] — when you limit our ability to fight at night, where you restrict the ground commanders’ ability to react quickly without having to go up the chain of command and also when you’re forced to bring along the Afghan forces who are notorious for the lack of security — then I think it puts troops in greater risk.”

Mr. Kerry said last week that the security deal demonstrates to Mr. Karzai that Washington is listening to his concerns about civilian deaths.

“It’s very important for President Karzai to know that the issues that he’s raised with us for many years have been properly addressed, and it’s very important for us to know that issues we have raised with him for a number of years are properly addressed,” the secretary of state said.

Lisa Curtis, a foreign-policy analyst at the Heritage Foundation, said there are some pluses. Mr. Karzai is agreeing to some night raids, and Mr. Obama’s language “does leave room for the U.S. to conduct counterterrorism missions against high-value targets,” she said.

Left unsettled is Mr. Karzai’s call for a delay in signing the agreement until the spring.

“That would almost certainly be a deal-killer from the U.S. perspective, as the U.S. needs to begin planning for any residual force presence no later than January 2014,” Ms. Curtis said.