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Did the captured Zero influence the Hellcat's design?
Short answer
Not decisively: tests of the recovered Zero did not drastically influence the Hellcat's design. The XF6F-1 prototype first flew on June 26, 1942, four days before Koga's Zero went down. They did provide knowledge of its handling, including its limits when rolling to the right and in a dive.

A timeline that doesn't add up
The prototype contract was signed on June 30, 1941. The XF6F-1 prototype first flew on June 26, 1942, four days before Koga's Zero crashed. The more powerful variant, the XF6F-3, flew on July 30, 1942, shortly after the Zero was found, but before it was thoroughly tested.
The recovery and the trials
On July 10, 1942, an American PBY Catalina piloted by Lieutenant William "Bill" Thies spotted the wreckage. The aircraft was loaded onto the USS St. Mihiel and shipped to Seattle, arriving on August 1. On September 20, 1942, two months after the capture, Lieutenant Commander Eddie R. Sanders flew it on its first test flight. Between September 20 and October 15, 24 test flights were made. Wind tunnel tests at the Langley Research Center began in March 1943.
What was learned
The tests did not drastically influence the Hellcat's design. They did convey knowledge of the Zero's handling characteristics, including its limitations when rolling to the right and when diving.
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