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Your LimeSurvey version: Version 5.3.18+220530
Own server or LimeSurvey hosting: Own server
Survey theme/template: LimeSurvey Bootwatch Theme
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We are running into a 504 Gateway Time-out error at 30 seconds when attempting to load the survey logic file on our largest survey. All other survey logic files load as expected, and no issues have been observed in use of the survey where the logic file is timing out with a 504. Any suggestions on what setting we should extend to allow more processing time?

I would have a look at the maximum execution time setting in PHP as the first point.

www.simplified.guide/php/increase-max-execution-time

It is probably at 30s (php default?) and Limesurvey probably needs more time to process the survey logic file of this large survey.

See what happens if you increase it.

It's something else -- max_execution_time is set to 1800.

Gateway timeout is an generic error message. What webstack is used?
Webserver, php handler or an reverse proxy can cause the timeout.

The 30seconds is an indication for an Apache2 timeout.

Syntax: TimeOut seconds
Default: TimeOut 300
Context: server config, virtual host

Consult the documentation for your apache server.

It is indeed Apache2, however it is a cluster of web nodes behind a load balancer (cloud site). The load-balancer has a hardcoded timeout of 30 seconds.