The fastest walking route from T4 to LAXit is probably thru the parking garages to T3 then north side terminal sidewalk. This is basically two straight lines and avoids all red lights and the slight detours required walking from T6 to T2 via theme building.

Fastest route: T4 departure (upper) level to T4 garage connector, go down a level and walk to T3 garage to T3 garage connector, go up a level to T3 departure level, then to curb side sidewalk to T2 to T1 to LAX-it.

Shortest route: T4 arrival (lower) level to sidewalk to T5 to crosswalk between T5 and T6, then cut thru the parking lot in front of T6 to East Way (east side or right of theme building), the north to crosswalk between T1 and T2, the terminal curb side sidewalk to LAX-it.

Note that fastest route is not likely the shortest route because the shortest route require 4 street crossings that will each take at least 2 minutes wait time.