Using the method described in Recipe File: Soft Boiled Eggs where the water is brought to a boil, pot moved off the heat source, refrigerated large eggs placed into the water, lid replaced, and eggs shocked in ice water after a set duration of time, I tested various amounts of time spent in the just boiling water.
After 2 minutes: The thin albumen (egg white closest to the shell) has turned white and is semi-solid but the thick albumen (egg white layer nearest to the yolk) is still clear and liquid.
After 3 minutes: The thin albumen has solidified. The thick albumen is liquid but turning white.}?>
After 4 minutes: The thin albumen has fully solidified. The thick albumen is semi-solid.
After 5 minutes: The whites are solid, but when tilted they still flow.
After 6 minutes: The whites are solid but not stiff (moves a little when the egg is tilted). The yolk has begun to thicken, but the egg still can't be peeled intact.
After 7 minutes: The whites are solid but not stiff. The yolk has thickened but flows well.
After 8 minutes: The whites are solid. The yolk has thickened and parts are beginning to no longer flow.
After 9 minutes: The whites are solid. Half the yolk has gelled (solid but still translucent - not yet the opaque yellow of a hard boiled egg) and half continues to thickly flow.