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I got no clue what's your point here. My point is that the calculated minimum force does not move the block if the block is initially at rest. If initially (before we apply the force) it is moving then what's the point of the problem, minimum force to move a block that is already moving?In the real world, if a system is static then we can deduce the forces exactly balance, but we cannot argue it the other way.

When we write that the limit of static frictional force is ##\mu_sF_N## we mean that as a threshold: if the other forces tending to make the bodies slide against each other add up to less than this then they won’t slip, and if they add up to more they will. If we attempt to apply a force of exactly that we cannot tell what will happen.