How to get a precision stable 4.01v output?
Depending on what accuracy you want and the amount of power you want, a precision reference at a common 4.096V can be used to get a few mA at a voltage as precise as a couple of ppm. There are a few such ICs around which are typically used to set reference voltages for ADCs or DACs (hence the weird power-of-2 voltage)

For a power supply situation (where you need more than a couple of mA) you can use LDO regulators (which have a drop out voltage of only 300mV or so) such as the TPS7A85 or similar. These will give you typical accuracies of 1% or so.

Given your comment of 10% tolerance, pretty much any LDO you can find will do the job. There are many fixed voltage versions, probably around 4V or so, or adjustable versions which will be as accurate as the resistors you use. An adjustable LDO using 1% or 0.5% tolerance resistors to set the value will easily give you the desired tolerance and stability.