What is the precise reason for the 5 minute limit at 5800?

To the original poster, I think I understand your question. It sounds like you are wondering what particular weakness caused Rotax to set the max power output where they did. I don't think there is one single thing. As others have stated, it is just where Rotax decided to draw the line that balances power output against reliability of the entire engine system.

If there was one particular thing that caused Rotax to set the power limitations where they did, that would indicate a poorly designed engine, because it would be telling us that the other engine components were over built, relative to whatever it is that is susceptible to failure.

I just read that the FAA requirement for testing certified engines involves running it for 50 hours at maximum power (maximum rated MAP and RPM) with all temperatures in the green, then another 50 hours at maximum power but with all temperatures right at redline, then another 50 hours alternating between max power and 65-75% power. At the end of the test the engine is disassembled and all parts must be within the tolerance for new parts. This makes me believe that the engine could handle more in theory but the 5 minute limit is there to give us ample safety margin and to ensure it lasts until TBO.

https://www.takeflightsandiego.com/assets/documents/EngineCare.pdf

Peak HP is a factor in this formula (Torque X RPM) /5252 = HP

Torque is the cylinder pressure, a function of its size, its compression ratio as well as what type of induction you have.  (it may be just carb working on pressure differential or turbo for example) 

RPM is self-explanatory, in this formula we use the crankshaft speed.

5252 is a distance used for the calculation.  (In countries other than the USA there is a very similar version for power in Kilowatt power.  The formula then is basically the metric version of the standard HP one.  

Your Rotax is tested to its design limit, in this case 5800 is the full RPM.  This can be used when you have a constant speed or inflight adjustable propeller however with fixed pitch we need to address the issue of propeller unloading at flight speeds that could lead to over RPM when in a dive for example.  The maximum continuous power limit is 5500 which in theory you can use all day long but would be very costly in fuel burn and accelerated wear in the engine.  

In flight testing the FAA and EASA prescribe the same methods to evaluate an engines durability.  This is called block testing and engines must show successful completion of the block testing before consideration for certification.  While there is undoubtably more performance to be had the manufacturer must consider durability and market dependability.  Given this they will select the RPM and best overall long-term health of the engine when it is in design.  A few take always from all of this is that the design, at least in the case of Rotax, is that for the most part the engine will be used in the 75% power range most of its life.  This gives good economy and best overall life of the components.  So simply speaking we look at the max power, 5800 (limited to 5 mins WOT) and come down the curve to find 75% of that max peak.  In the case of the 9 series Rotax this is generally in the 5000 to 5200 RPM which happens to be the happy place for smooth running, and peak torque point of the engine.  

A word about fixed pitch propellers.  Settings, especially for ground adjustable propellers, should load the engine enough at level flight to be a governor to control the engine RPM.  Each airframe design, propeller design, will have variables.  in addition, regional issues that affect power such as density altitude can greatly affect the settings.  Work with your local technicians and get recommendations from the OEM on what works best for that aircraft design.  just my 2 cents worth, good luck. 

So back to you direct question...5800 is the design RPM.  Only use this for 5 mins as it has not been approved for more due to increased wear and stress that might limit the enigne life.  It is only useful if you have a inflight adjustable system, otherwise I suggest simply forget about it and set your fixed pitch to prevent overspeed in normal flight operations. 

"So back to you direct question...5800 is the design RPM.  Only use this for 5 mins as it has not been approved for more due to increased wear and stress that might limit the enigne life."

I know that. This was just my own curiosity. I like knowing the details about the wear and stress (thermal? mechanical? what parts of the engine?). The answer about piston temps was what I was looking for (tho that one is speculation).

Like I said it's all about longevity and reliability plus liability to Rotax. 

We can't pull over to the curb like a car and when over mountains it's nice to have an engine that keeps running  and you trust. Mnt. landings or tree landings without a strip hurt. :)