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Stair Question

There is no maximum gap. IBC 1014.7 establishes a minimum gap of 1-1/2 inches. IBC 1014.8 limits the projection of a handrail to 4-1/2 inches into the required width of a stairway. If the stair is wider than the minimum required width, the handrail can project more, leaving a wider gap to the wall or guard.
 
Can't quite picture your situation but if like between end of guard and end of a vertical piece supporting handrail, 4" as in the 4" sphere rule. But maybe you don't mean a vertical gap between different railings.
 
at the top of the stairs, what is the max gap between the guardrail and handrail for the stairs??
Not sure if this would be helpful to you or not, but the Stairbuilders and Manufacturers Association (stairways.org) has several books they describe as “visual interpretations” of the residential code requirements. They have a few free downloads (most recent version is from the 2009 IRC,) I don’t see an option to download digital copies of their interpretations for newer versions of the code.
 
I contacted the Stairbuiders and Manufacturers Association, their executive director, Terra Erb, responded with the following:

“Thank you for reaching out to the SMA regarding the posting of our Visual Interpretation of the Stair Building Code. You are welcome to share the 2012 VI by linking to the SMA website here: https://stairways.org/2012-VI. If you have need for printed copies, please note that they are available for FREE, you only pay shipping. Newer digital versions of the VI are not currently available to the public.”
 
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