Is it mandatory to submit your exact accrued leave days when filling out DA 31? If so, where is the Army Regulation that directs it?

Compromising your integrity, or permitting another Soldier to do so to benefit from a 4-day pass is selling one's self very short.
Not saying you or anyone else compromised their integrity, but asking whether it is mandatory to provide accurate information on any form, and saying S1 has access to it, seems to be seeking justification for delibeeately providing false information.

Step 4 specifically states to provide a copy of most recent LES. There is you answer. Easy research. Why would you lie about the number of days you have anyway? Integrity, its a thing!!!!!!!!! And yes every place is different, because if you read this regulation its states that every commander WILL have a leave and pass program. But never lie about the number of days leave you have accrued SSG. Bad business.

AR 600-8-10

4–4. Steps to request ordinary leave
The steps to request ordinary leave are shown in table 4–2.
Table 4–2
Request ordinary leave
Step Work center Required action
1 Soldier On DA Form 31 (Request and Authority for Leave) (chap 12), complete blocks 2
through 11.
2 Soldier On DA Form 31, block 7, check ordinary.
3 Soldier Read instruction on reverse side of DA Form 31. If needed, request clarification.
4 Soldier Provide the latest LES to first person in chain of command with leave request (if
applicable, annotate corrections to ending leave balance and provide reason for
discrepancy).
5 Soldier Obtain chain of command recommendation in block 12 and forward to unit commander (or designee) for approval.
6 Unit Receive leave requests and follow processing procedures in paragraphs 12–1
and 12–3.
AR 600–8–10 • 15 February 2006 9
Table 4–2
Request ordinary leave—Continued
Step Work center Required action
7 BNS1 Receive leave request from unit and follow processing procedures in paragraphs

Others have already pretty much said the why as to you should enter what your LES says for amount Accrued. Lets say SGT Snuffy requests 10 days leave fills out DA 31 requesting 10 days but SGT Snuffy only has accrued 7 days to date. The Commander can only approve what you have in the bank in 99% of the requests( Some special circumstances can be dealt with on a case by case basis) Most commanders do not take every single Request and run all the entries through the system to make sure they are accurate with days. So SGT Snuffies leave form squeaks through and he goes on leave and is carried on the unit report as being on leave. SGT Snuffy returns ten days later signs of of leave and the form is processed and then big red Flagg pops up when the data is entered into the system. So now SGT snuffy lied on his request, CPT Smith catches multiple heat rounds from higher for not micromanaging this troops leave requests and everyone is pissed off. So its best to just do it for all concerned.
Ona. side not a number of years back I was coming off of a deployment and using up my remaining leave. I requested 15 days, which is what I knew I had. But my LES said I never took any leave earlier which I did. The folks at demob tried to get me to take the additional 15 days. I declined both because it was not right and I knew down the line my earlier request would be found and entered and I would be in deep Doo-Doo. Mistakes are made with days entry and tracking all the time and back before computers certain NCO's/Clerks used to charge troops $50 bucks to Lose the Request when the leave was over. Which is a really sorry thing to to do and leads to all sorts of career ending BS. End of war story.