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Question: Which function is more credible?

Question:Which function is more credible?

sursumCorda 1169 Maple 2024
polynomial realdomain
October 30 2024
1

Here are two systems over the reals:

sys__1:=And(r*(387*r+52)+2<r*(226*q+121*s)+9*q*(q*(2*q-5)-3*s+2)+6*s,4*q^3+r*(27*r+4)+s^2=q*(q+18*r),q>=0,r>=0):
sys__2:=And((392-1739*q)*r+4*(2-9*q)**2+2151*r**2<75*r*s,4*q**3+r*(27*r+4)+s**2=q*(q+18*r),q>=0,r>=0):

The following results indicate that both sys__1 and sys__2 are satisfiable 

QuantifierElimination:-QuantifierEliminate(exists([s,q,r],sys__1));
                              true

QuantifierElimination:-QuantifierEliminate(exists([s,q,r],sys__1));
                              true

but RealDomain:-solve simply returns an empty list (that is, no solution exists) in both cases

RealDomain:-solve(sys__1,[q,s,r]); # ⟹ sys1 cannot be satisfied
                               []

RealDomain:-solve(sys__2,[q,s,r]); # ⟹ sys2 cannot be satisfied
                               []

As discussed in the previous problem, in contrast to using QuantifierElimination:-QuantifierEliminate, the use of RealDomain:-solve is unsafe. Nevertheless, the above output suggests that even the much-more-sophisticated QuantifierElimination:-QuantifierEliminate is still not always reliable (since the correct returned-value appears to be false in lieu of true). So, what is the right command to handle polynomial systems over real domains in Maple? 
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