# Should I believe this numerical trivariate integration result involving Liouville's constant to fifteen decimal places?

I have two constraints

jj1 = Q1 > 0 && Q2 > 0 && Q3 > 0 && Q1 + 3 Q2 + 2 Q3 < 1 && Q1^2 + 3 Q2 Q1 + (3 Q2 + Q3)^2 < 3 Q2 + 2 Q1 Q3


and

S1 = (256 (Q1 - Q3)^6 (1 - 9 Q2 - 6 Q3 +
3 (Q1^2 + 9 Q2^2 + 6 Q2 Q3 + 4 Q3^2 + Q1 (-1 + 3 Q2 + 4 Q3))))/6561 > (8192 Sqrt[2/91])/16209796869


If I issue the command

36 Integrate[Boole[jj1], {Q1, 0, 1}, {Q3, 0, 1}, {Q2, 0, 1}]


I obtain the result

(8 Pi)/(27 Sqrt[3])


I can also obtain this result using the command

36  Integrate[1, {Q1, 0, 1/3}, {Q3, 0, (1 + Q1)/4}, {Q2, 1/6 (1 - Q1 - 2 Q3) - 1/6 Sqrt[1 - 2 Q1 - 3 Q1^2 - 4 Q3 + 12 Q1 Q3], 1/6 (1 - Q1 - 2 Q3) + 1/6 Sqrt[1 - 2 Q1 - 3 Q1^2 - 4 Q3 + 12 Q1 Q3]}]


where the integration limits are obtained from

GenericCylindricalDecomposition[jj1, {Q1, Q3, Q2}]


Now, if I modify the three-dimensional integration problem immediately above by replacing Integrate by NIntegrate and replacing the integrand of 1 by Boole[S1] and use Working Precision->20, I obtain a result of

0.076083637421679183037

which WolframAlpha informs me coincides to fifteen decimal places with $$$$\frac{30 (10 \mathcal{L}_{Li}+1)}{64 \mathcal{L}_{Li}+821} \approx 0.076083637421679151988,$$$$ where $$\mathcal{L}_{Li}$$ is Liouville's constant https://mathworld.wolfram.com/LiouvillesConstant.html (I substituted the constant $$\approx 0.11000100000000000000000100000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000001$$ and the formula does hold.)

The source of my sceptism as to this result (which was totally unexpected/surprising to me) is that I have a third constraint

S2 = (4 Sqrt[(Q1 - Q3)^2] +
4/3 Sqrt[
1 - 9 Q2 - 6 Q3 +
3 (Q1^2 + 9 Q2^2 + 6 Q2 Q3 + 4 Q3^2 + Q1 (-1 + 3 Q2 + 4 Q3))]) >4/3


Now, if I replace Boole[S1] in the numerical integration by Boole[S2], I get 0.018903515328655050694 which to high precision is $$\frac{2}{81} \left(4 \sqrt{3} \pi -21\right) \approx 0.018903515328657140917$$. (This $$\frac{2}{81} \left(4 \sqrt{3} \pi -21\right)$$ is obtainable by exact integration, so I have full confidence in it. I, however, have no comparable exact integration result based on S1.)

Now, what I find immediately puzzling is that I am engaged in a related quasi-Monte Carlo (quasirandom) procedure (discussed in Given measures on sets and on certain Boolean combinations of the sets, can one check their consistency and/or extend them to other combinations? and How can one generate an open-ended sequence of low-discrepancy points in 3D?)

Based on more than twelve billion points, I get an S2-based estimate of 0.018903823--matching the known exact value to six places--while for the S1-based estimate, I currently have 0.07912772, agreeing to only two decimal places with the Liouville's constant-related one of 0.07608363.

Of course, I would be pleased to have an exact computation using Boole[S1]--which would definitively settle the question. Further, subject-matter (quantum-information-theoretic) considerations make an exact computation of Boole[S1||S2] of central interest.

Well, the coincidence between $$$$\frac{30 (10 \mathcal{L}_{Li}+1)}{64 \mathcal{L}_{Li}+821} \approx 0.076083637421679151988472257219968531634399,$$$$ where $$\mathcal{L}_{Li}$$ is the Lioville's constant, and 36 NIntegrate[Boole[S1], {Q1, 0, 1/3}, {Q3, 0, (1 + Q1)/4}, {Q2, 1/6 (1 - Q1 - 2 Q3) - 1/6 Sqrt[1 - 2 Q1 - 3 Q1^2 - 4 Q3 + 12 Q1 Q3], 1/6 (1 - Q1 - 2 Q3) + 1/6 Sqrt[1 - 2 Q1 - 3 Q1^2 - 4 Q3 + 12 Q1 Q3]}]
So, my basic search for the exact integration results employing Boole[S1] and Boole[S1||S2] ($$\approx 0.081502827$$) has yet to be completed.