So taking eq. (18) in this

paper and squaring the integrand results in the following formula. We have relabelled the A's and B's so easier to check with GAViewer and perhaps easier to follow.

Step 1:

(A1B1 + A1B2 + A2B1 - A2B2) (A1B1 + A1B2 + A2B1 - A2B2) =

A1B1A1B1 + A1B2A1B1 + A2B1A1B1 - A2B2A1B1 + A1B1A1B2 + A1B2A1B2 + A2B1A1B2 - A2B2A1B2 + A1B1A2B1 + A1B2A2B1 + A2B1A2B1 - A2B2A2B1 - A1B1A2B2 - A1B2A2B2 - A2B1A2B2 + A2B2A2B2

And it is obvious that the expansion is correct. Now, the terms in the result can be collected as follows but keeping the A orders and B orders the same.

Step 2:

A1A1B1B1 + A1A1B2B1 + A2A1B1B1 - A2A1B2B1 + A1A1B1B2 + A1A1B2B2 + A2A1B1B2 - A2A1B2B2 + A1A2B1B1 + A1A2B2B1 + A2A2B1B1 - A2A2B2B1 - A1A2B1B2 - A1A2B2B2 - A2A2B1B2 + A2A2B2B2

Then since a bivector squared is -1 we further reduce to;

Step 3:

1 - B2B1 - A2A1 - A2A1B2B1 - B1B2 + 1 + A2A1B1B2 + A2A1 - A1A2 + A1A2B2B1 +1 + B2B1 - A1A2B1B2 + A1A2 + B1B2 +1

Which further reduces to;

Step 4:

4 - A2A1B2B1 + A2A1B1B2 + A1A2B2B1 - A1A2B1B2

We saw earlier that this is the result we want and is verified by GAViewer to be correct and after taking the square root and applying many iterations gives us

. Now the original dependency between expectation terms is shown by this;

A1B1 + A1B2 + A2B1 - A2B2

So if an independency is creeping in here, I would say that it has to be in step 2. Otherwise this is a mystery as to how (A1B1 + A1B2 + A2B1 - A2B2) with dependent expectation terms can return

since it is mathematically proven that it can't be larger than 2.

However, I think Joy claims it is because the original math proof doesn't use bivectors and GA. Joy, please correct me if I am wrong about that.

PS: The reason I think it has to be in Step 2 because we saw earlier without the rearrangement, the result came out to be 2 since the scalar cancelled out.