To simplify the expression \( A \), we will break it down into parts.

1. **Calculate the first term:**

\[
2 \left| \frac{1}{2} - \frac{3}{4} \right| = 2 \left| \frac{2}{4} - \frac{3}{4} \right| = 2 \left| -\frac{1}{4} \right| = 2 \cdot \frac{1}{4} = \frac{1}{2}
\]

2. **Calculate the second term:**

\[
\sqrt{\frac{4}{9}} = \frac{2}{3}
\]

3. **Calculate the third term:**

\[
2^2 \cdot 10 + 2^2 \cdot 6 = 4 \cdot 10 + 4 \cdot 6 = 40 + 24 = 64
\]

\[
2^2 \cdot 15 - 2^6 = 4 \cdot 15 - 64 = 60 - 64 = -4
\]

Therefore, this fraction becomes:

\[
\frac{64}{-4} = -16
\]

4. **Combine the second and third terms:**

\[
\frac{2}{3} - 16 = \frac{2}{3} - \frac{48}{3} = \frac{2 - 48}{3} = \frac{-46}{3}
\]

5. **Combine all parts:**

\[
\frac{1}{2} + \frac{-46}{3}
\]

To add these fractions, we need a common denominator, which is 6:

\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{-46}{3} = \frac{-92}{6}
\]

Thus, the sum is:

\[
\frac{3 - 92}{6} = \frac{-89}{6}
\]

6. **The final term:**

Now multiplying this with \( (0 \cdot (3) - \frac{1}{4} - \frac{1}{12})^{1023} \):

Since \( 0 \cdot (3) = 0 \):

\[
0 - \frac{1}{4} - \frac{1}{12} = -\frac{1}{4} - \frac{1}{12}
\]

Finding a common denominator (12):

\[
-\frac{1}{4} = -\frac{3}{12}
\]

Therefore,

\[
-\frac{3}{12} - \frac{1}{12} = -\frac{4}{12} = -\frac{1}{3}
\]

Raising it to the power 1023 will still yield:

\[
\left( -\frac{1}{3} \right)^{1023} = -\frac{1}{3^{1023}}
\]

Finally, combine the terms:

\[
A = \frac{-89}{6} \cdot \left( -\frac{1}{3^{1023}} \right) = \frac{89}{6 \cdot 3^{1023}}
\]

Thus, the final result is:

\[
\boxed{\frac{89}{6 \cdot 3^{1023}}}
\]