Kraft's inequality and Shannon's noiseless coding theorem for an encoding
I don't know what a "compact instantaneous binary encoding" is, but I'm guessing it's a prefix code that saturates Kraft's inequality. If so, your numbers don't correspond to a compact prefix code, since $1/4+1/4+1/16+1/16+1/32+1/32 &lt; 1$.

Shannon's noiseless coding theorem states that when optimally encoding large blocks, in the limit the average encoding size per data item is the entropy. You can prove the upper bound part of the theorem using Huffman coding applied to large blocks. Huffman coding produces a code whose average bit length is at most $H(X)+1$, where $X$ is the source. If you use Huffman coding on blocks of size $n$, the average bit length per data item becomes $H(X)+1/n$ (why?), and by letting $n\to\infty$, you can get as close to the entropy as you want.