The logistics of -iLog(i)

The Shannon formulation for the share of a constrained channel to some message happening at rate i, where the sum of i is one.  The short answer is a hidden assumption, the system is packing the channel such that is follows the shortest path toward packed.  It reaches equilibrium and meet Euler conditions. He inserts this assumption when SNR is required to have bell shaped disributions.

Then, he shows that if he has a fair sample of the oath, a finite set of points from the process, he can walk the shortest path taking finite, and irregular steps. There is an algorithm that will find the optimum stepping spots given a finite sample. The log of something is generically the number of bits needed to encode the value. That is logistically equivalent to a transportation channel.

This is where the natural log comes from, the euler conditions specify a natural packing process.

The finite Huffman encoder

Shannon shows that the optimum decoder can always be a binary system, log base 2, up to a given accuracy. hat led to the Huffman encoding of a single, typical message being sent through the SNR limited channel.

Abstract algebra says there is a dual of the Huffman encoder, a generator which will take a channel packed with a set of integer indices and generate reordered sequences of the original message. his is the generator of the channel. What happens when the original sequence grows in length?

The generator begins to look like the natural log, links and nodes become dense. The uniform index becomes a long integer set and begins to generate more of the real number line. In the sense of a constrained channel, the effective SNR is dropping  rising, matching error is dropping, or the gaussian noise in the original formulation is dropping.

Just to handwave the rest of the story

We have proven, above, an algebra exists on generators, they have a distance measuring property, which can be accomplished in the dual as operations on two (or more) sets of indices, like dividing them and watching the remainder. Hmw many packed deposit queues fit into the oack loan queues, it becomes node by node algebra, we have found common denominators ots between two finite channels.

What has changed in Shannon's original is that we reconstruct the SNR, and bit error becomes the one, as in SNR+1.  Noise has meaning, it is the emptying queues. The 1 is retained bit error variance. We get the matching process assuming all agents pack sphere.