Map size in minimum redundancy encoding

It goes as nLog2(n) in Shannon because his channels are all single, bitstream pipes, and signal has two degrees of freedom. The map has to be transmitted.  It is just an anomaly of where Shannon worked. But if he had considered three pipes, then his map size goes down, to nLog3(n), and signal has three degrees of freedom.

There is nothing sacred about SNR going as the square of frequency. SNR can easily go as the third power, no  problem. We just get a different basis function. In a three pipe system, each pipe has root2(noise) , but when combined they have root3/2(noise) (and 3 times signal). In physics it makes a lot of difference, noise is kinetic energy, the more degrees of freedom kinetic energy has, the more stable is packed matter.