Nyquist in reverse

 The rate says we can recreate a signal if we sample it twice.. This can be extended to dimensions, as in:

x^2 + y^2 + z^2, each sample twice.  What signals are at maximum entropy Signals that have commutative multiplications. as on:

3xyz.

They are maximum entropy because they are the just sufficient axis of primes and no more, to make concentric rings about the Lie circle. Thus, any Markov n-tuple is a integer set that  multiplies modulo n.  That defines the bandwidth of round off error, a small set of small integers.

The modulo error should be interpreted as a curvature on the Lie, making it a salad bowl.

So this gal drops by the Lie paper store.

She has a bunch of products taken from a three dimensional system.  She wants the schematic paper to draw a three dimensional hut. The clerk manager finds the nearest Markov 3-tuple sheets. These are transparent graph paper with the maximum entropy points about the inner circle/ She can mark one or two up, then stack them. She can get an idea of the geodesics she can make from those mixed products.

But, all points on the n tuples are maximum entropy, the step most entropy maximizing node may not be up. There is a point where the n+1 tuple tree is needed, the system supports another dimension. Relative prime theory, theory of everything.

The theory of relative prime.

The sets of relative primes count integers in all the commutative integer rings in steps up a spiraled lie stair case.  The maximum entropy set always chooses n+1 when appropriate and counts up from there. The multiplicative sets never include duplicates, this is minimum redundancy, hence maximum entropy.  Modulo n, it is guaranteed to be relative prime.  Spatially I can convert this into a Lie circle exercise, fit three concentrics into the circle and meet both objectives, count the angles modulo n. That as Shannon's thing, this should boil down to a Lie circle packing.