Lagrange, Shannon: Same sampling theory different basis functions?

I noticed that Shannon information theory was simply a restatement of Shannon sampling theory for trig functions, and that Lagrange theory is the same for Hyperbolic functions.

Given a Gaussian channel, the spectrum can be organized in integer units of trig functions and a  ring established. Same with Lagrange, just change the basis set? The trigs define the bounded multipliers, the Hyperbolic the bounded quotients.

Look, they both want deviation to fall as the square, that is the 4 on the right which is 2^2. One assumes a hyperbolic noise channel, the other a trig noise channel. So they both have canonical twos binary representations. The one wants sampling to twice the unit circle, the other to 1/2 the unit circle.

And they both really define the finite value of the natural base given the noise spectrum. Both of them all about finite number lines.  If the natural base represents noise dropping at the square, the sample rate is 2, when dropping as the cube, the sample rate is 3.

It is the natural log, or natural exponent because that is what defines the full set of math operations, especially addition of powers, in the  number line.  In real cases, the problem is the finite log in a finite number line. Without powers defined there is no digit system. So Shannon and Lagrange are about defining the length of a digit system that matches the integer errors in counting. The number of integers one gets is the minimally redundant set.

The sphere
It is a third degree polynomial in r, whih generates three orthogonal bases which make up a 16 digit worm counter, as near as I can tell.  The value one in that worm counter is maximally packed, it is the quarks and electron.  The three independent bau rates must be composted of compound shift of the Phi sequence. A
shift (shift(shift (Fn,Fm),shift (Fn,Fm),shift(shift (Fn,Fm),shift (Fn,Fm))
The compound braid concept, or spiral to the third power, and it walks the Markov tree maintaining the unit zero where it can, at the sample rate of light. The whole proton must be filled with the compound braid system. The Unit zero is the sum of the 16 bit fraction, plus one. In other words, it has to be a complete, compound worm gear system.