The Lucas numbers do not generate sinh(ax) and cosh(ax), they alternate sinh and cosh. So the tanh(ax) that processes use is really sinh(na-a)/cosh(na); where a is usually some low order polynomial of (ln(ph()) + f), the f are the sum of angle offsets that define charge,spin and mass I presume. That difference is what makes an imprecise finite number line out of locally additive systems, and the solution set corrects that imprecision for some finite set of points.
When the hyperbolic differential is solved, the second derivative of that polynomial appears as a multiplier on the tanh, and has to be equal to: sinh(na)/sinh(n*a-a), In other words, the laws of the system are designed to create the group solutions that make the finite number line work. The solutions are the energy quantizations.
I am not doing much on this, but I just want to make sure that the dots are mostly connected and not leave anyone astray.
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