The general soltion, then

If I have it. Say the electron orbital.  Write all the equations of motion using r^k, leaving coefficients to the initial conditions. That is recursive. Match coefficients with a equally long tanh power series creating Tanh(k*ln(r)). Then use the differential which says, the combination of things made by a recursive r^k is bound in density, in a ring group sense. Diagonalize the sequence.Solve the matched coefficient finite power series for r, the natural base for the equations of motion which bounds the entropy. Bound by the coefficient matching with the finite tanh. That functions guarantees that the number of combinations of events drops faster than the average number of events, plus a band gap.

So that is a condition on the equations of motion. Finite recursive and diagonal.  The tanh goes to constant with j as tanh(j*r) gets big. Change dimensionality and solve more complex band limits, its all in power series of Tanh, and the polynomial length in the differential equation. Add the the values ln(r) as a finite recursion in r^n, add it to your custom tanh spectral tool.

Matching  recursive motion into Bournoull power series is a linear transform, one to one. The coefficients on the equatio  of motion are mapped to fill the tanh coefficients from n small to large. Coefficients become r^[logr(a)], the a being the set of coefficients. So ypur sequence has to be upward bound as the integer goes up. And the sequence has to go to zero toward integer down. Uniformly unidirectional, except negative symmetrical about 0. Your equations of motion are now written in units of combinations, of N things taken each of k times and recursive by m.

Once the system is mapped to the finite tanh; all the rules of Newtons grammar are available in the hyperbolics. I can construct any realization, in units of quant. Or start with equations in recursive version of Tanh. Write the equations in tanh. Set whatever band stops you want, do a polynomial divide and find a bunch of roots.

My equations of motion parameter  on quant, and time, distance can get generated, using large tanh x arguments.  Trascendentals have to be finite recursive defined.  But the values tanh(n), are always locally valid, no relativity.
Cosh(x) = cos(ix) , so the system collapses imaginary numbers.

Within the system find the recursions for X and t.  Cycle through the quants and generate those values.  Then run the equations as integrals on x and t. great for economics, model the user agent as a recursive process with coefficients. Compare wave functions with Poisson, se if there is queueing.