The Tanh expansion

The top one. The B2n coefficients are Bournouli numbers.  They are coefficients in a finite order polynomial that computes the term on the left.


That term, Sm, by my interpretation, is the total number of marks on a ruler for each of m subdivisions over the k th integer subdividers.
The tanh(x) is a power series of  of possible total marks on all rulers, for all the possible subdivision exponents. Tanh(x) is also the bounded slope of the ruler measuring error, when x is sequence F+1/F/(F-1/F).  Each Sm(n), can be formulated as recursive in a polynomial of order n.   n is he total number of things being rearranged. So tanh(x) giving the maximum error when we have an infinite number of integers having m subdivisions. But tanh x is also a power series solution to differential equations on the sphere, and we are mostly limited to 3rd power differential equations. If we limit x to quantized values, then we can limit ourselves to local solutions bounded by 3. A general system for spherical differential hyperbolic wave. A recipe system for bounding the measurement error using local estimation.
So with x integerized then the bounded measuring error is a integer sum of things in groups of order three.  x can be represented by a base with integer exponent.