Conservation Of Homotopic Residue

Let us initially consider a non-perturbative proximal locus, that is of a Lagrangian-based discrete energy, that is at a relatively tight instance.  Take a certain general genus of a differential equation, that is here to be relatively Laplacian-based -- that is here to be in terms of two different spatial parameters that are to be signified by two different respective variables.  Let's say that the said differential equation here, is to at least partially represent the quantific condition of both the loss and/or the gain of homotopic residue,--  that is to be present at a "snapshot" of extrapolation.  Solve one of the parameters, in terms of the other -- while then solving for the main variable, that is of this special general genus of a case.  Set this just extrapolated differential solution to the following two different things.:
1)  The inverse hyperbolic cotangent -- in order to work to determine the convergent homotopic residue.
2)  The inverse hyperbolic tangent -- in order to work to determine the divergent homotopic residue.

If this so-eluded-to differential equation, is of an evenly-gauged tense of a relatively Laplacian-based set of Ward-Cauchy conditions, then, the net resultant Reimman-related component of the said homotopic residue, when this is to be determined by subtracting the divergent homotopic residue from the convergent homotopic residue -- will be either of the nature of being 1, 0, or -1.
Take both the correlative Reimman-based angle that is to be derived from the said inverse hyperbolic cotangent as well as the correlative Reimman-based angle that is to be derived from the said inverse hyperbolic tangent.  Next -- subtract the latter angle from the first angle.  The resultant angle should tend to either be of the nature of being either 90 degrees, 0 degrees, or -90 degrees.

Also, if this so-eluded-to differential equation, is of an evenly-gauged tense of a relatively Laplacian-based set of Ward-Cauchy conditions, then, the net resultant Nijenhuis-related component of the said homotopic residue, when this is to be determined by subtracting the divergent homotopic residue from the convergent homotopic residue -- will be either of the nature of being i, 0, or -i.
Take both the correlative Nijenhuis-based angle that is to be derived from the said inverse hyperbolic cotangent as well as the Nijenhuis-based angle that is to be derived from the said inverse hyperbolic tangent.  Next -- subtract the latter angle from the first angle.  The resultant angle should tend to either be of the nature of being either 90i degrees, 0 degrees, or -90i degrees.

One may then call such an extrapolation of evidence, as being of the nature of the Conservation Of Homotopic Residue.
I will continue with the suspense later!  To Be Continued!  Sincerely, Samuel David Roach.