Although the assumption of an equilibrium distribution for the transition matrix led a good fit to the data it ultimately failed because its gain function did not agree with the assumed gain function. I am willing to admit that it was a blunder but it also points out the need to be skeptical about what one is doing. Sleep deprivation can make someone more error prone and takes its toll on results.
Curve fits can be difficult since there is often more than one minimum and a number of them close together can interfer with each other and prevent a computer program from converging to a solution even if one has an approximate solution. Even if one can find a local minimum that does not guarantee that the minimum is a global minimum, that is, the lowest of all minimums. So there may be competing mechanisms.
Conclusion: Equilibrium Distribution = FAIL