"[3] The first consequential re-evaluation of the mathematical modeling behind the critical positivity ratio was published in 2008 by a group of Finnish researchers from the Systems Analysis Laboratory at Aalto University (Jukka Luoma, Raimo Hämäläinen, and Esa Saarinen).
[12][non-primary source needed] A formal retraction for the mathematical modeling elements of the Losada and Fredrickson (2005) paper was issued by the journal, American Psychologist, concluding that both the specific critical positivity ratio of 2.9013 and its upper limit were invalid.
[14][9][15] Sokal later stated, "The main claim made by Fredrickson and Losada is so implausible on its face that some red flags ought to have been raised",[9] as would only happen broadly in graduate student Brown's initiating the collaboration that resulted in the Brown-Sokal-Friedman rebuttal.
[9] Building on research by Barbara Fredrickson suggesting that individuals with a higher ratio of positive to negative emotions tend to have more successful life outcomes,[16] and on studies by Marcial Losada applying differential equations from fluid dynamics to human emotions,[citation needed] Fredrickson and Losada proposed as informative a ratio of positive to negative affect derived from nonlinear dynamics modelling (based on Lorenz systems), which appeared in 2005 in a paper in American Psychologist.
[10] Brown, Sokal, and Friedman state that one can:only marvel at the astonishing coincidence that human emotions should turn out to be governed by exactly the same set of equations that were derived in a celebrated article several decades ago as a deliberately simplified model of convection in fluids, and whose solutions happen to have visually appealing properties.
An alternative explanation – and, frankly, the one that appears most plausible to us – is that the entire process of "derivation" of the Lorenz equations has been contrived to demonstrate an imagined fit between some rather limited empirical data and the scientifically impressive world of nonlinear dynamics.
[11] The American Psychologist proceeded to formally retract as invalid the mathematical modeling elements of Fredrickson and Losada's paper, including the specific critical positivity ratios of 2.9013 and its upper limit.
Nickerson, an independent scholar formerly at the University of Colorado, Boulder,[citation needed] concurred with the Brown-Sokal-Friedman rebuttal conclusion of the lack of empirical evidence for a critical positivity ratio, and noted the necessity of distinguishing between within-person-across-time versus within-time-across-persons theories.
[8] In 2014, the rebuttal authors also responded to comments from others on their 2013 work, Among the purely mathematical errors clearly noted... are Fredrickson and Losada's assertion that the r = 22 data (alleged to be characteristic of “medium-performance teams”) end up in a limit cycle... and their implicit claims concerning the absence of chaotic attraction at large values of r...
"[18] As of January 2014, as Andrew Anthony notes from his preparation for his article in The Observer from that period, Fredrickson continued to maintain "on empirical grounds" that "tipping points [in relation to positive emotions and flourishing] are highly probable", as communicated to him via email.
[3] In 2018, the Journal of Humanistic Psychology published a special issue focused on the aftermath to the rebuttal of the original Fredrickson and Losada article, where Harris L. Friedman and Nicholas J. L. Brown served as monitoring editors.
[14][9][15] Sokal would state that the paper's "main claim... is so implausible on its face that some red flags ought to have been raised",[9] as would only happen broadly with graduate student Brown's initiating the collaboration that resulted in the Brown-Sokal-Friedman rebuttal.