Fredrickson responded to the critique by agreeing that Losada's mathematical modelling was "questionable" and did not show that there are precise values of the ratio, but also arguing that the evidence for the benefits of a high positivity/negativity ratio is solid. Fredrickson noted that Losada declined to respond to the criticism. 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. Other respondents In a follow-up to the 2013 papers—the Brown-Sokal-Friedman rebuttal, and the Fredrickson response—American Psychologist published further scholarly responses, mostly supportive, but some critical of at least some aspects of the rebuttal. The series of responses culminated in a further response to these from Brown, Sokal, and Friedman. C.A. Nickerson, an independent scholar formerly at the University of Colorado, Boulder, 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. Emeritus professor Raimo Hämäläinen and colleagues responded, passing over the Brown-Sokal-Friedman rebuttal claims of failed criteria for use of differential equations in modeling, instead arguing that there were no fundamental errors in the mathematics itself, only problems related to the model's justification and interpretation. Follow-up from Brown, Sokal, and Friedman The original rebuttal authors were openly critical about Fredrickson's partial retraction, and American Psychologist published their response to it in 2014, where they emphatically argued that there was no evidence whatsoever, as of that date, for the existence of a critical positivity ratio (i.e., a tipping-point for positivity). In 2014, the rebuttal authors also responded to comments from others on their 2013 work, noting Nickerson's concurrence regarding the lack of empirical evidence for a critical positivity ratio, and lauding her distinction between the within-person and within-time types of theories, noting that "[b]oth types of theories are valuable... but... conceptually distinct and by no means equivalent" and that they believed that "this distinction deserves to be more widely discussed in the literature on research methods". noting that Hämäläinen, Luoma, and Saarinen "concede our main point, namely the complete lack of justification for the use of the Lorenz equations in modeling the time evolution of human emotions", but confronting the "no clear mathematical errors" assertion, stating thatAmong the purely mathematical errors clearly noted... are Fredrickson and Losada's assertion that the r =