The critical positivity ratio is a discredited concept in positive psychology positing an exact ratio of positive to negative emotions which distinguishes "flourishing" people from "languishing" people. The ratio was proposed by Marcial Losada and psychologist Barbara Fredrickson, who identified a ratio of positive to negative affect of 2.9013 as separating flourishing from languishing individuals in a 2005 paper in American Psychologist. The concept of a critical positivity ratio was embraced by both academic psychologists and the lay public. Fredrickson wrote: "Just as zero degrees Celsius is a special number in thermodynamics, the 3-to-1 positivity ratio may well be a magic number in human psychology."The first critical evaluation of the mathematical modeling behind the critical positivity ratio was published by a group of Finnish researchers Luoma, Hämäläinen and Saarinen in 2008. The authors noted that "Only limited explanations are given about the modeling process and the meaning and interpretation of its parameters.

Thus, the reasoning behind the model equations remains unclear to the reader." Moreover, "the model produces strange and unreported behavior under certain conditions. Thus, the predictive validity of the model becomes problematic." In 2013, the critical positivity ratio aroused the skepticism of Nick Brown, a graduate student in applied positive psychology, who felt that the paper's mathematical claims underlying the critical positivity ratio were fundamentally flawed. Brown collaborated with physicist Alan Sokal and psychologist Harris Friedman on a re-analysis of the paper's data, they argued that Fredrickson and Losada's paper contained "numerous fundamental conceptual and mathematical errors", as did Losada's earlier work on positive psychology, which invalidated their claims. In their response, Hämäläinen and colleagues argued that there were no fundamental errors in the mathematics itself, but the problems relate to the interpretation and justification of the use of the model. Losada declined to respond to the criticism, indicating that he was too busy running his consulting business.

Fredrickson wrote a response in which she conceded that the mathematical aspects of the critical positivity ratio were "questionable" and that she had "neither the expertise nor the insight" to defend them, but she maintained that the empirical evidence was solid. Brown and colleagues, whose response was published the next year, maintain that there is no evidence for the critical positivity ratio. In response, American Psychologist formally retracted the mathematical modeling elements of Fredrickson & Losada's paper, including the specific critical positivity ratio of 2.9013, as invalid. The problems with the paper went unnoticed for years despite the widespread publicity surrounding the critical positivity ratio, contributed to a perception that social psychology as a field lacked scientific soundness and rigorous critical thinking. Sokal stated: "The main claim made by Fredrickson and Losada is so implausible on its face that some red flags ought to have been raised." Building on research by Barbara Fredrickson indicating that individuals with a higher ratio of positive to negative emotions tend to have more successful life outcomes, on studies by Marcial Losada applying differential equations from fluid dynamics to human emotions and Losada used nonlinear dynamics modelling to argue that the ideal positivity/negativity ratio lies between 2.9013 and 11.6346.

They argued that those with ratios within this range will "flourish", whereas those with values outside it will "languish". They claimed that their model predicted cut-off points for the maximum and minimum positivity ratios within which one should observe qualitative changes in an individual's level of flourishing. Losada's article was critiqued by Andrés Navas in a Note to the French website of the CNRS "Images des Mathématiques"; the whole theory of the critical positivity ratio was critiqued by Nicholas Brown, Alan Sokal, Harris Friedman, in a 2013 article published in American Psychologist. Brown et al. argue that Losada's conclusions in previous papers using modelling from fluid dynamics, those in his paper co-authored with Fredrickson, are not only based on poorly reported experiments – they argue that it is difficult to draw any conclusions from some previous studies by Losada because critical details are omitted, "interpretations of results are made with little or no justification" – but are based on elementary errors in the use of differential equations.

Among the severe flaws claimed by Brown et al. in the positivity-ratio theory and its presentation were: That the data used by Losada in several analyses do not meet the basic criteria for the use of differential equations. That the differential equations used by Losada to calculate the critical positivity ratio use parameters taken directly from Lorenz's simplified and most arbitrary models for fluid dynamics. Losada gives no rationale for his choice of parameters. Using different arbitrary parameters gives different positivity ratios, thus the precise values for the ratio given by Fredrickson and Losada are meaningless; that the butterfly-like Figure 1 provided by Fredrickson and Losada is not a model of the data taken from their human participants, but "the results of computer simulations of the Lorenz equations, nothing more". That if one takes the

Shulan is a city in northern Jilin province, Northeast China. It falls under the administration of 71 kilometres to the south-southwest. Shulan's historical sites include the tombs of Wanyan Xiyin, the erstwhile "Chief Shaman" of the Jurchens and the chief minister of the early Jin Empire, his family members. Since 1961, they have been listed on the provincial register of the protected historical sites. During the Qing Dynasty, the so-called Outer Willow Palisade, separating the Mongolian and Manchurian lands to the west and east ended near the town of Fate within today's Shulan. Subdistricts: Beicheng Subdistrict, Nancheng Subdistrict, Huancheng Subdistrict, Jishu Subdistrict Towns: Baiqi, Shangying, Ping'an, Fate, Xiaocheng, Jinma Townships: Lianhua Township, Liangjiashan Township, Xin'an Township, Qili Township, Tiande Township Shulan City Government official site