The critical positivity ratio (also known as the "Losada ratio" or the "Losada line") is a largely 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 psychologists Barbara Fredrickson and Marcial Losada, who believed that they had identified an experimental measure of affect whose model-derived positive-to-negative ratio of 2.9013 defined a critical separation between flourishing and languishing individuals, as reported in their 2005 paper in American Psychologist. This concept of a critical positivity ratio was widely embraced by academic psychologists and the lay public; Fredrickson and Losada's paper had been cited more than 320 times by January 2014, and Fredrickson wrote a popular book expounding the concept of "the 3-to-1 ratio that will change your life". In it she 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 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). The authors noted that "only very limited explanations are given about the modeling process and the meaning and interpretation of its parameters... [so that] the reasoning behind the model equations remains unclear to the reader"; moreover, they noted that "the model also produces strange and previously unreported behavior under certain conditions... [so that] the predictive validity of the model also becomes problematic." Losada's 1999 modeling article was also critiqued by Andrés Navas in a French language publication, a note in the CNRS publication, Images des Mathématiques. Neither of these articles received broad attention at the times of their publication. Later, but of critical importance, the Fredrickson and Losada work on modeling the positivity ratio aroused the skepticism of Nick Brown, a graduate student in applied positive psychology, who questioned whether such work could reliably make such broad claims, and perceived that the paper's mathematical claims underlying the critical positivity ratio were suspect. Brown contacted and ultimately collaborated with physics and maths professor Alan Sokal and psychology professor Harris Friedman on a re-analysis of the paper's data (hereafter the Brown-Sokal-Friedman rebuttal). They argued that Losada's earlier work on positive psychology and Fredrickson and Losada's 2005 critical positivity ratio paper contained "numerous fundamental