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Established in the locally conformally flat case by Schoen [43,44] and for n\\leq 24 by Khuri-Marques-Schoen [26], it has revealed to be generally false for n\\geq 25 as shown by Brendle [8] and Brendle-Marques [9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1) Scal_g, Scal_g being the Scalar curvature of (M,g). 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Established in the locally conformally flat case by Schoen [43,44] and for n\\leq 24 by Khuri-Marques-Schoen [26], it has revealed to be generally false for n\\geq 25 as shown by Brendle [8] and Brendle-Marques [9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1) Scal_g, Scal_g being the Scalar curvature of (M,g). 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