Three Quarters: The Exponent at the Border of Two Laws

Every student of life meets the same fraction early. Basal metabolic rate rises with body mass M as B proportional to M^a, and the exponent a is taught as 3/4: Kleiber's law, fitted by Max Kleiber in 1932. This paper censuses what actually supports that number. The standing null needs no biology: Rubner's surface geometry of 1883 derives 2/3 from dimensional analysis alone. The one famous derivation of 3/4, West, Brown and Enquist's 1997 model of space-filling fractal distribution networks, does not survive its own corrections: Kozlowski and Konarzewski (2004) showed the model is internally consistent only if the exponent is 1 or its size-invariant capillary is given up, and Savage, Deeds and Fontana showed the model's 3/4 holds only in the limit of infinite network size, while across the eight orders of magnitude of mammalian sizes the same model predicts 0.81, at odds with the data it was built to explain. The data decline to convict the null: Dodds, Rothman and Weitz (2001), re-analyzing the classic mammal and bird compilations, found little evidence for rejecting 2/3 in favor of 3/4, with one cleaned compiled set at 0.72 plus or minus 0.02; and White, Cassey and Blackburn's census of 127 published interspecific exponents (2007) found no universal exponent at all. Across the mammalian mass range the two laws' predictions separate by a factor of about 3.8. Grade: **SURROUNDED, NOT DERIVED — THE NULL STANDS AT TWO-THIRDS, THE CLAIM AT THREE-QUARTERS, AND THE MODEL THAT WOULD SETTLE IT PREDICTS A THIRD NUMBER AT REAL SIZES.** This paper belongs to the breadth program; it touches no claim of the physics core.