The Dimensional Content of Variational Free Energy: A Buckingham π Partition in a Cellular Sensing System

The free energy principle holds that any system persisting across a statistical boundary can be described as though it minimizes a single quantity, the variational free energy, and it has been offered as a unifying account of mind and life. Its status is contested: critics have questioned whether the Markov blanket construction is well defined and whether the principle explains anything or merely redescribes it. This paper brings a different kind of test to that debate. Because the variational free energy is built from probabilities it is dimensionless, and dimensional analysis supplies a unit-independent, theory-neutral criterion for physical content: a quantity that genuinely couples to the world must appear as a function of the physical dimensionless groups of the system, whereas a quantity whose value can be changed without moving any such group is not tracking the physics. Applied in a concrete, physically bounded system, bacterial concentration sensing, whose Markov blanket is an actual membrane, the criterion shows that the variational free energy partitions into three parts: a leading term equal to a known physical quantity, the Berg-Purcell measurement number; a dominant correction that is independent of every physical group and can be moved at will by the agent's priors; and a subleading term that does couple to the physics but vanishes as sensing improves. In this system the principle is a bookkeeping device laid over a coupling law that dimensional analysis exhibits directly. The test generalizes: any postulated coupling quantity can be asked whether its dimensional content collapses onto a physical group or floats free of them all.