Recognition cost fixes two-way traffic: reciprocity as a theorem, and a measured lower bound on the recognition scale

One physical premise about an elementary chemical step yields a symmetry of its rates that no thermodynamic bookkeeping supplies, and existing measurements then return a number the premise had left free.

The premise is that the mean activation barrier of the step is the recognition cost of the step's own distinction, expressed in thermal units. The recognition cost is J(x) = ½(x + x⁻¹) − 1, forced inside the framework by a composition law and one quadratic calibration, and satisfying J(x) = J(1/x). Nothing further is assumed. Additivity of cost forces the conversion to thermal units to be linear, and nonnegativity of cost forces its constant to be nonnegative, so both the shape and the sign of the resulting rate law are consequences.

The consequence is reciprocity. Writing F and G for the logarithmic forward and reverse rates at dimensionless drive A, with local detailed balance F(A) + G(A) = A, and α = F′ for the differential transfer coefficient, the premise forces

α(A) + α(−A) = 1,

equivalently that the two states are interchangeable, k₊(A) = k₋(−A). This is distinct from the detailed-balance constraint α(A) + β(A) = 1, which relates two directions at one drive where ours relates one direction at two opposite drives. It is refuted by any elementary step with a constant transfer coefficient differing from one half, and by any nonzero even part of the response, j(A) + j(−A), in a system where both directions are sampled in the same environment. Both falsifiers are free of every remaining constant.

The one quantity the framework does not derive is the scale at which the drive counts as one unit of recognition. Writing it as κ times the reorganization scale, the law's entire disagreement with Marcus theory is an extra logarithmic rate suppression Λr⁴/(48κ²), with r the drive in units of the reorganization scale. Every reported inverted region falls short of the Marcus parabola rather than exceeding it, while κ = 1 demands about twelve decades of suppression beyond it at r = 2.4. The conclusion needs only the sign of that discrepancy, so κ = 1 is excluded and κ ≳ 5 is required. The recognition scale of an electron-transfer channel is therefore not its reorganization energy but several times larger, which is the first quantitative statement about that scale the framework has received from outside itself.

We also prove why a weaker premise would not do. The hypothesis that the traffic is some function of the recognition cost is equivalent to the conclusion that it is even, so it establishes nothing; and for any odd response the induced mobility is automatically even, so weakening the transport law establishes nothing either. Linearity is the least hypothesis that is not a restatement of its own conclusion.