The de facto design standard in DEX protocol development prescribes a static invariant which limits liquidity providers to a narrow range of predefined pricing curves. This work argues that these limitations are not fundamental; they are vestiges of a design paradigm inhertied by modern protocols from their progenitors. I propose a new, more powerful, inverted approach for engineering bonding curves and their incorporation into DEX protocol designs, including Automatic Market Makers.

The core of this invention is the Mean-of-Derivatives (MoD) Property, a novel predicate for characterizing bonding curves. A function satisfies this property if the effective exchange rate over any interval (i.e., the slope of the secant line) is equal to a specified mathematical mean (e.g., arithmetic, geometric, logarithmic, power difference) of the marginal exchange rates (i.e., the tangent line slopes) at the interval's endpoints. This principle inverts the traditional constructions; rather than defining a curve and deriving its rate behavior, instead, the desired rate behavior is defined first and the unique bonding curve which satisfies it is constructed thereform. This single integral template is shown to be a universal constructor; by taking the limits of the shape parameter, the weighted Hoelder mean degenerates into classical means, and the resulting bonding curves converge to the canonical exponential, logarithmic, and hyperbolic forms that underpin existing DEX theory.

The framework's internal consistency is ensured by a robust system of parameter-induced duality, where a simple algebraic constraint on the parameters of a function guarantees the existence and validity of its inverse within the same class.