The Joint Clearing Bargain: A Derived Objective for Two Coupled Variables, and the Constant It Reveals

A recurring problem has one shape: two coupled quantities in tension — more of one forgoes some of the other — and an external knob moving the pair along the attainable frontier. The Joint Clearing Bargain (JCB) answers with a scoring surface derived rather than fitted, z = mᵖ·(1−r²)ᶣ, pinned by boundary conditions with nothing estimated. The two need not share a unit, or carry any unit at all: both are normalized per period onto [0,1]², so the surface ranks configurations, not quantities. Three major claims sustain novelty. First, the objective: the surface treats the two sides as complements under a balanced mix and substitutes under a lopsided one, a regime transition no fixed-elasticity aggregator can carry. Second, the parameter: it carries one governing number, the clearing bond p, its returns to scale — not chosen but measured, by a non-circular estimator converging to a fixed point the system operates at and reveals. Third, the finding: that level belongs to the machinery-and-knob configuration, not to the domain it is applied to. The demonstration is a two-sided venue: the pair is committed capital against executed flow, the knob a (price, fee) found inside the clearing — no oracle, no posted curve. Three synthetic markets sharing only that configuration return the same fixed point: p ≈ 0.60 with the fee alone, p ≈ 0.76 with the price control on. Against the same flow, a constant-product market maker ties up roughly eighteen times the capital and returns its providers about a hundredth per dollar. Any tension with a lever is an admissible domain — a premium against delivered funding flow, say, or a price against delivered volume.