MMP.11:5.2 - Retain an exchange balance without inventing its mechanism

Two nonnegative amounts x and y exchange a conserved total N. The forward and reverse rates are unknown. Use locally Lipschitz nonnegative rate functions a(x,y) and b(x,y), defined on a neighborhood of the nonnegative states being used, and construct

q=x*a(x,y)-y*b(x,y),

x_dot=-q, y_dot=q.

Adding the two equations gives zero change in x+y for every admitted a and b. At x=0, x_dot=y*b(0,y)>=0; at y=0, y_dot=x*a(x,0)>=0. With these regularity conditions the continuous-time solution preserves nonnegativity. These are properties of the coupled construction. The applicability of conserved exchange to the subject remains a premise.

Even complete knowledge of q need not identify the gross transfers. For any nonnegative locally Lipschitz h, define

a_new=a+y*h, b_new=b+x*h.

The two added contributions to q are x*y*h and -y*x*h, which cancel. The whole observed evolution is unchanged. This is an algebraic family of alternatives, not merely several successful numerical fits.

For a dimensionless instance, take a=b=1. The alternative h=1 gives a_new=1+y and b_new=1+x, yet both models have q=x-y. At x=2, y=1 they both predict x_dot=-1.

Change the question: a proposed intervention suppresses only the reverse transfer while leaving the forward rate law applicable. Under that causal premise, C.28.MR replaces the reverse contribution by zero. The first model gives x_dot=-2; the second gives x_dot=-4 at the same state. Ordinary observations of x and y under the unchanged mechanisms cannot choose between these accounts. A prediction under the old operation can still be useful; the proposed intervention needs a contribution that distinguishes the mechanisms or a sufficient bound covering them.

For dimensional quantities, a and b have inverse-time units, while h has inverse-amount-inverse-time units. Restoring units prevents treating the added terms as arbitrary dimensionless corrections.