MMP.18:5.1 - Preserve a transfer across different time resolutions

Two components model stores A and B. Material flows from A to B with the supplied rate q(t)=k*t, where t is time since the interval’s start and k=1 unit per minute squared. The stores have enough material and capacity for the stipulated transfer over T=1 minute. Initially A=10 units and B=0.

The donor computes the interval amount:

Q = integral from 0 to T of k*t dt = k*T^2/2 = 0.5 units.
A(T) = 10 - 0.5 = 9.5 units.

The receiving simulator takes the initial rate q(0)=0 and holds it throughout the minute. It obtains B(T)=0. The combined stores now total 9.5 units, although the model contains no external removal.

The error is in the exchange. Both components must use the same transferred amount over the same interval. Sending Q=0.5 units and applying A(T)=10-Q, B(T)=Q gives a total of 10 units. C.29.BB supplies the balance; this construction makes the exchange between the two component representations satisfy it.

Now change the requested result: when does B first reach 0.125 units? Under the supplied continuous rate, B(t)=k*t^2/2, so it reaches the threshold at t=0.5 minutes. A receiver that inserts the entire amount only at T=1 minute reports a different crossing time despite preserving the final balance.

For that question, send or reconstruct the cumulative transfer Q(t)=k*t^2/2 over the interval, or use a computation with adequate intermediate and event resolution. Exchanging only the interval amount is sufficient for the final stores but insufficient for the crossing. The changed question reopens the temporal representation, not the already correct conservation argument.

A nonlinear receiver can likewise need more than an averaged input. Suppose two equally weighted fine cells supply x values (0,2), and the receiver requires the average of x^2. The mean input is 1, but the required response is (0+4)/2=2. Sending only the mean and squaring it gives 1. Supplying variance 1 restores 1^2+1=2; a justified closure could supply the same missing contribution in a larger model.