B.5:5.2 - A physical question changes the representation

In a constructed engineering case, a cabinet contains two components. The intended question concerns the hottest component, but a proposed lumped model reports only their arithmetic mean temperature.

Let the modeled component temperatures be T1 and T2, with mean m = (T1 + T2)/2. The states (20,80) and (50,50), in degrees Celsius, both give m = 50. Their maxima are 80 and 50. A stipulated threshold of 60 therefore gives different classifications. On a state set containing both pairs, no function of m alone can recover the maximum or that classification.

A constructive repair retains the signed contrast d = (T1 − T2)/2. Then T1 = m + d, T2 = m − d, and max(T1,T2) = m + |d|. If an independently supported bound |d| ≤ Δ holds for the physical conditions, m + Δ is an upper bound within that model. Otherwise the contrast has to be estimated or measured, or the inference restricted.

The next physical inquiry is concrete: which temperature differences are possible under the actual loading, coupling and time window; how well does one temperature represent each component; and what measurements and error bounds cover the hottest relevant region? Compare a direct measurement, a two-state model and a qualified conservative bound by the evidence each requires and the decision each can support. Use existing adequate physical knowledge before selecting another experiment.

The mathematical argument exposes the lost information. For physical use, justify the two-temperature approximation. A more accurate fit to the same mean cannot settle that lost distinction. The first useful return is the changed quantity and representation question plus the physical premise that would distinguish the continuations.