A Kirchhoff Prediction from the QEC Sum: Sum 1/R_n Cannot Exceed 2R

The QEC (Johnson) sum of all the unit fractions is 2. In a resistor network, the unit fractions appear as the conductances 1/R, 1/2R, 1/3R, ... of a harmonic ladder. This paper first shows that Kirchhoff's laws already produce the 2R of the QEC sum. A source of internal resistance R drives the harmonic ladder in parallel. The loop resistance R + (Sum_{n=1}^{N} 1/nR)^(-1) never exceeds 2R for any number of rungs N, and the delivered power never exceeds V^2 / 4R. Both bounds are reached at the first rung alone, the unit fraction 1/1, which is the matched load of the maximum power transfer theorem, a measured result. From the QEC sum, the paper then predicts that the parallel sum itself, Sum_n 1/R_n with n → ∞, cannot exceed 2R. The prediction requires that siemens and ohms become equivalent somewhere in the circuit. A bench test with resistors and a meter is specified, with its pass and fail criteria fixed in advance.