Apparent higher-order coupling from state-dependent impedance in energy-harvesting measurements

On a complete three-factor design, the three-way inclusion-exclusion contrast measures exactly the part of a joint response that no combination of lower-order responses reconstructs, and a positive value is naturally read as an irreducible three-body coupling. This note shows that the reading is not safe when the contrast is computed from power delivered into a fixed external load. For a linear source, delivered power is the available power multiplied by the load-matching efficiency eta(m) = 4m/(1+m)^2, with m the ratio of internal to external resistance. Because eta is nonlinear in m, a device whose available power is strictly additive in the three stimuli, and whose internal impedance responds to them with main effects only, still produces a nonzero three-way contrast in the fixed-load data. The contrast is given in closed form, and since eta'''(m) = 24(3-m)/(1+m)^5 vanishes only at the isolated point m = 3, the artefact is generic rather than exceptional. In a worked counterexample with ground truth known exactly, the available-power contrast is exactly zero while the fixed-load contrast is +2.33 microwatts. Across a bounded illustrative parameter sweep the artefact reaches a median of 4.0 per cent and an upper decile of 39.7 per cent of the measured span. The effect is third order in the impedance excursion, so it is a conditional rather than universal objection. When the available power is additive, two factors acting on the impedance are sufficient provided the third changes the available power, and the result holds whatever main effects the available power carries. A single impedance-moving factor cannot manufacture a contrast from an additive response, but it promotes a lower-order interaction into a three-way one. Two diagnostics follow. The first repeats the decomposition on maximum available power. The second requires no additional instrument: repeating the lattice at a second reference load can reverse the sign of a load-dependent contrast. A reversal establishes that the contrast contains a load-dependent component; it does not establish that the whole of it is artefact. The note shows that such a contrast can arise with no coupling whatever in the available power. It does not claim that any particular published result does so arise. Every computed value is regenerated by the accompanying script, which depends on nothing beyond the Python standard library.