Two Slow Variables: A Quantitative Bound on Heat-Budget Driving of the Earth System

A single slow variable cannot carry the Earth system's Phanerozoic variability. Several records - geomagnetic reversal rate, sea level, the deep-water budget, plate motion - rise and fall on the same slow timescale, and the most economical reading of that co-variation is one slow variable: the global heat-budget ratio G_E = [Q_rad + Q_cool + Q_ic]/Q_loss, which stands at 41/45 = 0.91 +/- 0.2 today. Made decidable against the reversal record, that framework fails in a specific, quantified way. Over the interval carrying reversal data G_E is exactly linear in age (r = 1.0000), so its apparent agreement with the inverted barrier is an agreement between two monotone curves; as a predictor it adds nothing to a constant rate (delta AIC = +1.2, p = 0.38); and spanning the observed range of rates would require an elasticity d ln f / d ln G_E >~ 51 against a fitted beta = 1.8 +/- 2.0. The exclusion is of the dynamic range, not of the sign, and no reweighting of the same curve closes a factor-of-thirty gap. What the record does require is a second, non-monotone axis (chi-square = 62.4 on two degrees of freedom, p = 2.8e-14). Its period is measurable - 185 Myr, profile interval 175-190 - and it is not the framework's own: the 200-250 Myr supercontinent half-cycle is disfavoured by delta AIC = 30-125. The measurement rests on a single cluster contrast, so it constrains a frequency without establishing a cycle. results generalise beyond this framework: any driver interpolated through a few anchors outside the data window is affine in age inside it, hence untestable by correlation; and inverting a reversal sequence into a barrier height adds no information, the two differing by at most 5e-5. As an independent planetary application, the lunar dynamo is shown to have ceased at 1.0 +/- 0.3 Ga while its budget ratio was still above unity (median 1.26), so it died of decoupling rather than starvation, and G_E > 1 is necessary but not sufficient for dynamo operation. The dataset that would settle the remaining question is a continuously sampled Paleozoic polarity timescale. paper is organised so that the reusable results lead rather than follow: the identifiability diagnostic and the lunar discrimination both stand independently of whether the proposed single variable is correct. 14 pages, 4 figures, 2 tables, 18 references.