Exact Conservation Laws of the Lorenz Attractor: Classification and Deterministic Prediction of Lobe-Switching Events
Predicting when a chaotic trajectory will switch between the lobes of the Lorenz attractor is a long-standing challenge in nonlinear dynamics. This work shows that algebraic conservation laws, constructed by augmenting phase space with history-accumulating auxiliary variables, provide a deterministic solution. Systematic enumeration identifies eighteen valid invariants in three classes, each tied to a nullcline of the Lorenz flow, while six candidates fail, proving that the dynamics constrains which conservation laws are admissible. One class generates sharp spikes synchronized with lobe-switching events, achieving $99.2\%$ sensitivity with $0.3\%$ false-positive rate ($\mathrm{AUC} = 0.9995$) as a continuous Poincaré section analogue. The spike amplitude predicts switching latency via $Δt = t_{\min} + C\mathcal{A}^{-n}$ with $R^2 > 0.95$ across all parameter combinations tested. At canonical parameters $(σ, ρ, β) = (10, 28, 8/3)$, $n = 2.14 \pm 0.17$ with $R^2 = 0.93$ for individual events; the exponent increases with $β$ and decreases with $ρ$, while the $σ$-dependence is non-monotonic. The latency distribution reveals a topological gap of width $Δt_{\mathrm{gap}} \approx 0.68 \pm 0.01$ for $ρ$ sufficiently above the onset of chaos, explained by the Shilnikov passage map. Under stochastic perturbations, lobe-sensitive invariants are ${\sim}\,10^3$ times more robust than their smooth counterparts. In the Rayleigh-Bénard convection context, the auxiliary variables correspond to integrated heat-flux anomalies. Conservation is verified to $O(10^{-36})$.