Schr¨odinger's Repetitorium, Revisited: The Multigrade MSD Vector for Single-Trajectory Ergodicity from Lennard-Jones Gases to Global Inequality Dynamics

Schr¨odinger’s question of when the time average of an ob- servable along a single realization equals its ensemble av- erage has remained largely conceptual outside biophysics. We show that the classical Mean Square Displacement (MSD) slope ˆ αraw on the raw series, on its own, agrees with documented ergodicity verdicts on a wide range of canonical systems but does not discriminate qualita- tively distinct mechanisms of non-ergodicity that share the same scalar slope. We therefore introduce the Multigrade MSD Vector: the collection of MSD slopes ˆ α (c) 16 c=1 computed channel-by-channel on the Clifford- algebra moment-tensor signature C (y; t) ∈R16 derived from the kinematic embedding q (t) = (yt, ∆yt, ∆2yt) ∈ Cl (3, 0). We validate the diagnostic on five canonical syn- thetic processes (recovering α= 2Hexactly for fractional Brownian motion) ; on a Lennard-Jones gas at three tem- peratures (yielding α≈0, the quantitative footprint of Boltzmann ergodicity) ; on a 3D Edwards–Anderson spin glass at T <Tg (where two replicas of the same Hamil- tonian give α= 0. 01 and α= 1. 01, the operational sig- nature of Parisi replica symmetry breaking) ; on the 14 Nelson–Plosser macroeconomic series (13/14 agreement with the original 1982 unit-root verdict) ; on Argentine income deciles 1991–2024 (panel α= 1. 11, all individual deciles non-ergodic except deciles 8–9) ; on Panel Study of Income Dynamics 1968–2023 (panel αfalls from 0. 75 pre- 1985 to 0. 35 post-1985, directionally consistent with the fading absolute mobility documented by Chetty et al. ) ; and on the World Inequality Database covering 200+ countries (where∼92% of national Gini trajectories are non-ergodic and the prediction that Nordic economies are ergodic is quantitatively refuted). The mean MSD slope on the bivector sub-block, ⟨ˆ α (2) ⟩, separates classical in- tegrated processes (random walk, fBm; ⟨ˆ α (2) ⟩≈0) from structurally non-ergodic processes (P´olya urn; ⟨ˆ α (2) ⟩< 0) that share the same ˆ αraw. On real data, the vec- tor reveals mechanistic distinctions invisible to the scalar test: Sweden and Norway, comparable in ˆ αraw, occupy different mechanistic regimes (Sweden: bivector-negative structural; Norway: I (1) -pure) ; PSID 1968-1985 vs 1986- 2023, similar in scalar dynamics, separate into qualita- tively distinct mechanism classes. The two-dimensional plane (ˆ αraw, ⟨ˆ α (2) ⟩) is the central diagnostic of the pa- per. The construction has two operational advantages over the scalar slope: (i) interpretability — each compo- nent of the vector is the MSD slope of an algebraically- meaningful object (level, velocity, scalar product, bivec- tor, trivector), so a series is classified not just as “ergodic / non-ergodic” but as a specific mechanism class (I (1) - pure, P´olya-like structural, anomalous-diffusive) ; and (ii) short-trajectory power — the kinematic whitening ˜ q = Σ−1/2 q q on which the channels are built decorrelates the three components of the embedding and materially improves the discriminative power at small T, lifting Co- hen’s d for P´olya-vs-RW separation from∼0 to 0. 77 at T = 50 where the non-whitened diagnostic fails.