Observable Matrix Dynamics of Stocks

Abstract: The Observable Matrix Dynamics (OMD) approach monitors the time development of complex non-linear systems through the trajectory of a fixed-size distance matrix and its spectrum. We apply it to the S&P 500 cross section over three crisis decades, the 2001 dot-com bust, the 2007--2008 financial crisis, and the 2020 Covid crash, with three fixed-size observables on a fixed universe. The arccos distance matrix of the rolling return correlations reads the correlation geometry: its effective dimension collapses at the 2008 and 2020 crises, while the 2001 bust is a dispersed unwind. Read against machine-learning distance matrices, its spectrum stays in the un-relaxed, pre-learning regime with no low-dimensional manifold, so the market never learns its correlation structure or relaxes to a stationary geometry. Subtracting the market factor exposes a coherent sector rotation, whose name-level attribution identifies which stocks drive each crisis and in what order. At a short lookback these signals resolve precursors and forecast the endogenous 2008 crisis, though not the exogenous 2020 shock. The other two observables model the daily return and volatility rankings as Markov chains on their ranking spaces. The return chain has persistent, defensive-led bellwethers and near-reversible dynamics. The volatility chain is far more persistent, led by the financial sector, and is the only one to carry a weak, episodic arrow of time, flaring at market stress and matching volatility clustering and the Zumbach effect. All three matrices show coherent changes during market crashes.

• The paper applies a novel observable matrix dynamics framework to reveal high-dimensional, non-equilibrium market behavior across crises.
• It leverages rolling return correlations and ranking-based Markov chains to extract spectral, geometric, and dynamical diagnostics of market turbulence.
• Findings highlight persistent market structure, absence of low-dimensional collapse, and coherent sectoral rotations during major financial crises.

Observable Matrix Dynamics of Stocks: An Authoritative Technical Analysis

Overview and Methodology

"Observable Matrix Dynamics of Stocks" (2607.19005) applies the Observable Matrix Dynamics (OMD) paradigm to the cross-sectional analysis of the S&P 500 over three crises: the 2001 dot-com bust, the 2007–2008 financial crisis, and the 2020 Covid crash. The methodological innovation is to reduce the complex non-linear system of stock returns to a trajectory of three fixed-size matrix observables: the arccos distance matrix built from rolling return correlations, and two Markov transition matrices defined by return and volatility rankings, respectively.

The OMD framework, building on tools from random matrix theory (RMT) and Euclidean random matrices [bogomolny2003, halperin2026omd], is leveraged to extract spectral, geometric, and dynamical diagnostics rather than relying solely on scalar summaries of correlation matrices. The key insight is that a market’s high-dimensional geometry, encoded in the distance matrix, does not relax or settle into a stationary manifold, in contrast to the learning trajectories of neural networks. This persistent non-equilibrium is quantified through the spectral properties of the trajectory, sectoral decompositions, and entropy-based measures of disequilibrium and time-irreversibility.

Distance Matrix and Spectral Diagnostics

The central observable is the arccos distance matrix $M_{ij} = \arccos(C_{ij})$ of rolling correlations, interpreted via the Bogomolny–Bohigas–Schmit (BBS) random distance matrix framework. Key spectral diagnostics extracted include:

• The Perron eigenvalue $\Lambda_1$: scales with average angular separation in returns.
• The delocalized exponent $\beta$, read from the decay $|\lambda_K| \sim K^{-\beta}$ of ranked eigenvalues: in principle encoding an effective latent dimension.

In all three crises, the empirical spectra fundamentally deviate from the equilibrium predictions of BBS and the relaxation seen in learned neural representations:

• The market spectrum never displays low-dimensional manifold structure or shoulder/multiplet features, and $\beta$ remains in the unrelaxed regime $(\beta \lesssim 0.7)$ throughout.
• Relaxation phase transitions, analogues of “grokking” in machine learning, are absent. The market factor dominates, but both pre- and post-deflation spectra lack the geometric hallmarks of learning-induced dimensional collapse.

  Figure 1: BBS spectrum of the distance matrix $M(t)$, showing persistent unrelaxed, high-dimensional behavior across three crises and under market-factor removal.

Additionally, contrasting these dynamics with learning systems (e.g., transformers or MNIST experiments) shows that only trained, relaxed systems acquire structured, low-dimensional geometry detectable via $\beta > 1$ and multiplet shoulders. The equity market does not.

Figure 2: Spectral evolution in learning systems (grokking, MNIST) displays phase transitions absent from real market distance matrices.

Crisis Diagnostics: Dimensional Collapse and Sector Geometry

The onset of major crises is marked by sharp spectral changes:

• 2008 and 2020 crises display collapses in effective factor count (participation ratio) and rises in the market-factor share.
• The 2001 dot-com bust is orthogonal, with increased dispersion and decorrelation among technology names.

A crucial novelty arises when market-factor removal is performed:

• Post-deflation, sector geometry (notably utilities, technology, energy, financials) exhibits coherent rotations, captured quantitatively by projector drift and commutator diagnostics, distinguishing genuine rotation from simple eigenvalue growth.

  Figure 3: Market-factor share and effective factor count evolution, highlighting sharp contraction at crisis onsets for 2008/2020 versus dispersion for 2001.

  

  Figure 4: Covid period, showing trajectories of mean correlation, market share, effective factors, and geometric diagnostics.

  

  Figure 5: Comparison of raw and market-factor-removed spectral dynamics reveals sectoral reorganization on independent timescales.

The ‘market’ eigenvector direction remains stable across crises; it is the leading sectoral mode that rotates. This rotation is coherent (i.e., much less than random-subspace drift), with distinct sectoral composition identifying each crisis:

Figure 6: Eigenbasis rotation diagnostics — market direction is stable, but sectoral leading eigenvectors exhibit coherent rotation at crises.

Figure 7: Sector composition and concentration of the rotating market-removed leading eigenvector, pinpointing crisis identities (e.g., energy in 2008, utilities in Covid).

The name-level analysis resolves the order, synchronization, and drivers of these rotations, with strong correspondence between sectoral movers and the forecastability of endogenous versus exogenous events.

Figure 8: Name-level attribution of loading changes, quantifying which stocks reorient most at crisis onsets and their temporal synchronization.

Ranking Chains: Mixing, Entropy, and Irreversibility

Ranking-based Markov chains (for return and volatility orderings) provide a market-neutral, complementary tool:

• Return chains: Moderate persistence (mixing time ~1 week), persistent sectoral leaders/laggards, nearly reversible dynamics (low entropy production).
• Volatility chains: High persistence (mixing time ~1 month), led by financials, episodic but significant time-irreversibility at stress events, aligning with the Zumbach effect.

  Figure 9: Spectra and mixing timelines of Markov chains, measuring order persistence and volatility clustering.

  

  Figure 10: Entropy production in ranking chains, with only volatility displaying significant irreversibility at periods of market stress.

Time-resolved entropy production diagnostics demonstrate:

• Volatility ranking is essentially reversible in stable periods, but flares ($z\sim 4-8$) in response to sustained crises (2002, 2008), matching irreversible volatility clustering rather than brief symmetrical shocks.
• The return ranking never accumulates irreversibility, consistent with weak time-reversal asymmetry at the level of returns.

  Figure 11: Time-resolved entropy production “arrow of time” quantifying market irreversibility; strong in volatility during crises, absent for return ordering.

Transfer entropy analysis uncovers directed sectoral and name-level information flow:

• Utilities and defensives are leading indicators in return ranking during crises (flight to quality); financials lead in the volatility channel during financial distress.

  Figure 12: Net transfer entropy between sectors for return and volatility orderings, revealing crisis-phase leadership reversals.

  

  Figure 13: Name-level lead-lag network; transfer entropy pinpoints bellwethers and followers in both risk and performance channels.

Crash Anatomy: Co-movement and Dispersion

Crash diagnostics are twofold:

• Distance matrix: Elevated co-movement (mean correlation), contraction of effective dimension.
• Ranking chain: Enhanced cross-sectional dispersion and migration rate.

  Figure 14: Simultaneous rise in co-movement and dispersion at crashes, tightly coupled in their dynamics.

Joint dynamics across crises show coupled but temporally staggered diagnostic responses: spectral collapse occurs at crash onset; market-neutral ranking-based indicators (entropy production, transfer-entropy leadership) rise more gradually post-event.

Figure 15: Co-evolution of spectral, geometrical, and ranking-based diagnostics, highlighting time lag between correlation collapse and ranking chain response.

Implications and Theoretical Insights

Empirical Contradictions to Low-Rank or Manifold Learning in Market Geometry

Unlike machine learning systems or relaxed physical models, financial cross sections fail to develop or retain a low-dimensional manifold structure, maintaining persistent high-dimensional spectral signatures and never approaching the BBS equilibrium regime. The distance matrix remains an unrelaxed, dynamically evolving object throughout crises and recovery; this underscores the appropriateness of non-equilibrium statistical mechanics descriptions (possibly NESS) for markets.

Conditional Early Warning and Forecastability

Fragility signals (rising mean correlation, falling participation ratio) only provide early warning for endogenously building crises, not for exogenous events or decorrelated unwinds. Name-level synchronization in sectoral eigenvector rotation matches the presence or absence of predictability.

Practical Uses and Extensions

Diagnostics from distance matrices and ranking chains offer potential for:

• Dynamic covariance and risk modeling: the matrix-based observables can serve as state variables in regime-sensitive optimization and risk management.
• Enhanced regime recognition: conditionality in diagnostic response to types of shocks or crises.
• Stochastic portfolio theory: integration of transfer entropy and ranking stationarity for dynamic allocation.

Further research is directed toward the incorporation of exogenous covariates into ranking-chain dynamics and the real-time cointegration of correlated diagnostic signals for portfolio management.

Conclusion

This study delivers an integrated matrix-based framework for real-time and historical monitoring of market dynamics, revealing that financial markets, as measured by rolling distance matrices and ranking-based Markov chains, retain high-dimensional, non-equilibrium geometries even during rapid and profound regime transitions. Market crises manifest as dimensional collapse and coherent sectoral rotations—but without the geometric phase transitions characteristic of learning systems. Entropy- and information-theoretic analysis of ranking dynamics uncovers the non-equilibrium signatures of risk clustering and identifies sectoral hierarchy in crisis propagation. These results have substantial implications for market modeling, risk estimation, and future AI-driven financial decision-making.