Geometric Dynamics of Consumer Credit Cycles: A Multivector-based Linear-Attention Framework for Explanatory Economic Analysis

TL;DR

This paper develops a geometric framework for consumer credit dynamics by embedding economic states as multivectors in Clifford algebra and decomposing interactions into projection ($a \cdot b$) and rotation ($a \wedge b$) components. A linear attention mechanism operating on these multivector embeddings identifies historically analogous configurations, yielding time-varying, interpretable bivector parameters that quantify feedback loops among unemployment, savings, consumption, and revolving credit. The approach addresses limitations of correlation-based analyses by distinguishing crisis amplification from normal cyclical stress, providing exact impulse-response formulas, stability guarantees, and a structured regularization scheme. Empirically, the model applied to US data from 1980–2024 reveals crisis-specific geometric signatures (e.g., the 2008 feedback spiral vs. 1990–91 sequential dynamics; 2020 policy-driven decoupling) and offers practical tools for risk monitoring, policy calibration, and scenario analysis. Overall, the work contributes a principled, interpretable synthesis of geometric algebra, linear attention, and macroeconomic time series with implications for explanatory analysis and real-time policy assessment.

Abstract

This study introduces geometric algebra to decompose credit system relationships into their projective (correlation-like) and rotational (feedback-spiral) components. We represent economic states as multi-vectors in Clifford algebra, where bivector elements capture the rotational coupling between unemployment, consumption, savings, and credit utilization. This mathematical framework reveals interaction patterns invisible to conventional analysis: when unemployment and credit contraction enter simultaneous feedback loops, their geometric relationship shifts from simple correlation to dangerous rotational dynamics that characterize systemic crises.

Figures (7)

• Figure 1: Historical fit using GA embedding, linear attention, and MLP head. Shaded bands mark major crises. Blue line: actual charge-off rate; red dashed line: model predictions.
• Figure 2: Attended context trajectory (PCA of context vectors). Color intensity denotes charge-off rate; red circles mark crisis quarters. The bottom arc represents the 2008 crisis; the top cluster represents the COVID crisis.
• Figure 3: Component evolution heatmap (log scale). Rows represent scalar, vector, and bivector components; columns represent quarters from 1980Q1--2024Q2. Crisis periods exhibit sharp bivector spikes with different interaction structures: unemployment--credit/consumption dominance in 2008; savings-related bivectors in 2020.
• Figure 4: Attention score heatmap (log scale). Columns: current time; rows: lookback lags ($t-1$ to $t-8$). The dashed line marks perfect recency weighting. Crisis periods show broader weight dispersion across lags; normal times exhibit stronger recency focus.
• Figure 5: Attention score distributions for selected quarters. Normal periods show balanced attention across lags; crisis onset exhibits sharp short-horizon focus; crisis peaks demonstrate broad historical search patterns.

Theorems & Definitions (10)

• Proposition 1: Geometric Invariance
• proof
• Proposition 2: Bounded Outputs
• proof
• Proposition 3: Lipschitz Continuity
• proof : Proof Sketch
• Remark 1: Theoretical Justification for Structured Regularization
• Proposition 4: Exact Impulse Response
• proof
• Corollary 1: Generalization Bound