A Riemannian Geometric Reconstruction of Quantitative Social Science: Lie Derivatives, Morse Theory, and a Geometric Answer to the Lucas Critique (Methodological Proposal)

This paper proposes a systematic reformulation of the principal analytical methods of quantitative social science within the Riemannian geometric framework of Geometric Intelligence (GI) theory. Japanese, French, and Russian editions are forthcoming under separate DOIs. SCHOLARLY CONTRIBUTIONS The paper establishes explicit correspondences between ten widely used methods in econometrics, computational political science, quantitative sociology, and cliometrics — vector autoregression (Hamilton, 1994), Granger causality (Granger, 1969), the Chow test (Chow, 1960), simultaneous equations models (Greene, 2018), the Lucas critique (Lucas, 1976), the synthetic control method (Abadie, Diamond, and Hainmueller, 2010), the NOMINATE spatial model (Poole and Rosenthal, 1997), Bayesian structural time-series models (Brodersen et al. , 2015), social network analysis (Granovetter, 1973), and agent-based models (Epstein and Axtell, 1996) — and their natural counterparts on a Riemannian manifold constructed from data via the pullback metric of a variational autoencoder: Lie derivatives, Lie brackets, scalar curvature, Morse functions, manifold embeddings, and vector fields. In each case, the Riemannian extension reveals structural information that the existing method, by construction, cannot capture. The central contribution is a geometric reformulation of the Lucas critique. Lucas (1976, Carnegie-Rochester Conference Series on Public Policy) demonstrated that policy-regime changes alter the expectation-formation functions of economic agents, rendering fixed-parameter structural models systematically invalid. We restate this in the language of Riemannian geometry: the Lucas critique is the error of employing a fixed-metric model when the metric is not a Killing field — that is, when the Lie derivative of the metric with respect to the policy vector field is non-vanishing (LX g ≠ 0). The Lie derivative of GI theory measures structural change in a manner that is free of the structural model assumptions (utility functions, production functions, rational expectations) upon which dynamic stochastic general equilibrium (DSGE) models rely. This constitutes a response to the Lucas critique that operates on a different plane from that of DSGE. The paper further observes that whilst the method of constructing a manifold is discipline-dependent (from physical theory in physics and engineering; from data in the social sciences), the analytical apparatus available on the manifold once constructed — Lie derivatives, covariant derivatives, scalar curvature, geodesics, Morse functions — is discipline-independent. This observation implies that quantitative social scientists may share a common mathematical language with physicists, chemists, and engineers for the analysis of data on manifolds, in much the same way that Fogel and North imported the econometric toolkit into the discipline of history. CHARACTER OF THE PAPER AND THE AUTHOR'S POSITION ON ANTICIPATED OBJECTIONS This is a methodological proposal, not a proof or an empirical study, and the author wishes to state the following explicitly. (1) The precise scope of "model-free. " The Lie derivative of GI theory is free of structural model assumptions: it does not presuppose a utility function, a production function, or the rationality of expectations. It is, in this sense, structural-model-free. It is not, however, assumption-free in toto: the construction of the manifold depends on a variational autoencoder, and the VAE is itself a model with an architecture and hyperparameters. The precise characterisation is therefore "structural-model-free but generative-model-dependent. " This dependence is, we contend, qualitatively different from the structural-model dependence of DSGE: the VAE architecture can be varied and the robustness of the results verified empirically, whereas the structural assumptions of a DSGE model (rational expectations, specific functional forms for utility and production) are not easily subjected to the same kind of empirical robustness check. (2) The character of the correspondences. The correspondences proposed in this paper (e. g. Granger causality ↔ Lie bracket, Chow test ↔ scalar curvature) are functional analogies — not formal equivalence theorems. Granger causality is a statistical test; the Lie bracket is a differential-geometric operation. We do not claim to have proved that the former is a special case of the latter. We claim, more modestly, that these correspondences are illuminating, that they reveal structural information invisible to the existing methods, and that their rigorous mathematical formalisation is a worthy task for future work. (3) Empirical validation. This paper is a theoretical and methodological proposal; large-scale empirical validation is deferred to individual case studies. For an illustrative application of GI theory to historical data, the reader is referred to "The Manifold of Peter the Great" (Étale Cohomology, Zenodo, 2026), which constructs a Riemannian manifold from the policy environment of Peter the Great's reign (1687–1706). The dataset in that case study was constructed by the author on the basis of historical records and is not extracted directly from archival sources; application to real-world social science data remains a task for further research. GI Theory (Vols. 1–2): https: //doi. org/10. 5281/zenodo. 19140530 https: //doi. org/10. 5281/zenodo. 19158228 The Manifold of Peter the Great — Chapter 1 (English version): https: //doi. org/10. 5281/zenodo. 19470900 The Manifold of Peter the Great — Chapter 1 (Japanese version): https: //doi. org/10. 5281/zenodo. 19461689 The Manifold of Peter the Great — Chapter 2 (Japanese version): https: //doi. org/10. 5281/zenodo. 19476503