Geometrical Macroeconomics

From Mach and Einstein to Curvature as a Fragility Indicator

This tutorial walks you through the complete theory behind this dashboard, step by step. No prior knowledge of differential geometry is assumed. By the end, you will understand exactly what the curvature numbers on the dashboard mean, how they are computed, and why they capture information that simpler statistics miss.

Modern economics rests on mathematical foundations that, while sophisticated, may be fundamentally mismatched to the phenomena they describe. When economists build models, they implicitly assume that moving from one economic state to another is like walking across a featureless plain--that economic space is flat. But what if this assumption is wrong?

Consider what happens during a financial crisis. Small perturbations trigger catastrophic cascades. The path from stability to collapse runs through territory that looks nothing like either endpoint. The very act of trying to escape a crisis can change what "escape" means. These phenomena suggest that economic space is not flat but curved--bent and warped in ways that standard models cannot capture.

This tutorial introduces Geometrical Macroeconomics: a framework that treats the economy as a curved manifold, where discrete Ricci curvature on financial correlation networks serves as a geometric indicator of systemic fragility. Drawing on ideas from Ernst Mach's relational ontology and Albert Einstein's geometrization program, we construct a minimal foundational structure upon which specific models can be built, tested, and refined.

How to Read This Tutorial

The tutorial is structured in layers. Each part builds on the previous one:

• Parts I-II give you the conceptual foundation -- why geometry is the right language
• Part III combines the ideas into a formal framework -- what the mathematical objects are
• Parts IV-VI address practical details -- how to make the math rigorous, with a worked example
• Parts VII-VIII derive testable hypotheses and a principled metric choice
• Parts IX-X explain the software implementation and how this compares to alternatives

If you just want to understand the dashboard, skip to Part VII and Part IX. If you want the full picture, read sequentially.

1.1 The Man and His Ideas

Ernst Mach (1838-1916) was an Austrian physicist and philosopher whose influence far exceeded his technical contributions. Though he made important discoveries in shock waves (the "Mach number" is named after him), his lasting legacy was philosophical: a radical critique of Newtonian mechanics that would later inspire Einstein's revolution.

Mach's central insight was deceptively simple: there is no such thing as absolute space. When we say a train moves at 100 kilometers per hour, we must ask: moving relative to what? The ground? But the ground moves relative to the Sun. The Sun moves relative to the galaxy. There is no fixed, immovable background against which all motion can be measured--only the relationships between things.

1.2 Mach's Principle

Mach went further. He argued that even inertia--the resistance of objects to acceleration--is not an intrinsic property of matter but emerges from gravitational relationships with all other matter in the universe.

"No one is competent to predicate things about absolute space and absolute motion; they are pure things of thought, pure mental constructs, that cannot be produced in experience."

1.3 The Economic Translation

Now apply this principle to economics. When we say a stock is worth $100, we must ask: worth relative to what? Dollars? But dollars are only valuable relative to what they can buy. Other stocks? Those are valuable relative to earnings, which are valuable relative to... the problem becomes circular.

Mach's Economic Lesson: There are no absolute values in an economy. Every price, every productivity measure, every "fundamental" only makes sense in relation to everything else.

This has three profound implications:

• No Absolute Numeraire: The choice of measurement unit (dollar, gold, labor-hour) is purely conventional. No unit of account has privileged status.
• Holistic Determination: Each economic variable is determined by all others simultaneously. Partial equilibrium analysis is, at best, an approximation.
• Path Dependence: Since relations constitute values, changing the path through economic space can change the values themselves. History matters intrinsically, not just through hysteresis effects.

The Malaney-Weinstein work on gauge theory and the index number problem is a rigorous implementation of this Machian view: price indices transform under numeraire change exactly like gauge potentials transform under gauge transformation.

What this means for the dashboard: When we build a correlation network, we are expressing Mach's insight computationally. Each asset's position in the network is defined entirely by its relationships (correlations) with other assets. There are no "absolute" coordinates--only relative distances.

2.1 From Philosophy to Mathematics

Albert Einstein (1879-1955) transformed Mach's philosophical critique into revolutionary physics. While Mach argued against absolute space, Einstein showed how to do physics without it. The result was General Relativity (1915), one of the greatest intellectual achievements in human history.

Einstein's key insight was that gravity is not a force acting across space but a manifestation of spacetime curvature. Mass and energy curve the fabric of spacetime; objects then follow the natural contours of that curved space.

2.2 The Rubber Sheet Analogy

Imagine a stretched rubber sheet. Place a heavy bowling ball on it, and it creates a depression. Roll a marble nearby, and it curves toward the bowling ball--not because the bowling ball is pulling it, but because the marble is following the curved surface. Geometry replaces force.

2.3 Three Key Ideas for Economics

• Intrinsic Geometry: Riemannian geometry describes curved spaces "from the inside," without reference to an embedding in higher-dimensional flat space. We cannot step outside the economy to view it from an absolute vantage point -- just like ants on a sphere can measure curvature without ever leaving the surface.
• Curvature as Information: Positive curvature makes nearby paths converge (like lines of longitude meeting at the poles). Negative curvature makes them diverge (like paths on a saddle). In economic terms, curvature encodes whether small perturbations are self-correcting or self-amplifying.
• Local vs. Global: Local curvature has global consequences. A small dent in spacetime near the Sun bends the path of light from distant stars. Similarly, local economic disturbances can have global effects.

2.4 Understanding Curvature Visually

Curvature	What Happens	Physical Example	Economic Meaning
R > 0	Nearby paths converge	Surface of a sphere	Perturbations self-correct
R = 0	Paths stay parallel	Flat plane	Independent agents, no coupling
R < 0	Nearby paths diverge	Saddle shape	Perturbations amplify

2.5 The Economic Translation

The mathematical apparatus of Riemannian geometry -- metrics, connections, curvature tensors -- is a general framework for describing any space with a notion of distance. That economic systems possess such structure is a hypothesis about appropriate mathematical language, not a claim about deep ontological similarity to physical spacetime.

• Economic Space: The space of all possible economic states
• Economic Distance: The aggregate "cost" of transitioning between nearby states (transaction costs, information costs, coordination friction)
• Economic Curvature: Whether small perturbations self-correct or self-amplify
• Economic Geodesics: Paths of least resistance given current constraints encoded in the metric

2.6 A Word of Caution

The transfer of physical concepts to economics carries well-documented risks. Philip Mirowski's More Heat than Light showed how neoclassical economics borrowed formalism from 19th-century physics without the underlying conservation laws. We are acutely aware of this danger. Mathematical isomorphism does not equal ontological identity. The economy is not literally a curved spacetime.

Note on Extensibility: The manifold M may include coordinates beyond directly observable market variables, representing institutional quality, informational state, or expectational coordination. We treat this as a methodological option rather than a foundational axiom.

What this means for the dashboard: The metric tensor is what turns raw correlations into geometric distances. When you see the correlation network on the Network page, each edge weight is a distance derived from the correlation between two assets. The network is a discrete approximation of the economic manifold.

3.1 Two Problems, Two Solutions

The combination of Mach and Einstein addresses two specific deficiencies in conventional economics:

Thinker	Core Idea	Problem Solved	Contribution
Mach	Relationalism	Absolutism of value	Axiom 1
Einstein	Geometrization	Flatness assumption	Axiom 2

The Absolutism of Value: Standard economics treats prices as properties of goods. Mach says: no, prices are relations between goods. The whole system matters.

The Flatness Assumption: Standard economics assumes all transitions are equally easy. Einstein says: no, the space of states can be curved, making some transitions systematically harder than others.

3.2 The Economic Manifold -- Formally Defined

The economic manifold is a tuple (M, g) where M is an n-dimensional smooth manifold with Riemannian metric g_uv. Each point represents a complete macroeconomic state. In a minimal specification with n assets:

where rⁱ is the log-return of asset i. Notice these are dimensionless -- this matters (see Part IV).

3.3 Why Curvature Sees What Correlation Cannot

This is the key insight that justifies the whole framework. Consider two scenarios:

• Scenario A: All 20 assets become 80% correlated with each other. The correlation network is dense and uniform.
• Scenario B: Assets become highly correlated within clusters (tech stocks with tech, banks with banks), with a few critical "bridge" assets connecting the clusters.

Both scenarios might have the same average correlation. But Scenario B is far more fragile -- break those bridges, and the system fragments. Curvature captures this structural difference. Negative curvature identifies bottlenecks -- edges where the network is vulnerable, where redundancy is low, where a single failure can cascade.

What this means for the dashboard: When the Risk Gauge shows negative curvature, it does not just mean "correlations are high." It means the structure of correlations is fragile -- there are bottlenecks that, if broken, would fragment the network.

3.4 Core Components Summary

Component	Mathematical Object	Economic Interpretation	Dashboard Location
State space	Manifold M	Space of macroeconomic configurations	--
Distance	Metric guv	Aggregate transition friction	Network edge weights
Interdependence	Curvature tensor	Self-correcting vs. amplifying dynamics	Risk Gauge, Curvature Trend
Efficient paths	Geodesics	Friction-minimizing trajectories	Geodesics page

The Problem

Before we can compute curvature, we need a well-defined metric. But here is a problem: what does it mean to add a change in price (dollars) to a change in quantity (units)? The expression ds² = dP² + dQ² mixes incompatible dimensions. It is mathematically meaningless.

The Solution: Work in Dimensionless Coordinates

We resolve this by using only dimensionless quantities:

• Log-returns: r_i(t) = ln[P_i(t)/P_i(t-1)]. These are pure numbers -- the ratio of two prices is dimensionless. This is what the dashboard computes from raw price data.
• Ratios: X/X₀ where X₀ is a reference value.
• Z-scores: (X - mean) / standard deviation. Also dimensionless.

The Correlation-Based Distance

For the dashboard's financial network, distances are derived from correlations:

This formula has nice properties:

• d = 0 when rho = 1 (perfectly correlated assets are "at the same point")
• d = sqrt(2) ~ 1.41 when rho = 0 (uncorrelated assets are at intermediate distance)
• d = 2 when rho = -1 (anti-correlated assets are maximally distant)
• The triangle inequality is satisfied (it is a valid metric)

With dimensionless coordinates, the metric, curvature, and all derived quantities are also dimensionless. The framework is mathematically well-defined.

What this means for the dashboard: The edge weights you see on the Network page are these correlation-based distances. Short edges mean strongly correlated assets; long edges mean weakly correlated or anti-correlated assets.

In physics, Einstein's equations are powerful because they are constrained by energy-momentum conservation. Without analogous constraints, geometric economics would be empty formalism -- you could fit anything. We propose three candidates for economic conservation laws:

Candidate 1: Accounting Identities

The equation Y = C + I + G + (X - M) is not a theory -- it is a definition. Income equals expenditure by construction. At the sectoral level, private, government, and foreign balances sum to zero. These identities constrain which points on the manifold are valid economic states.

Candidate 2: No-Arbitrage Conditions

In financial markets, if you could make risk-free profit by trading in a loop (buy A, sell for B, exchange for C, sell for more A), that is arbitrage. No-arbitrage conditions are geometric: they say certain curvature components must vanish. Arbitrage is curvature, in a precise mathematical sense.

Candidate 3: Budget Constraints and Walras' Law

The sum of excess demands across all markets is identically zero. This ensures the price system is internally consistent -- a flow conservation law.

Honest assessment: We have not solved the conservation problem. Deriving economic "field equations" from first principles remains the central open problem. But these candidates show that conservation-like constraints do exist in economics.

Let us compute curvature for the simplest possible case: a two-sector economy. This demonstrates that the framework produces concrete numbers, not just philosophy.

Step 1: Define the Metric

Consider two sectors with log-returns (r¹, r²). We choose a metric where sector 1's state affects the cost of adjusting sector 2:

When sector 1 has positive returns (r¹ > 0), the exponential factor grows, making sector 2 adjustments "costly." When sector 1 has negative returns, sector 2 adjustments become "cheaper." The parameter alpha controls how strong this coupling is.

Why this metric and not a simpler one? A naive metric ds² = f(r¹)(dr¹)² + g(r²)(dr²)² gives zero curvature in 2D. You need cross-coordinate coupling (g₂₂ depending on r¹) to get non-trivial curvature. This mirrors reality: sectors are coupled, not independent.

Step 2: Compute Christoffel Symbols

These encode how the coordinate system "twists" from point to point. For our metric, the only non-zero ones are:

• Gamma¹₂₂ = -(alpha/2) e^alpha*r1
• Gamma²₁₂ = Gamma²₂₁ = alpha/2

Step 3: Compute Curvature

For this metric, the Gaussian curvature works out to:

Step 4: Interpret

The curvature is constant and negative everywhere on the manifold. This means:

• Stronger coupling (larger alpha) produces more negative curvature
• Negative curvature means nearby geodesics diverge -- small perturbations amplify
• The asymmetric coupling between sectors creates inherent instability

This is a minimal model of how sectoral coupling generates systemic fragility. Real applications use empirically calibrated metrics, but the mathematical machinery is identical.

Connection to the dashboard: The dashboard does not compute continuous curvature like this example. Instead, it computes the discrete analogue (Ollivier-Ricci curvature) on a network -- but the geometric intuition is the same: negative curvature = diverging paths = amplifying perturbations = fragility.

This is the part that connects theory to the dashboard. We now make a concrete, testable claim about what curvature reveals.

From Continuous to Discrete

We cannot directly observe the continuous economic manifold. But we can build networks from correlation data and compute discrete curvature on them. The bridge from theory to data:

1. Collect prices for N assets over a time period
2. Compute log returns: r_i(t) = ln[P_i(t)/P_i(t-1)]
3. Estimate correlations rho_ij over a 60-day rolling window
4. Convert to distances: d_ij = sqrt(2(1 - rho_ij))
5. Keep edges where |rho| > 0.3 to filter noise
6. Compute Ollivier-Ricci curvature on each edge

Ollivier-Ricci Curvature -- What the Dashboard Computes

For each edge (i, j) in the network:

In plain English: Take two connected nodes. Each has a "neighborhood" -- the other nodes it connects to. Now ask: how expensive is it to rearrange node i's neighborhood to look like node j's neighborhood? This cost is the Wasserstein distance W₁ (also called the "earth mover's distance" -- imagine shoveling dirt from one pile configuration to another).

• If the rearrangement is cheap relative to the direct distance d(i,j), curvature is positive. The neighborhoods overlap -- the connection is part of a cluster.
• If the rearrangement is expensive relative to d(i,j), curvature is negative. The neighborhoods are far apart -- the connection is a bottleneck.

Forman-Ricci Curvature

A computationally simpler alternative. Edges in many triangles (3-cliques) have positive curvature; bridge edges connecting otherwise disconnected regions have negative curvature. We recommend computing both for robustness.

Note on predictive claims: Recent empirical work suggests curvature may be a "crash hallmark" (contemporaneous indicator) rather than a leading indicator. We frame our hypothesis conservatively: curvature captures structural fragility, and whether this fragility precedes or accompanies crises is to be determined empirically. Either result is scientifically valuable.

What to Look for on the Dashboard

• Cross-sectional: On the Curvature page, edges with the most negative curvature are the most fragile connections. These are the bottlenecks.
• Time-series: On the Curvature Trend chart, declining curvature indicates the network is becoming more fragile over time.
• Regime: Compare curvature during calm periods vs. stress periods. The hypothesis predicts curvature is more negative during stress.

So far we have used a correlation-based metric. But is there a principled way to choose the metric? Yes -- the Fisher information metric.

The Idea

Imagine you observe economic data generated by some process with unknown parameters theta. The Fisher information matrix measures how much your observations tell you about theta:

This is a Riemannian metric. Two states are "close" if their data distributions are hard to distinguish; "far" if observations easily discriminate between them.

Why This Metric Is Special

Chentsov's theorem proves that the Fisher metric is the unique Riemannian metric on statistical manifolds (up to a constant) that is invariant under sufficient statistics. The choice is not arbitrary -- it is forced by mathematical requirements. This is the strongest possible theoretical foundation for picking a metric.

A Concrete Example

For Gaussian returns with parameters (mu, sigma), the Fisher metric gives constant negative curvature: R = -1/2. This means the space of Gaussian distributions is hyperbolic -- small parameter changes lead to diverging probability distributions. Financial instability is, in a sense, baked into the geometry of Gaussian returns.

Current status: The dashboard currently uses the simpler correlation-based metric. The Fisher metric provides stronger theoretical grounding and is a natural refinement for subsequent iterations.

Here is what happens when you load the dashboard, step by step:

The Data Pipeline

Stage	What Happens	Parameters	Dashboard Page
1. Data Ingestion	Fetches 2 years of adjusted close prices from Yahoo Finance for 20 large-cap US stocks (AAPL, MSFT, GOOGL, AMZN, META, NVDA, TSLA, JPM, V, JNJ, WMT, PG, XOM, UNH, MA, HD, DIS, PYPL, BAC, ADBE). Computes log-returns.	730 days of history	Data
2. Correlation	Computes pairwise Pearson correlations over a 60-day rolling window. This gives a correlation matrix at each time point.	60-day window	Network
3. Network	Converts correlations to distances via d = sqrt(2(1-rho)). Keeps edges where |rho| > 0.3. Result: a weighted graph at each time point.	Threshold: |rho| > 0.3	Network
4. Curvature	Computes Ollivier-Ricci curvature for each edge using Wasserstein optimal transport on uniform neighborhood measures.	Uniform measure	Curvature
5. Aggregation	Mean curvature across all edges gives a single scalar kappa(t) for each time point. 7-day trend and alert levels are derived from this.	4 alert levels	Dashboard gauges

Why These Specific Stocks?

The 20-stock sample spans technology, financials, healthcare, consumer, and energy. This cross-sector mix is intentional: curvature is most informative when the network includes assets from different sectors. Intra-sector correlations are expected to be high (tech stocks move together). The interesting signal is in the cross-sector connections -- are they robust clusters or fragile bridges?

Technology Stack

Component	Technology	Role
Language	Python 3.10+	All computation
Data	pandas, numpy, yfinance	Price fetching, returns, correlations
Networks	NetworkX	Graph construction and analysis
Curvature	GraphRicciCurvature	Ollivier-Ricci and Forman-Ricci
Transport	POT	Wasserstein distance computation
Visualization	Plotly	Interactive charts on dashboard

Approach	Mathematics	What It Measures	Limitation
Standard DSGE	Optimization	Equilibrium deviations	Crises are "exogenous shocks"
Network Economics	Graph theory	Connectivity, centrality	Topology only, not geometry
Agent-Based	Simulation	Emergent cascades	Hard to calibrate, many parameters
Gauge-Theoretic	Fiber bundles	Arbitrage, price consistency	Focuses on arbitrage, not stability
This framework	Riemannian geometry	Structural fragility	Empirically unproven (open research)

What Does Geometric Economics Add?

• vs. Network economics: Standard network measures (degree, centrality, clustering) describe topology -- who is connected to whom. Curvature describes geometry -- whether connections are part of resilient clusters or fragile bridges. Topology tells you the map; geometry tells you the terrain.
• vs. Correlation analysis: Average correlation tells you "markets are moving together." Curvature tells you how they are moving together -- uniformly (resilient) or through bottlenecks (fragile). Same average, very different structure.
• vs. Gauge-theoretic economics: Complementary approaches. Gauge curvature detects arbitrage opportunities (price inconsistencies). Ricci curvature detects structural fragility (network bottlenecks). Different questions, same mathematical language.

The framework presented here offers a geometrically grounded approach to measuring systemic fragility in financial networks. By applying discrete Ricci curvature to correlation-based network structures, the implementation provides a quantitative indicator that captures structural vulnerabilities not visible through conventional correlation statistics or simple network topology measures.

Metric Choice

The choice of metric specification -- whether correlation-based or derived from Fisher information -- affects the resulting curvature values, though the qualitative behavior should remain consistent. The correlation-based metric offers computational simplicity and immediate applicability to standard financial data, while the Fisher information metric provides stronger theoretical grounding at the cost of additional complexity in parameter estimation. For initial implementation, the correlation-based approach is recommended, with Fisher information serving as a refinement for subsequent iterations.

Practical Considerations

The framework's handling of dimensional consistency through non-dimensionalization ensures mathematical well-definedness, but practitioners should remain attentive to the choice of reference values when constructing ratios and the sensitivity of results to windowing parameters. The rolling window approach introduces a trade-off between responsiveness and stability: shorter windows capture rapid changes in network structure but introduce noise, while longer windows provide smoother signals at the cost of delayed detection.

Epistemic Humility

It is important to maintain appropriate epistemic humility regarding the framework's predictive capabilities. The curvature measure captures structural fragility -- the degree to which perturbations are likely to amplify rather than dissipate -- but does not predict the timing or direction of market moves. Empirical evidence suggests that curvature may function as a contemporaneous indicator of stress rather than a leading indicator with positive lead time. This limitation does not diminish the framework's value; understanding the geometric structure of financial interdependence remains valuable for risk assessment and portfolio construction even without precise predictive power.

Implementation

The implementation builds on mature, well-tested libraries including GraphRicciCurvature for discrete curvature computation and POT for optimal transport calculations. This reliance on existing tools reduces implementation risk and allows practitioners to focus on the application-specific aspects of data ingestion, network construction, and result interpretation. The modular architecture separates concerns cleanly, enabling independent testing and validation of each component.

Theoretical Limitations

From a theoretical standpoint, the framework inherits certain limitations from its foundations. Economic processes are typically non-ergodic, as Peters (2019) has emphasized, and exhibit fundamental asymmetries between gains and losses that the geodesic framework does not fully capture. The transition from micro-level agent decisions to macro-level manifold structure involves aggregation assumptions that remain implicit rather than derived. These theoretical gaps do not preclude practical utility but should inform interpretation of results.

The conservation principles proposed -- accounting identities, no-arbitrage conditions, and budget constraints -- provide partial grounding for the geometric approach but fall short of the complete field equations that would fully specify dynamics. This incompleteness is acknowledged rather than concealed; the framework is offered as a practical tool for measuring network fragility rather than a complete theory of economic dynamics.

In conclusion, this paper has presented a complete specification for implementing geometric fragility measurement in financial networks. The theoretical foundations connect the approach to established traditions in both physics and economics, while the implementation architecture provides a concrete path to deployment. The core hypothesis -- that discrete Ricci curvature correlates with financial stress and captures structural information beyond simple statistics -- is empirically testable with the tools and methods described. The framework represents a practical application of differential geometric concepts to financial risk assessment, grounded in rigorous mathematics while remaining computationally tractable for real-world use.

References

• Ernst Mach, The Science of Mechanics (1883)
• Sean Carroll, Spacetime and Geometry
• Malaney, The Index Number Problem: A Differential Geometric Approach (1996)
• Vazquez and Farinelli, "Gauge invariance, geometry and arbitrage" (2009)
• Yann Ollivier, "Ricci curvature of Markov chains on metric spaces" (2009)
• Sandhu et al., "Graph curvature for differentiating cancer networks" (2016)
• Samal et al., "Comparative analysis of two discretizations of Ricci curvature" (2018)
• Peters, "The ergodicity problem in economics" (2019)
• Mirowski, More Heat than Light (1989)