Contents Chapter 1: The Scalar Economy →

Geometric Economics: Decision Manifolds, Equilibria, and the Geometry of Markets

Andrew H. Bond Senior Member, IEEE Department of Computer Engineering San Jose State University andrew.bond@sjsu.edu

Spring 2026

Preface

Every economics textbook begins with a simplification: reduce the agent to a utility maximizer, reduce the outcome to a number, find the number that is biggest. This simplification has been extraordinarily productive. It built the edifice of modern economics — general equilibrium, welfare theorems, mechanism design, auction theory — and its practitioners have earned more than their share of Nobel prizes.

This book argues that the simplification is also the source of economics’ most persistent failures. Not because humans are irrational — the behavioral economists have made that case convincingly — but because the space on which economic decisions are made has geometric structure that scalar utility cannot represent. When we compress a nine-dimensional decision into a single number, we lose precisely the information that determines actual behavior: which values trade off against which, where the moral boundaries lie, how trust and fairness interact with price, and why the same monetary gamble produces different choices depending on whether it is framed as a loss or a gain.

The mathematical name for this structure is geometry. Not the geometry of triangles but the geometry of curved spaces, tensor fields, and symmetry groups — the same mathematics that describes how matter curves spacetime, how electromagnetic fields transform under change of coordinates, and how conservation laws emerge from symmetry. This book applies that mathematics to economic decision-making and demonstrates that it resolves, from first principles, puzzles that have resisted scalar analysis for decades.

The central construction is the Bond Geodesic Equilibrium (BGE), a generalization of Nash equilibrium to the full decision manifold. We prove that Nash equilibrium is the scalar projection of BGE — the special case obtained by collapsing all dimensions except monetary payoff. The projection is lossy: the Scalar Irrecoverability Theorem (Chapter 6) proves that the information destroyed by this collapse is mathematically irrecoverable. Homo economicus is not wrong. It is a shadow on the wall of a nine-dimensional cave.

The framework is empirically grounded. The companion library eris-econ (published on PyPI) implements the full pipeline — decision manifold, A* pathfinding, BGE computation — and validates against published experimental data from game theory and prospect theory across multiple cultures. The results: 16 of 16 behavioral predictions correct at 2.70% mean absolute error, including cross-cultural variation in ultimatum game offers explained by metric differences rather than “cultural preferences.”

This is the fourth book in the Geometric Series. Geometric Methods in Computational Modeling (Bond, 2026a) provides the mathematical toolkit. Geometric Ethics (Bond, 2026b) develops the moral manifold. Geometric Reasoning (Bond, 2026c) generalizes from ethics to all reasoning, establishing the search-geometry connection, the heuristic field formalism, and the failure taxonomy. The present book instantiates that general framework on the economic decision manifold.

I thank my colleagues in the SJSU Department of Computer Engineering, the researchers whose experimental data make the empirical validation possible — especially Fraser, Nettle, Ruggeri, and their collaborators — and the computational resources provided by the Atlas workstation and San Jose State University.

Andrew H. Bond San Jose, California March 2026

Core Objects at a Glance

Object What It Is Where Developed
Economic decision complex \mathcal{E} 9-dimensional weighted simplicial complex of economic states and transitions Ch. 3
Decision manifold metric Mahalanobis distance + boundary penalties: d(s,t) = \sqrt{(s-t)^T \Sigma^{-1} (s-t)} + \sum_k \beta_k \cdot \mathbb{1}[\text{boundary}_k] Ch. 4
Price heuristic h(x) The price signal as scalar field guiding economic search; admissible = efficient market Ch. 5
Economic tensor Obligations as vectors, interests as covectors, GDP as a specific contraction Ch. 6
Bond Geodesic Equilibrium (BGE) Multi-agent equilibrium on the full manifold; Nash is the d₁-only projection Ch. 7
Economic gauge group \mathbb{R}^+ \times S_n (scaling × relabeling); invariance under unit changes Ch. 8
Value conservation Noether’s theorem applied to economic BIP; value ≠ created by relabeling Ch. 9
Scalar Irrecoverability No continuous \phi: \mathbb{R}^9 \to \mathbb{R} is injective; GDP destroys 8 dimensions Ch. 6

Key Results at a Glance

Finding Source
Nash equilibrium is the scalar projection of BGE (Theorem 21) eris-econ
Loss aversion varies 1.0–3.53 with dimensional activation (contradicts constant λ) eris-econ
16/16 behavioral predictions correct at 2.70% MAE eris-econ validation
Kahneman & Tversky 17 problems: 13/17 predicted (76%) eris-econ
Cross-cultural ultimatum offers explained by metric variation Henrich et al. data
Public goods: 4.3% out-of-sample prediction error Fraser & Nettle
Endowment ratio: 1.66 (consistent with experimental WTA/WTP) eris-econ

How to Read This Book

  • The theoretical path: Chapters 1–9. From the scalar failure through the full geometric framework.
  • The empirical path: Chapters 3, 4, 7, and the worked examples throughout. For readers who want data and predictions.
  • The applied path: Chapters 1, 7, 10, 13–15. From motivation through equilibrium to applications.
  • The fast path: Chapters 1, 6, 7, 10. The scalar failure, irrecoverability, BGE, and market failures. The core argument in four chapters.

Each chapter opens with a running example — Maria, who owns a small coffee shop in a gentrifying neighborhood — that grounds the abstract framework in the decisions of a real business owner navigating a multi-dimensional economic landscape.

Epistemic Status Classification

Tag Meaning Approx. Count
[Established Mathematics.] Standard results from game theory, differential geometry, topology ~25
[Empirical.] Claims supported by eris-econ validation data, published experimental results ~20
[Modeling Axiom.] Structural choices (9 dimensions, Mahalanobis metric, BGE as equilibrium concept) ~15
[Conditional Theorem.] Results that follow from stated assumptions (BGE-Nash, irrecoverability) ~15
[Speculation/Extension.] Interpretive claims beyond what data directly supports ~10

Table of Contents

Part I: The Problem

  1. The Scalar Economy
  2. Historical Precursors — Geometry Before Geometry

Part II: The Framework

  1. The Economic Decision Manifold
  2. The Economic Metric
  3. The Price Signal as Heuristic Field
  4. Economic Tensors and Scalar Irrecoverability

Part III: Dynamics and Symmetry

  1. The Bond Geodesic Equilibrium
  2. Market Symmetries and Gauge Invariance
  3. Conservation Laws for Economics

Part IV: Failure Modes

  1. Market Failures as Geometric Pathologies
  2. Financial Crises as Curvature Singularities
  3. Inequality as Metric Distortion

Part V: Applications

  1. Trade as Geodesic Exchange
  2. Regulation as Boundary Enforcement
  3. The Discount Rate as Geodesic Curvature

Part VI: Horizons

  1. Open Questions

Appendices

A. Mathematical Prerequisites B. The eris-econ Library C. Empirical Validation Data

Notation

Symbol Meaning
\mathcal{E} Economic decision complex
d_1, \ldots, d_9 Nine economic dimensions (consequences through epistemic)
\Sigma 9×9 covariance matrix encoding dimension interactions
\beta_k Boundary penalty for constraint k
\gamma^* Bond geodesic (optimal path on the decision manifold)
h(x) Price heuristic (scalar field on \mathcal{E})
f(x) = g(x) + h(x) A* evaluation function (g = System 2, h = System 1)
\text{BF} Behavioral friction (total edge-weight cost of a path)
\Gamma^+ Augmented game (multi-agent decision problem on \mathcal{E})
\alpha_i, \kappa_i Self-separation and cross-sensitivity for agent i
\lambda Loss aversion coefficient (varies with dimensionality in our framework)
BGE Bond Geodesic Equilibrium
BIP Bond Invariance Principle
Contents Chapter 1: The Scalar Economy →