The Geometry of Investor Irrationality: λ × FAR × O = e² —— Euler-Mehta Financial Spacetime and the Advantage of Patient Capital

For nearly fifty years, behavioral finance has measured investor irrationality without recognizing its invariant structure. Loss aversion, fear asymmetry, and overconfidence have been treated as independent cognitive errors. They are not independent. Their product is Euler’s number squared.

This paper introduces a geometric framework in which the loss-recovery asymmetry of multiplicative returns generates a hyperbolic manifold with constant Gaussian curvature K = −1. On this surface, three independently measured cognitive biases (loss aversion λ = 2.25, fear asymmetry ratio FAR = 2.50, overconfidence O = 1.31) combine to yield a single invariant: λ × FAR × O = 7.369 versus e² = 7.389, verified to 0.27% with no fitting performed.

Investor irrationality has a characteristic scale, and that scale is e².

The invariant has a deeper structure. The eigenvalue e² governs the curvature of capital deployment on the manifold; its square root, Euler's number e, governs the rate. These are the second and first derivatives of the same eigenfunction. Independently, the behavioral square root √(λ × FAR × O) = 2.7146 converges on e = 2.7183 to within 0.14%, and the dual-process decomposition reveals that System 1 overshoots e by 8.4% while System 2 undershoots it by 8.0%, bracketing the constant with near-perfect symmetry. Geometry and psychology meet twice, at e and at e², because the exponential structure requires both simultaneously.

The invariant is predictive. Geodesic optimization on the manifold derives an exponential deployment rule, the EM Ladder, whose net advantage per drawdown-recovery cycle is predicted to equal ℰ_M = e(e − 1) ≈ 4.67%. This prediction uses zero free parameters. Across 4,498 rolling windows spanning up to 54 years in 13 mega-cap securities, the empirical mean advantage is +4.84% (p = 0.438, win rate 83.6%). After Newey-West adjustment for overlapping windows (effective sample size reduced from 4,498 to 67), the 95% confidence interval widens to [1.33%, 8.35%]; the predicted value remains near its center.

The strategy is antifragile: its advantage increases monotonically with market disorder, reaching an Antifragility Ratio of 12.9× across behavioral intensity quintiles (Spearman ρ = 1.00). The causal chain is specific: higher investor irrationality produces deeper drawdowns, larger deployment opportunities, and amplified returns. The advantage is generated entirely by volatility itself.

The behavioral premium is not a market inefficiency that competition eliminates. It is a structural property of human cognition interacting with multiplicative dynamics: markets cannot become "more behavioral" or "less behavioral" in aggregate, because the invariant is a property of the mind, not of the market. The premium is as permanent as the cognitive architecture that produces it.

Because the framework derives from a single geometric surface, its extensions are not separate models but natural consequences of the manifold’s curvature. The same geometry that produces the deployment rule also produces an optimal portfolio size (N* ≈ 15 from a Spectral Resolution Principle), competitive moat thresholds at powers of e, a geometric Euler-Mehta Safe Withdrawal Rate of 3.57% achieving a 100% survival rate across every historical cohort, with near-perfect survival (99.5% at 30 years, 95.5% at 50 years) across 10,000 Monte Carlo paths at each horizon, and sovereign wealth fund architecture derived entirely from Euler's number.

Financial Spacetime is precise description: the manifold's metric ds/df = 1/(1 − f) is simultaneously the geometry of price dynamics and the Weber-Fechner law governing how every human mind perceives proportional change, unifying spatial allocation and temporal distribution under a single constant.

No prior work appears to have identified the product of independently measured cognitive biases as a mathematical invariant, demonstrated that this product equals the eigenvalue of a differential operator on a Riemannian manifold derived from the multiplicative structure of returns, or established convergence between a geometric and a behavioral derivation of the same fundamental constant. Each of these components draws on established literatures (Riemannian geometry, behavioral economics, eigenvalue theory, geodesic optimization), but their specific synthesis, in which the eigenvalue of the deployment function on a hyperbolic manifold with  K = −1 equals the product λ × FAR × O to within 0.27% with no fitting performed, does not appear to have a direct precedent.