Gravity of Risk

A Barrier-Coordinate Theory of Existential Risk Measurement

Overview

Linear risk measures treat probability as a flat coordinate: every increment of p is equal regardless of proximity to an absorbing boundary. This paper shows this is a systematic measurement error with a specific, quantifiable structure.

Key Results

Part I: Structural Layer

Using the ruin probability framework of classical ruin theory (Cramér, 1930; Lundberg, 1903), we define a barrier coordinate s(p) = −log(1 − p) derived from the one-sided restriction of KL divergence to the absorbing boundary.

Two structural results are proved, requiring no dynamical or directional assumptions:

• Proposition 1 (Barrier-Coordinate Underestimation): Any affine approximation in p systematically underestimates barrier-coordinate increments, with exact error ∑ δⁿ/n for n ≥ 2, where δ is the fraction of remaining survival margin consumed. Bounded between δ²/2 and δ²/2(1−δ).

• Proposition 2 (Bregman Asymmetry): The induced Bregman discrepancy is asymmetric, with the asymmetry cubic at leading order (δ³/3). The underestimation error of Proposition 1 is exactly the forward Bregman divergence — two views of the same object.

These results hold regardless of dynamics, direction, or domain.

Part II: Dynamic Specialization

Applying the structural layer to maintenance-limited non-equilibrium systems using the viability-theoretic framework (Aubin, 1991):

• Maintenance ratio ρ = m/E (minimum maintenance power over total available power)
• Local maintenance frontier at ρ = 1 serving as a local energetic proxy for the viability kernel boundary
• Shadow-value amplification of additional power by (1 − ρ)⁻¹ relative to flat models
• Kernel capture — the condition where the system remains functional but every admissible trajectory terminates at the absorbing boundary

Under a thermodynamically motivated sign assumption (uncontrolled worsening, a(z) > 0), superlinear growth in energy output becomes necessary to keep ρ bounded away from 1 in regimes where maintenance burden scales superlinearly near the boundary.

Central Thesis

Standard linear risk measures are not wrong because human cognition is biased. They are wrong because they linearize the wrong coordinate. The error is one-sided, quantifiable, and grows without bound as the absorbing boundary is approached.

What This Paper Does Not Do

• It does not determine whether any specific system is moving toward or away from the boundary
• It does not resolve whether AI, or any other technology, poses an existential threat
• It does not prescribe policy
• The sign of the drift — whether AGI causes or prevents viability capture — cannot be determined from within the framework

Connections

The barrier coordinate s(p) = −log(1 − p) is simultaneously:

• The negative log-survival (survival analysis)
• The cumulative hazard function (actuarial science)
• The self-information of survival (information theory)
• The canonical self-concordant barrier on [0, 1) (interior-point optimization)

The dynamic specialization is in significant part a formal mathematical treatment of Tainter's thesis: civilizations collapse when the marginal cost of maintaining complexity exceeds available energy. The maintenance ratio ρ = m/E and the frontier at ρ = 1 are Tainter's argument expressed as a control problem.

References

• Amari, S. (1985). Differential-Geometric Methods in Statistics. Springer-Verlag.
• Ames, A. D., et al. (2019). Control barrier functions: Theory and applications. 18th European Control Conference.
• Aubin, J.-P. (1991). Viability Theory. Birkhäuser.
• Bregman, L. M. (1967). The relaxation method of finding the common point of convex sets. USSR Computational Mathematics and Mathematical Physics, 7(3):200–217.
• Cramér, H. (1930). On the mathematical theory of risk. Försäkringsaktiebolaget Skandias Festskrift.
• Lundberg, F. (1903). Approximerad Framställning af Sannolikhetsfunktionen. Doctoral thesis, Uppsala.
• Merton, R. C. (1974). On the pricing of corporate debt. Journal of Finance, 29(2):449–470.
• Nesterov, Y. & Nemirovski, A. (1994). Interior-Point Polynomial Algorithms in Convex Programming. SIAM.
• Scheffer, M., et al. (2009). Early-warning signals for critical transitions. Nature, 461:53–59.
• Tainter, J. A. (1988). The Collapse of Complex Societies. Cambridge University Press.

Citation

If you use this work, please cite:

Alioto, J. P. (2026). Gravity of Risk: A Barrier-Coordinate Theory of
Existential Risk Measurement.

License

This work is licensed under CC BY 4.0.