The Climate Hazard Yield Surface: A Geometric Framework for the Economics of Climate Action, with Per-Household Dollar Translation and the Doomsday Clock Isocurve

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Climate action is an investment: costs are front-loaded and benefits arrive continuously over decades. Standard economic communication presents only the cost side, because the benefit structure has not been given a precise mathematical form. We introduce the Climate Hazard Yield Surface $\Phi(t_{inv}, t_{pay})$—the damage-avoided per dollar of mitigation investment made at $t_{inv}$, realised at $t_{pay}$.

The surface is strictly two-dimensional, not a curve. Its domain is a triangular region in the $(t_{inv}, t_{pay})$ plane (causality).

We establish three core geometric properties of this surface:

1. Nonlinear Convexity: It exhibits positive curvature near the $3^\circ C$ tipping-point threshold.

2. Non-Separability: There is no discount-factor decomposition $\Phi = f(t_{inv}) \cdot g(t_{pay})$ because investment and redemption are inextricably entangled through the physical atmospheric carbon budget.

3. The Doomsday Clock Isocurve: We prove that the breakeven isocurve $\{\Phi = 1\}$—the set of investment dates that just recover their cost by a given payoff horizon—is well-defined, computable, and constitutes a literal "doomsday clock": a precise, scenario-specific deadline for capital deployment before mitigation yields mathematically collapse below 1.0.

By mapping climate mitigation directly onto the term structure mathematics of the Pacioli Manifold (Economic Gauge Theory), this framework provides a rigorous foundation for integrating physical climate risk directly into standard fixed-income and derivative pricing models.