The Mortality Input Problem: Trajectory-Dependent Death and the Lifecycle Model

Heterogeneous-agent macroeconomics has reformed the income and wealth sides of the household problem. The HANK program established that aggregate dynamics require the cross-sectional distribution of marginal propensities to consume, balance-sheet exposures, and permanent income. Every model in this program inherits, without examination, a smooth actuarial mortality hazard calibrated to population life tables. This is the last unreformed input, and it is structurally wrong. The expected residual life of an agent is a value function defined on a manifold whose boundary is death. That function has a geometric property rarely stated in the economic literature: its curvature diverges at a specific power law rate as the agent approaches the boundary. Every lifecycle model that represents mortality as a smooth hazard imposes a bounded-curvature approximation on a function whose curvature is unbounded. No function in the smooth hazard class can encode the shock structure of catastrophic diagnoses that dominate individual mortality trajectories near the boundary, regardless of how many parameters the hazard contains. We establish three structural consequences. First, the population-level response to symmetric interventions is asymmetric: the worsening direction systematically exceeds the improving direction by a factor governed by a single cross-sectional moment, the covariance between boundary curvature and intervention exposure. Second, this covariance is not a small correction in clinical settings: unlike the borrowing-constraint analog in macroeconomics, where the boundary binds for a minority of agents, the mortality boundary is universal, and the population-level curvature integral diverges in a way the macroeconomic scaling intuition cannot accommodate. Third, the representation class that correctly encodes the boundary geometry exists: trained networks with piecewise-linear activation produce value functions that are exactly tropical polynomials in the max-plus semiring, with the density of the piecewise structure near the boundary encoding the curvature divergence that smooth functions cannot. A constellation of puzzles that the lifecycle literature has documented and not resolved, covering wealth decumulation, bequest dispersion, annuitization, retirement timing, portfolio composition, health expenditure at end of life, long-term care insurance, Social Security claiming, and pension tax choices. These are projections of a single geometric fact. Correcting the mortality input generates each of them as equilibrium properties of the model rather than as calibrated parameters or behavioral anomalies.