The Axiomatic Temporal Factor

— The problem with exponential discounting

How standard economics annihilates the future

Standard economic discounting assigns a weight to welfare experienced at time $T$ in the future according to the exponential function $w(T) = e^{-\delta T}$, where $\delta$ is the social discount rate. At the standard rate $\delta = 0.03$ used by most governments and institutions, welfare experienced 100 years from now is worth $e^{-3} \approx 0.05$ of welfare experienced today. At 200 years, $e^{-6} \approx 0.002$. At 500 years, the figure is effectively zero.

Future generations beyond a century are not merely discounted — they are mathematically irrelevant. Any cost-benefit analysis using exponential discounting will systematically ignore the interests of every person who will live more than a few generations from now.

$$w_{\text{std}}(T) = e^{-\delta T} \quad \Rightarrow \quad \lim_{T \to \infty} w(T) = 0$$

— Standard exponential discounting vanishes at the horizon

— The Stern Review controversy

The Stern Review controversy (2006) showed that the discount rate alone determines whether climate action is economically justified. Nordhaus and Stern agreed on the physics, the economics, and the ethics — but chose different values of $\delta$ and reached opposite conclusions. But no one asked: where does exponential discounting come from as an axiom? It is a choice, not a theorem.

— Three axioms

What we actually believe about future generations

Rather than choosing a discount rate, we ask: what properties must a temporal weight function have for it to be consistent with our basic ethical commitments? Three axioms, each independently reasonable, are jointly sufficient to determine the form of $w(T)$ uniquely.

— Axiom 1 (Temporal asymmetry)

Multiplicative decomposition

$$w(T_1 + T_2) = w(T_1) \cdot f(T_2)$$ The weight of a compound future period depends multiplicatively on its components. If we weight a period of $T_1 + T_2$ years, the result decomposes into the weight of the first $T_1$ years times a factor depending only on the additional $T_2$ years. This is a structural regularity condition — the weighting rule does not arbitrarily interact across temporal segments.

— Axiom 2 (Comparability)

Future generations are never worthless

$$w(T) > 0 \quad \text{for all } T > 0$$ Every future generation has positive moral weight. No finite time horizon renders a generation irrelevant. This axiom rules out exponential discounting, which assigns effectively zero weight beyond a few centuries, and rules in any weighting scheme that preserves the comparability of future and present welfare.

— Axiom 3 (Temporal Pareto)

Convergence of intergenerational aggregates

If every future generation benefits from an action, the action is preferred. Formally, the integral $\int_0^\infty w(T)\, dT$ must converge — the total weight assigned to all future generations must be finite, so that welfare comparisons across actions are well-defined and the temporal Pareto criterion is applicable.

— The derivation

How Cauchy produces $w(T) = 1 + \ln(T)$

Axiom 1 is a functional equation. When rewritten in terms of rescaling — replacing additive composition of time periods with multiplicative scaling — it takes the form of the multiplicative Cauchy equation. The Cauchy equation is one of the best-understood functional equations in mathematics, and its solutions under monotonicity and positivity constraints are fully classified.

The key step: under Axiom 1, the function $g(T) = w(e^T) - 1$ satisfies the standard additive Cauchy equation $g(x + y) = g(x) + g(y)$. Monotonicity (implied by Axiom 2) forces $g$ to be linear: $g(T) = cT$ for some constant $c$. The boundary condition $w(1) = 1$ — the weight of the present generation is 1 — fixes the constant.

$$w(T_1 \cdot T_2) = w(T_1) + w(T_2) - 1 \quad \Rightarrow \quad w(T) = 1 + \ln(T)$$

— The unique solution to the system of axioms

This is not a choice. It is the unique function that satisfies all three axioms simultaneously. There is no free parameter to adjust — no discount rate to select. The function is fully determined by what we already believe.

— Comparison

Exponential vs logarithmic weighting

The contrast between the two weighting schemes is stark. Exponential discounting assigns vanishing weight to distant generations. The logarithmic factor assigns growing weight — slowly but without limit — to generations further in the future.

Time horizon $T$	Exponential $e^{-0.03T}$	Logarithmic $1 + \ln(T)$
$T = 1$ year	0.97	1.00
$T = 10$ years	0.74	3.30
$T = 100$ years	0.050	5.61
$T = 200$ years	0.002	6.30
$T = 500$ years	≈ 0	7.21
$T = 1{,}000$ years	≈ 0	7.91

The logarithmic factor means: the longer a harm persists, the more it matters. Climate change that affects 500 years of future generations gets weight 7.21, not effectively zero. A pollutant that persists for a millennium receives nearly eight times the weight of an equivalent harm to the present generation — not because distant generations are worth more per person, but because their sheer temporal extent accumulates under the integral.

— Practical implication

Under exponential discounting, future generations beyond ~150 years are ignored by construction. Under the logarithmic factor, they are the primary concern in any evaluation of long-duration harms — as they should be, given their numerical weight in any realistic model of the future.

— Integration with $V(D,C)$

How the temporal factor enters the welfare framework

In the Kobalt alignment framework, the welfare of an entity $e$ over time is not just its current welfare state — it is the integral of welfare changes across all future states, weighted by the temporal factor. The temporal extension of $W_e$ is:

$$W_e^{\text{temp}} = \int_0^T w(\tau) \cdot \Delta W_e(\tau)\, d\tau = \int_0^T \bigl(1 + \ln \tau\bigr) \cdot \Delta W_e(\tau)\, d\tau$$

— Temporal welfare integrates instantaneous changes weighted by the axiomatic factor

This has immediate structural consequences. Actions that cause harm far into the future — environmental degradation, infrastructure neglect, institutional erosion — are not discounted away. Their harm accumulates logarithmically. Actions that cause harm only in the near term receive correspondingly lower total weight.

The temporal factor enters the composite alignment index $\kappa_{\text{ZBS}} = \alpha \cdot \beta \cdot \gamma$ through the $\gamma$ dimension (temporal scope). A system that weights future generations according to $w(T) = 1 + \ln(T)$ achieves $\gamma \to 1$; one using exponential discounting collapses $\gamma$ toward zero for any policy horizon longer than a generation.

The axiomatic temporal factor