Decay Function

Role of decay in the Trust State Protocol

The decay function defines how trust state evolves in the absence of new verified events. Its role is to model the gradual loss of informational relevance as time passes without reinforcement. In Trust State Protocol, decay is not an auxiliary feature or an optional heuristic. It is a mandatory component of trust evolution and a primary mechanism by which the protocol avoids permanent or stale trust attribution.

Decay represents increasing uncertainty, not negative inference. The absence of recent information does not imply misconduct or unreliability. It implies only that prior outcomes become progressively less informative about future interactions as conditions, incentives, and behavior may change over time.

Trust state under temporal evolution

Let $T(t) \in [T_{\min}, T_{\max}]$ denote the trust state of an entity within a given context at time $t$. Let $t_i$ denote the time at which the most recent verified event was incorporated into the trust state, producing the value $T(t_i^+)$.

For all $t > t_i$ such that no new verified event occurs in the interval $(t_i, t]$, trust state evolves solely through decay. The decayed trust state at time $t$ is defined as

$$T(t) = D\big(T(t_i^+),\, t - t_i\big)$$

where $D$ is the decay operator.

Baseline uncertainty

Decay operates relative to a baseline uncertainty level, denoted $T_0$, which represents the trust state toward which confidence converges in the absence of information. $T_0$ is context specific and must satisfy

$$T_{\min} \le T_0 \le T_{\max}$$

Baseline uncertainty does not represent distrust, neutrality, or default reliability. It represents epistemic uncertainty in the absence of recent evidence. In most contexts, $T_0$ will correspond to an initialization value, but the protocol does not require this.

Functional requirements of the decay operator

The decay operator $D$ is constrained by several mandatory properties.

First, boundedness. For all admissible inputs, decay must preserve trust state within the defined bounds:

$$D(x, \Delta t) \in [T_{\min}, T_{\max}] \quad \forall x \in [T_{\min}, T_{\max}],\; \Delta t \ge 0$$

Second, continuity in time. Trust state must vary continuously as a function of elapsed time. Discrete jumps in trust state due solely to time progression are not permitted.

Third, monotonic convergence. For any initial trust state $x \ne T_0$, the decayed trust state must converge monotonically toward $T_0$ as elapsed time increases. Decay must not oscillate or overshoot the baseline.

Fourth, event independence. The decay process must depend only on the current trust state and elapsed time. It must not depend on the polarity, magnitude, or semantics of past events beyond their effect on the current state.

These properties ensure that decay behaves as a stabilizing force rather than as an implicit penalty or reward.

Canonical decay formulation

A canonical and admissible form of the decay operator is exponential decay toward baseline uncertainty:

$$D(x, \Delta t) = T_0 + (x - T_0)\, e^{-\lambda \Delta t}$$

where $\lambda > 0$ is the decay rate parameter.

This formulation satisfies all required properties. It is continuous, bounded, monotonic, and asymptotically convergent. The parameter $\lambda$controls the rate at which confidence erodes in the absence of new information. Larger values of $\lambda$ produce faster decay, while smaller values produce slower decay.

The protocol does not mandate exponential decay specifically, but any alternative formulation must satisfy the same constraints and produce equivalent qualitative behavior.

Interpretation of decay rate

The decay rate parameter $\lambda$ is context specific and reflects how quickly information becomes obsolete in that interaction domain. Contexts characterized by rapidly changing conditions or incentives may require higher decay rates, while contexts involving slower dynamics may justify lower rates.

Importantly, decay rate is not a moral or policy parameter. It encodes temporal uncertainty, not tolerance or punishment. The protocol requires that $\lambda$ be deterministic and declared for a given context, but it does not prescribe how it should be chosen.

Interaction between decay and events

Decay and event driven updates operate sequentially, not concurrently. At any time $t$ when a new verified event occurs, decay is first applied from the previous event time to $t$, producing a decayed trust state. The event update rule is then applied to that decayed value.

This ordering ensures that trust state always reflects both temporal distance and new information. It also prevents manipulation through delayed event reporting or artificial clustering of updates.

Stability and convergence behavior

In the absence of events, trust state converges asymptotically to $T_0$. Under repeated events with consistent impact, trust state converges to a stable equilibrium determined by the balance between event driven reinforcement and decay.

This behavior ensures long term stability. Trust does not diverge, oscillate, or lock into extreme values without continued reinforcement. The decay function therefore plays a critical role in ensuring that the trust state model remains well behaved under arbitrary interaction histories.

Resistance to stale trust accumulation

A primary objective of decay is to prevent stale trust accumulation. Without decay, early positive outcomes could dominate trust indefinitely, even as behavior, incentives, or context change. By enforcing decay, TSP ensures that trust reflects recent, relevant information rather than historical artifacts.

This property is especially important in systems with intermittent participation or long lived identifiers.

Contextual isolation of decay

Decay parameters are scoped to context. Different contexts may define different baseline uncertainty levels and decay rates. There is no global decay function and no shared temporal dynamics across contexts.

This isolation preserves semantic integrity and prevents implicit coupling between unrelated interaction domains.