State Transition Rules

Trust state as a bounded dynamical variable $\mathrm{<i>T</i>}$$($$t$$)$$\in$$[$$T_{}$${\min }_{}$${}_{}$$,$$T_{}$${\max }_{}$${}_{}$$]$$\subset$$\mathbb{R}$

State Transition Rules define the formal mechanism by which trust state evolves in the Trust State Protocol. They constitute the mathematical core of the protocol and provide the only normative specification governing how trust values change over time. All implementations of TSP must be reducible to the transition rules defined in this section. No other component of the protocol modifies trust state.

The objective of these rules is not to predict behavior, but to provide a stable, deterministic, and auditable method for updating confidence based on verified outcomes and elapsed time.

Within a given context, the trust state is modeled as a bounded real valued state variable. Let

denote the trust state of a specific entity at time $t$.

The bounds $T_{\min}$ and $T_{\max}$ are fixed per context and represent absolute lower and upper limits on confidence. These bounds are not derived from empirical probabilities and do not represent certainty or impossibility. They serve to constrain the state space and ensure stability and interpretability of the model.

Trust state is explicitly not a probability measure. It is a confidence state whose semantics are defined only relative to prior outcomes within the same context.

Discrete event driven updates

Trust state evolution is driven by a sequence of verified events occurring at discrete times
$\{ t_1, t_2, \ldots, t_n \}$.

Each event $e_i$ is associated with an event impact value
$I_i\in [T_{\min },T_{\max }],$ which represents the directional contribution of the event to confidence within the context. Event impact values are determined externally through the Event Model and are assumed to be bounded, deterministic, and context specific.

Let $T(t_i^-)$ denote the trust state immediately prior to event $e_i$, and $T(t_i^+)$ the trust state immediately after incorporation of that event.

Convex update formulation

The fundamental update rule of Trust State Protocol is defined as a convex combination of prior trust and event impact:

$$T(t_i^+) = (1 - \alpha_i)\, T(t_i^-) + \alpha_i\, I_i$$

where
$\alpha_i \in (0, 1]$
is a weighting parameter governing the sensitivity of trust state to the new event.

This formulation has several important properties.

First, it guarantees that trust updates remain within the convex hull of prior trust and event impact, ensuring boundedness even before explicit clamping is applied.

Second, it enforces incrementalism. No single event can fully determine trust state unless $\alpha_i = 1$, which is permitted only under explicitly declared contextual rules.

Third, it ensures linear interpretability. The contribution of historical behavior decays proportionally as new information is incorporated.

The protocol does not prescribe how $\alpha_i$ is selected, but it requires that the function determining $\alpha_i$ be deterministic, declared, and invariant under identical conditions. Typical influences may include verification strength, interaction criticality, or risk exposure, but these remain outside the protocol core.

Bound enforcement and saturation behavior

After each update, the resulting trust state is constrained to the admissible interval:

$$$$T(ti+)←min⁡(Tmax⁡,max⁡(Tmin⁡,T(ti+)))T(t_i^+) \leftarrow \min\left(T_{\max}, \max\left(T_{\min}, T(t_i^+)\right)\right)

This saturation operation ensures that trust state never escapes the defined state space, even under adversarial or extreme sequences of events.

Bound enforcement is not optional. It is a structural invariant of the protocol and must be applied uniformly across implementations.

Temporal evolution as a decay process

Between verified events, trust state evolves continuously as a function of time. Let $\Delta t$ denote the elapsed time since the most recent event. The trust state at time $t$ is given by

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

where $D$ is a decay operator satisfying the following minimal properties:

• monotonic convergence toward baseline uncertainty
• continuity in time
• boundedness within $[T_{\min}, T_{\max}]$
• independence from event polarity

Decay represents increasing uncertainty due to lack of recent information. It does not encode negative inference and does not imply degradation of behavior. Even highly verified events do not eliminate decay.

The explicit functional form of $D$ is defined separately, but the state transition rules require that decay be applied prior to any subsequent event update, ensuring temporal consistency.

Determinism and reproducibility

The state transition system defined here is deterministic. Given:

• identical initial trust state
• identical event sequence
• identical event impact values
• identical weighting parameters
• identical time progression

the resulting trust state trajectory must be identical across independent implementations.

This determinism is essential for auditability and cross system reasoning. Any stochasticity, heuristic adjustment, or discretionary override violates the protocol.

Stability and convergence properties

Under reasonable assumptions on the decay operator and bounded event impacts, the trust state model exhibits stability. In the absence of events, trust state converges toward a baseline determined by the decay function. Under repeated events with consistent impact values, trust state converges toward a fixed point determined by the balance between event influence and decay.

This behavior ensures that trust does not oscillate uncontrollably, diverge, or lock permanently into extreme states without continued reinforcement.

Contextual isolation of state dynamics

All variables, parameters, and functions defined in this section are scoped to a single context. Trust states in different contexts evolve independently and do not influence one another. There is no aggregation, normalization, or projection of trust across contexts within the protocol.

This isolation preserves semantic integrity and prevents cross domain inference.

Independence from policy and interpretation

State Transition Rules define how trust changes, not what trust means operationally. Trust values produced by these rules are signals, not decisions. Any interpretation, thresholding, or enforcement based on trust state occurs entirely outside the protocol.

This separation ensures that the mathematical model remains neutral and portable across systems with different objectives.