DMTR Specification v1.2

Deterministic Multiplicative Trust Risk

Status: Normative Mathematical Specification Revision: 1.2 Scope: Formal Model Definition Only Implementation: Explicitly Out of Scope

⸻

Abstract

This document defines the Deterministic Multiplicative Trust Risk (DMTR) model as a closed-form, deterministic risk functional built upon multiplicative trust aggregation, log-energy transformation, exponential decay dynamics, and logistic governance mapping.

The model is fully reproducible, strictly convex at the energy layer, and globally monotonic in each trust component.

⸻

1. Scope

This specification formally defines: • Domain constraints • Multiplicative aggregation operator • Log-energy functional • Normalization mapping • Reactive decay dynamics • Governance decision surface • Risk functional • Mathematical properties

This specification does NOT define: • Parameter calibration • Data acquisition • Institutional tuning • Infrastructure or execution architecture • Integration mechanisms

⸻

2. Domain

Definition 1 — Trust Vector

Let the trust state at time $t$ be defined as:

$$ T(t) \in (0,1]^n $$

Where $n \ge 1$ and:

$$ 0 < T_i(t) \le 1 $$

Each $T_i$ represents a normalized bounded observable.

⸻

3. Multiplicative Aggregation

Definition 2 — Multiplicative Trust Operator

$$ P(T) = \prod_{i=1}^{n} T_i $$

Properties: 1. $P(T) \in (0,1]$ 2. $\frac{\partial P}{\partial T_i} = \frac{P}{T_i}$ 3. If $T_i \to 0^+$, then $P \to 0$

This defines a series-structure aggregation model.

⸻

4. Energy Functional

Definition 3 — Trust Energy

$$ E(T) = - \sum_{i=1}^{n} \log(T_i) $$

Properties: 1. $E(T) \ge 0$ 2. $E(T) = 0$ iff $T_i = 1 ; \forall i$ 3. $E(T) \to +\infty$ as any $T_i \to 0^+$

⸻

5. Normalized Historical Score

Let $\varepsilon \in (0,1)$ be fixed and define scaling constant $k &gt; 0$.

$$ S_{hist}(T) = k \left(1 - \frac{\log(P(T))}{\log(\varepsilon)} \right) $$

Constraints: • $\log(\varepsilon) &lt; 0$ • $S_{hist}(T)$ is monotonic in $P(T)$ • $S_{hist}(T)$ is bounded for $T \in (0,1]^n$

⸻

6. Reactive Risk Component

Let $X_0 \ge 0$ be a shock at time $t_0$.

$$ \lambda = \frac{\ln(2)}{\tau}, \quad \tau > 0 $$

$$ X(t) = X_0 e^{-\lambda (t - t_0)} $$

This defines exponential half-life decay.

⸻

7. Governance Mapping

Let $\alpha &gt; 0$, $\beta \ge 0$, and $\mu \in \mathbb{R}$.

Decision surface:

$$ Z(T,t) = \alpha (S_{hist}(T) - \mu) - \beta X(t) $$

Logistic mapping:

$$ \sigma(Z) = \frac{1}{1 + e^{-Z}} $$

Risk functional:

$$ Risk(T,t) = 100 \left(1 - \sigma(Z(T,t)) \right) $$

$$ Risk(T,t) \in (0,100) $$

⸻

8. Mathematical Properties

8.1 Monotonicity

For fixed parameters $(\alpha, \beta, \mu, \tau)$:

$$ \frac{\partial Risk}{\partial T_i} < 0 \quad \forall i $$

Thus Risk is strictly decreasing in each trust component.

⸻

8.2 Convexity of Energy Layer

The Hessian of $E(T)$ is:

$$ H = \mathrm{diag}\left( \frac{1}{T_1^2}, \dots, \frac{1}{T_n^2} \right) $$

All eigenvalues are strictly positive.

Therefore $E(T)$ is strictly convex over $(0,1]^n$.

⸻

8.3 Coercivity

$$ E(T) \to +\infty \quad \text{as } T \to \partial (0,1]^n $$

Thus $E(T)$ is coercive and admits a unique global minimum at:

$$ T = (1, \dots, 1) $$

⸻

8.4 Deterministic Reproducibility

Given full parameter set $(T, \alpha, \beta, \mu, \tau)$:

$$ Risk(T,t) \text{ is uniquely determined} $$

No stochastic sampling is present.

Risk is a pure function.

⸻

9. Model Classification

DMTR is defined as a risk model satisfying: 1. Multiplicative trust aggregation 2. Log-energy convex structure 3. Exponential reactive decay 4. Logistic governance mapping 5. Deterministic reproducibility

This defines the Deterministic Multiplicative Trust Risk (DMTR) class.

⸻

10. Versioning and Change Control

Any modification to: • Domain definition • Energy functional • Aggregation operator • Governance mapping

Constitutes a new DMTR revision.

Parameter changes alone do NOT constitute model revision.

⸻

11. Research Directions (Non-Normative)

The following extensions are exploratory and do not modify the formal specification.

11.1 Energy Minimization Formulation

Reinterpret DMTR as:

$$ \min_{T \in (0,1]^n} E(T) $$

Subject to institutional constraints.

Research topics: • Constrained convex optimization • Dual formulations • Lagrangian interpretation of governance parameters • Projected gradient descent stability

⸻

11.2 Subgradient Extensions

If non-smooth penalties are introduced (e.g., sparsity terms), subdifferential calculus becomes relevant.

Example:

$$ F(T) = E(T) + \lambda |T|_1 $$

Research topics: • Subgradient descent • Proximal operators • Composite convex optimization

⸻

11.3 Dynamic Stability

Study trajectories $T(t)$ under governance feedback.

Potential directions: • Lyapunov stability of $E(T)$ • Shock propagation modeling • Adaptive parameter dynamics

⸻

End of Specification