1. Union-Based Ethics (UBE)
Union-Based Ethics partitions the stakeholder universe into seven nested categories, from individual to cosmic scales.
Theorem 1: Union Impact Aggregation
The total impact of a decision $d$ across all stakeholder unions is computed as:
Eq. 1.1
$$\text{Impact}(d) = \sum_{i=1}^{7} w_i \int_{\Omega} \mu_i(\omega) \cdot \text{effect}(d,\omega) \, d\omega$$
Union Impact Variables
$\Omega$
Universe of Entities
The complete set of all entities (persons, animals, ecosystems, future beings) that could be affected by decisions.
Type: Universal set
$\omega$
Individual Entity
A single entity within the universe $\Omega$ (e.g., a person, animal, or ecosystem).
Domain: $\omega \in \Omega$
$\mu_i(\omega)$
Membership Function
Degree to which entity $\omega$ belongs to union $i$. Uses fuzzy logic for overlapping memberships.
Type: $\mu_i: \Omega \to [0,1]$
$\text{effect}(d,\omega)$
Effect Function
Quantifies how decision $d$ impacts entity $\omega$. Can be positive (benefit) or negative (harm).
Type: effect: $\mathcal{D} \times \Omega \to \mathbb{R}$
$w_i$
Union Weight
Democratic weight assigned to union $i$, reflecting its relative importance in the specific decision context.
Constraint: $\sum_{i=1}^7 w_i = 1$
$\int_{\Omega} \cdot d\omega$
Integration Operator
Sums effects across all entities in the universe. For discrete entities, becomes $\sum_{\omega \in \Omega}$.
Type: Lebesgue integral
The Seven Unions Structure
Definition: Stakeholder Union Hierarchy
$$\mathcal{U} = \{U_1, U_2, U_3, U_4, U_5, U_6, U_7\}$$
with the nesting property: $U_i \subseteq U_{i+1}$ for $i \in \{1,2,...,6\}$
U₁
Self Union - Individual persons directly affected by the decision
U₂
Family Union - Kinship networks and chosen family relationships
U₃
Community Union - Local collectives, organizations, and neighborhoods
U₄
Nation Union - Political communities, states, and national identities
U₅
Humanity Union - All current human beings globally
U₆
Biosphere Union - All living systems and ecosystems
U₇
Future/Cosmos Union - Future generations and cosmic considerations
Weight Normalization Mathematics
Definition: Weight Vector Constraints
The weight vector $\mathbf{w} = (w_1, w_2, \ldots, w_7)^T$ must satisfy:
Eq. 1.2
$\begin{aligned}
\sum_{i=1}^{7} w_i &= 1 \quad &&\text{(Probability normalization)} \\[0.5em]
w_i &\geq 0 \quad &&\forall i \in \{1,2,\ldots,7\} \quad &&\text{(Non-negativity)} \\[0.5em]
w_i &\in [w_i^{\min}, w_i^{\max}] \quad &&\forall i \quad &&\text{(Democratic bounds)}
\end{aligned}$
Democratic Bound Variables
$w_i^{\min}$
Minimum Weight
Lower bound ensuring union $i$ maintains minimum representation.
Typical: $w_i^{\min} = 0.05$
$w_i^{\max}$
Maximum Weight
Upper bound preventing any union from dominating decisions.
Typical: $w_i^{\max} = 0.30$
Example: Factory Closure Decision
Consider a factory employing 500 workers facing potential closure. The membership functions yield:
$\begin{aligned}
\mu_1(\text{worker}_k) &= 1.0 \quad &&\text{for } k = 1, \ldots, 500 \quad &&\text{(direct impact)} \\
\mu_2(\text{family}_j) &= 0.8 \quad &&\text{for affected family members} \\
\mu_3(\text{business}_m) &= 0.4 \quad &&\text{for local businesses} \\
\mu_7(\text{future}) &= 0.6 \quad &&\text{due to pollution reduction}
\end{aligned}$
2. Ripple Logic
Ripple Logic models how decisions propagate through stakeholder networks over time and space, like waves in a multidimensional pond.
Theorem 2: Ripple Propagation Dynamics
The spatiotemporal evolution of decision impacts follows the damped wave equation:
Eq. 2.1
$$\frac{\partial^2 R}{\partial t^2} + \gamma \frac{\partial R}{\partial t} = v^2 \nabla^2 R + S(x,t)$$
This partial differential equation governs how impacts spread and decay through the network.
Wave Equation Variables
$R(x,t)$
Ripple Amplitude Field
The magnitude of impact at position $x$ in the network at time $t$. Represents how strongly a location feels the decision's effects.
Type: $R: \mathbb{R}^n \times \mathbb{R}_+ \to \mathbb{R}$
$\frac{\partial^2 R}{\partial t^2}$
Acceleration Term
Second time derivative representing how quickly the rate of change is changing.
Units: impact/(time)²
$\gamma$
Damping Coefficient
Controls how quickly ripples lose energy. Higher $\gamma$ means faster decay of impacts.
Domain: $\gamma > 0$, units: 1/time
$v$
Propagation Velocity
Speed at which impacts travel through the stakeholder network.
Domain: $v > 0$, units: distance/time
$\nabla^2 R$
Laplacian Operator
Spatial second derivative: $\nabla^2 = \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}$. Measures spatial curvature.
Type: Differential operator
$S(x,t)$
Source Term
Initial impact distribution from decision $d$. Represents where and when the "ripple" begins.
Type: $S: \mathbb{R}^n \times \mathbb{R}_+ \to \mathbb{R}$
Network Propagation (Discrete Version)
Definition: Discrete Network Dynamics
For a network graph $G = (V,E)$ with adjacency matrix $A$, the discrete ripple evolution becomes:
Eq. 2.2
$$R_j(t+\Delta t) = \sum_{i \in \mathcal{N}(j)} A_{ij} \cdot T(R_i \to j) \cdot R_i(t) + f_j^{\text{internal}}(t)$$
Network Variables
$G = (V,E)$
Network Graph
Graph structure where $V$ is the set of nodes (stakeholders) and $E$ is the set of edges (relationships).
Type: Graph structure
$A_{ij}$
Adjacency Matrix
Connection strength between nodes $i$ and $j$. Zero if no connection exists.
Domain: $A_{ij} \in [0,1]$
$T(R_i \to j)$
Transfer Function
Models how impact transfers from node $i$ to node $j$, accounting for attenuation and delay.
Type: $T: \mathbb{R} \to [0,1]$
$\mathcal{N}(j)$
Neighborhood Set
Set of all nodes directly connected to node $j$ in the network.
Type: $\mathcal{N}(j) \subseteq V$
$f_j^{\text{internal}}(t)$
Internal Forcing
Endogenous impact generation at node $j$ (e.g., secondary effects).
Type: Time-dependent function
$\Delta t$
Time Step
Discrete time increment for numerical simulation.
Domain: $\Delta t > 0$
Multi-Dimensional Distance Metric
Definition: Generalized Distance
Distance in the stakeholder space combines multiple dimensions:
Eq. 2.3
$$d(x_1, x_2) = \left[\sum_{k=1}^{m} \alpha_k (x_{1k} - x_{2k})^2\right]^{1/2}$$
Distance Metric Variables
$m$
Number of Dimensions
Count of different proximity measures (typically 5: geographic, economic, social, temporal, causal).
Domain: $m \in \mathbb{N}$
$\alpha_k$
Dimension Weight
Relative importance of dimension $k$ in determining overall distance.
Domain: $\alpha_k \geq 0$, $\sum \alpha_k = 1$
$x_{ik}$
Position Component
Location of entity $i$ along dimension $k$.
Domain: Dimension-specific
Example: Highway Construction Ripples
For a new highway, the noise impact ripple follows:
$$R_{\text{noise}}(r,t) = \frac{A_0}{4\pi r^2} \exp\left(-\frac{r}{L_d}\right) \cos(\omega t - kr)$$
Where the example-specific constants are:
- $A_0$: Initial amplitude (e.g., in decibels).
- $L_d$: The characteristic distance over which the ripple's effect decays (e.g., ~500m).
- $\omega$: The temporal frequency of the impact, such as daily traffic cycles (e.g., 2π/day).
- $k$: The spatial frequency, related to the wavelength of the ripple.
3. Bayesian Integration
Bayesian methods allow MathGov to learn from outcomes and update beliefs about uncertain parameters.
Theorem 3: Bayesian Learning Rule
Given prior beliefs and observed data, the posterior distribution follows Bayes' theorem:
Eq. 3.1
$$\pi(\theta | \mathcal{D}) = \frac{\mathcal{L}(\mathcal{D} | \theta) \cdot \pi(\theta)}{\int_{\Theta} \mathcal{L}(\mathcal{D} | \theta') \cdot \pi(\theta') \, d\theta'}$$
Bayesian Variables
$\theta$
Parameter Vector
Unknown parameters governing system behavior (e.g., elasticities, response rates, effect sizes).
Domain: $\theta \in \Theta \subseteq \mathbb{R}^p$
$\pi(\theta)$
Prior Distribution
Initial beliefs about parameters before observing data, elicited from stakeholders.
Type: Probability density
$\mathcal{D}$
Observed Data
Empirical observations of outcomes from implemented decisions.
Type: Dataset
$\mathcal{L}(\mathcal{D} | \theta)$
Likelihood Function
Probability of observing data $\mathcal{D}$ given parameters $\theta$.
Type: $\mathcal{L}: \mathcal{D} \times \Theta \to \mathbb{R}_+$
$\pi(\theta | \mathcal{D})$
Posterior Distribution
Updated beliefs about parameters after incorporating observed data.
Type: Probability density
$\Theta$
Parameter Space
Set of all possible parameter values.
Type: Subset of $\mathbb{R}^p$
$p$
Parameter Space Dimension
The number of dimensions of the parameter space, corresponding to the number of unknown parameters in the vector $\theta$.
Domain: $p \in \mathbb{N}$
Gaussian Conjugate Update
Definition: Normal-Normal Conjugacy
When prior and likelihood are both Gaussian, the posterior remains Gaussian with closed-form update:
Eq. 3.2
$\begin{aligned}
\text{Prior:} \quad \theta &\sim \mathcal{N}(\mu_0, \sigma_0^2) \\
\text{Likelihood:} \quad y | \theta &\sim \mathcal{N}(\theta, \sigma^2) \\[1em]
\text{Posterior:} \quad \theta | y &\sim \mathcal{N}(\mu_1, \sigma_1^2)
\end{aligned}$
where the posterior parameters are:
Eq. 3.3
$\begin{aligned}
\mu_1 &= \frac{\sigma^2 \mu_0 + \sigma_0^2 y}{\sigma^2 + \sigma_0^2} \quad &&\text{(Precision-weighted average)} \\[0.5em]
\sigma_1^2 &= \frac{\sigma^2 \sigma_0^2}{\sigma^2 + \sigma_0^2} \quad &&\text{(Reduced uncertainty)}
\end{aligned}$
Conjugate Update Variables
$\mu_0, \sigma_0^2$
Prior Parameters
Mean and variance of the prior Gaussian distribution.
Domain: $\mu_0 \in \mathbb{R}$, $\sigma_0^2 > 0$
$y$
Observation
Single observed outcome value.
Domain: $y \in \mathbb{R}$
$\sigma^2$
Observation Variance
Known measurement error or natural variability in observations.
Domain: $\sigma^2 > 0$
$\mu_1, \sigma_1^2$
Posterior Parameters
Updated mean and variance after incorporating observation.
Property: $\sigma_1^2 < \min(\sigma_0^2, \sigma^2)$
Sequential Learning Process
1
Prior Elicitation: Gather initial beliefs $\theta \sim \pi_0(\theta)$ from stakeholders
2
Data Collection: Observe outcomes $\mathcal{D}_t = \{y_1, y_2, \ldots, y_t\}$
3
Posterior Update: Calculate $\pi_t(\theta) = \pi(\theta | \mathcal{D}_t)$ via Bayes' rule
4
Decision Update: Reoptimize using expected utility $\mathbb{E}_{\pi_t}[U(d,\theta)]$
Example: Learning Employment Impact
A new policy's job creation is uncertain:
$\begin{aligned}
\text{Prior belief:} \quad & \theta_{\text{jobs}} \sim \mathcal{N}(500, 100^2) \quad &&\text{(expect 500 ± 200 jobs)} \\[0.5em]
\text{First observation:} \quad & y_1 = 450 \text{ jobs created} \\[0.5em]
\text{Updated belief:} \quad & \theta_{\text{jobs}} | y_1 \sim \mathcal{N}(475, 70.7^2) \quad &&\text{(refined to 475 ± 141 jobs)}
\end{aligned}$
The uncertainty reduced from σ = 100 to σ = 70.7, showing learning.
4. Sentience Gradient Protocol
The Sentience Gradient assigns moral weight to entities based on their capacity for subjective experience.
Definition: Sentience Score Function
The sentience mapping uses a sigmoid transformation of consciousness indicators:
Eq. 4.1
$$S(e) = L + \frac{U - L}{1 + \exp(-k(I(e) - I_0))}$$
This creates a smooth gradient from minimal to maximal moral consideration.
Sentience Function Variables
$S(e)$
Sentience Score
Moral weight assigned to entity $e$ based on consciousness level. Humans normalized to 1.0.
Domain: $S: \mathcal{E} \to [0,1]$
$\mathcal{E}$
Entity Space
Set of all entities with potential moral status (humans, animals, AI systems).
Type: Universal set
$L$
Lower Bound
Minimum sentience score assigned to simplest entities.
Typical: $L = 0$ or $L = 0.01$
$U$
Upper Bound
Maximum sentience score (assigned to humans as baseline).
Fixed: $U = 1.0$
$k$
Steepness Parameter
Controls how sharply sentience changes around the midpoint.
Domain: $k > 0$, typical: $k \in [0.5, 2]$
$I_0$
Midpoint Parameter
Consciousness indicator value where $S(e) = (L+U)/2$.
Domain: $I_0 \in \mathbb{R}$
Consciousness Indicator Aggregation
Definition: Composite Consciousness Score
Eq. 4.2
$$I(e) = \sum_{j=1}^{m} \beta_j \cdot \phi_j(e)$$
where consciousness is measured across $m$ evidence-based dimensions.
Consciousness Variables
$I(e)$
Total Consciousness
Aggregate consciousness score combining all measured dimensions.
Domain: $I(e) \in [0, m]$
$\phi_j(e)$
Dimension Score
Normalized score for consciousness dimension $j$ (e.g., neural complexity).
Domain: $\phi_j: \mathcal{E} \to [0,1]$
$\beta_j$
Dimension Weight
Evidence-based importance of dimension $j$ in determining consciousness.
Constraint: $\sum \beta_j = 1$
The Five Consciousness Dimensions
Eq. 4.3
$\boldsymbol{\phi}(e) = \begin{pmatrix}
\phi_1(e) & : & \text{Neural Complexity (Information Integration)} \\[0.3em]
\phi_2(e) & : & \text{Behavioral Flexibility (Adaptive Response)} \\[0.3em]
\phi_3(e) & : & \text{Social Cognition (Theory of Mind)} \\[0.3em]
\phi_4(e) & : & \text{Self-Awareness (Mirror Test, Metacognition)} \\[0.3em]
\phi_5(e) & : & \text{Evolutionary Distance (Phylogenetic Similarity)}
\end{pmatrix}$
Theorem 4: Sentience Monotonicity
The sentience function preserves consciousness ordering:
$$I(e_1) \geq I(e_2) \implies S(e_1) \geq S(e_2)$$
This ensures entities with higher consciousness indicators receive greater moral consideration.
Example: Comparative Sentience Scores
Calibrated sentience scores for representative entities:
$\begin{aligned}
S(\text{adult human}) &= 1.00 \quad &&\text{(normalization baseline)} \\
S(\text{great ape}) &= 0.85 \quad &&\text{(high cognition, self-awareness)} \\
S(\text{dog}) &= 0.55 \quad &&\text{(social cognition, emotions)} \\
S(\text{octopus}) &= 0.45 \quad &&\text{(complex but alien cognition)} \\
S(\text{bee}) &= 0.10 \quad &&\text{(simple but present awareness)} \\
S(\text{nematode}) &= 0.01 \quad &&\text{(minimal nervous system)}
\end{aligned}$
5. Rights Floor Constraints
The Rights Floor represents inviolable constraints that no optimization can override, protecting fundamental dignities.
Definition: Rights Floor Mathematical Structure
The rights floor is a collection of constraint functions:
Eq. 5.1
$$\mathcal{RF} = \{c_j: \mathcal{D} \to \mathbb{R} \mid j = 1, 2, \ldots, m\}$$
A decision $d$ is admissible if and only if:
Eq. 5.2
$$c_j(d) \geq 0 \quad \forall j \in \{1, 2, \ldots, m\}$$
Rights Constraint Variables
$\mathcal{RF}$
Rights Floor Set
Complete collection of all inviolable rights constraints.
Type: Set of functions
$c_j(d)$
Constraint Function
Measures satisfaction of right $j$ under decision $d$. Negative values indicate violation.
Type: $c_j: \mathcal{D} \to \mathbb{R}$
$m$
Number of Rights
Total count of protected rights in the system.
Domain: $m \in \mathbb{N}$
Lagrangian Formulation
Theorem 5: Karush-Kuhn-Tucker (KKT) Conditions
The constrained optimization yields the Lagrangian:
Eq. 5.3
$\begin{aligned}
\mathcal{L}(d,\boldsymbol{\lambda},\boldsymbol{\mu}) = & \sum_{i=1}^{7} w_i U_i(R(d,t)) S(u_i) \\
& + \sum_{j=1}^{m} \lambda_j c_j(d) \\
& + \sum_{k=1}^{K} \mu_k(b_k - r_k(d))
\end{aligned}$
The optimality conditions require:
Eq. 5.4
$\begin{aligned}
\nabla_d \mathcal{L}(d^*, \boldsymbol{\lambda}^*, \boldsymbol{\mu}^*) &= \mathbf{0} \quad &&\text{(Stationarity)} \\
c_j(d^*) &\geq 0 \quad &&\forall j \quad &&\text{(Primal feasibility)} \\
\lambda_j^* &\geq 0 \quad &&\forall j \quad &&\text{(Dual feasibility)} \\
\lambda_j^* c_j(d^*) &= 0 \quad &&\forall j \quad &&\text{(Complementary slackness)}
\end{aligned}$
KKT Variables
$\mathcal{L}$
Lagrangian Function
Combines objective with constraints using multipliers.
Type: $\mathcal{L}: \mathcal{D} \times \mathbb{R}^m \times \mathbb{R}^K \to \mathbb{R}$
$\lambda_j$
Rights Multiplier
Shadow price of right $j$. High values indicate binding constraints.
Domain: $\lambda_j \geq 0$
$\mu_k$
Resource Multiplier
Shadow price of resource $k$. Marginal value of additional resources.
Domain: $\mu_k \geq 0$
$d^*, \boldsymbol{\lambda}^*, \boldsymbol{\mu}^*$
Optimal Solution
Decision and multipliers satisfying all KKT conditions.
Type: Saddle point
$K$
Number of Resource Types
The total number of distinct resource categories being constrained (e.g., financial, environmental). It is the size of the set $\mathcal{K}$.
Domain: $K \in \mathbb{N}$
Fundamental Rights Categories
Eq. 5.5
$$\mathcal{RF} = \mathcal{RF}_{\text{subsist}} \cup \mathcal{RF}_{\text{dignity}} \cup \mathcal{RF}_{\text{particip}} \cup \mathcal{RF}_{\text{non-disc}} \cup \mathcal{RF}_{\text{environ}}$$
$\mathcal{RF}_{\text{subsist}}$
Subsistence Rights
Basic needs: water, food, shelter, healthcare access.
$\mathcal{RF}_{\text{dignity}}$
Dignity Rights
Human dignity: freedom from torture, slavery, degradation.
$\mathcal{RF}_{\text{particip}}$
Participation Rights
Democratic participation: voting, assembly, expression.
$\mathcal{RF}_{\text{non-disc}}$
Non-discrimination
Equal treatment regardless of identity characteristics.
$\mathcal{RF}_{\text{environ}}$
Environmental Rights
Clean air, water, and livable climate conditions.
Example: Water Access Constraint
The minimum water access right is formalized as:
$$c_{\text{water}}(d) = \min_{i \in \mathcal{P}} \{q_i^{\text{water}}(d)\} - q_{\text{min}} \geq 0$$
where:
- $\mathcal{P}$: Set of all persons.
- $q_i^{\text{water}}(d)$: Water access (liters/day) for person $i$ under decision $d$.
- $q_{\text{min}}$: The WHO minimum for subsistence, set at 50 liters/day.
This ensures no person falls below subsistence water access.
6. Master Optimization Problem
The complete MathGov system integrates all components into a unified optimization framework.
Theorem 6: MathGov Master Problem
The complete optimization incorporating all MathGov elements:
Eq. 6.1
$\begin{aligned}
\max_{d \in \mathcal{D}} \quad & \mathbb{E}_{\theta \sim \pi(\cdot|\mathcal{D})} \left[ \sum_{i=1}^{7} w_i \sum_{s \in U_i} S(s) \int_0^{T} e^{-\rho t} R(s,t,d) \, dt \right] \\[1em]
\text{subject to} \quad & V_r(d) = 0, \quad &&\forall r \in \mathcal{RF} \quad &&\text{(Rights violations)} \\
& \sum_{k=1}^{K} a_k(d) \leq b_k, \quad &&\forall k \in \{1,\ldots,K\} \quad &&\text{(Resource constraints)} \\
& \mathbb{P}[\mathcal{C}(d)] < \varepsilon \quad && &&\text{(Risk bounds)} \\
& d \in \mathcal{D} \quad && &&\text{(Feasible decisions)}
\end{aligned}$
Master Problem Variables
$\mathbb{E}_{\theta \sim \pi(\cdot|\mathcal{D})}[\cdot]$
Posterior Expectation
Expected value over uncertain parameters using updated beliefs.
Type: Expectation operator
$t$
Time
The time variable over which impacts are integrated, from 0 to the time horizon T.
Domain: $t \in [0, T]$
$T$
Time Horizon
Planning horizon for impact assessment (years).
Domain: $T > 0$
$\rho$
Discount Rate
Time preference parameter. Lower values give more weight to future impacts.
Domain: $\rho \geq 0$, typical: 0.01-0.05
$e^{-\rho t}$
Discount Factor
Exponential discounting of future impacts.
Range: $(0,1]$
$V_r(d)$
Rights Violation
Binary indicator: 1 if right $r$ is violated, 0 otherwise.
Domain: $V_r: \mathcal{D} \to \{0,1\}$
$a_k(d)$
Resource Consumption
Amount of resource $k$ used by decision $d$.
Type: $a_k: \mathcal{D} \to \mathbb{R}_+$
$\mathcal{C}(d)$
Catastrophic Event
Indicator for unacceptable outcomes (e.g., ecosystem collapse).
Type: Event in probability space
$\varepsilon$
Risk Tolerance
Maximum acceptable probability of catastrophic outcomes.
Domain: $\varepsilon \in (0,1)$, typical: 0.001-0.01
$\mathbb{P}[\cdot]$
Probability Measure
Probability of events under uncertainty.
Type: Probability operator
Hierarchical Decomposition Algorithm
Algorithm 1: Hierarchical MathGov Solver
Algorithm 6.1
$\begin{aligned}
&\textbf{function } \text{HierarchicalSolve}(\mathcal{P}, \epsilon): \\
&\quad \text{// 1. Union Decomposition} \\
&\quad \text{for } i = 1 \text{ to } 7: \\
&\quad \quad d_i^* \leftarrow \arg\max_{d_i} W_i(d_i) \\
&\quad \\
&\quad \text{// 2. Temporal Decomposition} \\
&\quad \text{Solve } T\text{-stage Bellman equation backwards} \\
&\quad \\
&\quad \text{// 3. Spatial Decomposition} \\
&\quad \mathcal{G} \leftarrow \text{GraphPartition}(\mathcal{N}, k) \\
&\quad \\
&\quad \text{// 4. Coordination Loop} \\
&\quad \textbf{repeat:} \\
&\quad \quad \boldsymbol{\lambda}^{(t+1)} \leftarrow \boldsymbol{\lambda}^{(t)} + \alpha \nabla \mathcal{L} \\
&\quad \quad \text{Update local solutions with } \boldsymbol{\lambda}^{(t+1)} \\
&\quad \textbf{until } \|\boldsymbol{\lambda}^{(t+1)} - \boldsymbol{\lambda}^{(t)}\| < \delta \\
&\quad \\
&\quad \text{// 5. Solution Assembly} \\
&\quad d^* \leftarrow \text{Merge}(\{d_i^*\}, \mathcal{RF}) \\
&\quad \textbf{return } d^*
\end{aligned}$
Algorithm Variables
$\mathcal{P}$
Problem Instance
Complete specification of the decision problem.
Type: Problem structure
$\epsilon$
Approximation Factor
Acceptable optimality gap for computational tractability.
Domain: $\epsilon \in (0,1)$
$\alpha$
Step Size
Learning rate for dual variable updates.
Domain: $\alpha > 0$
$\delta$
Convergence Threshold
Stopping criterion for coordination loop.
Domain: $\delta > 0$
Optimality Conditions
Eq. 6.2
$$\nabla_d W(d^*) + \sum_{r \in \mathcal{RF}} \lambda_r^* \nabla_d V_r(d^*) + \sum_{k=1}^{K} \mu_k^* \nabla_d a_k(d^*) = \mathbf{0}$$
Theorem 7: Approximation Guarantee
The hierarchical algorithm achieves near-optimal solutions efficiently:
$$W(d^*) \geq (1-\epsilon) \cdot W(d_{\text{opt}})$$
in computational time $\mathcal{O}(n^2 \log n)$ for $n$ stakeholders.
Example: Climate Adaptation Optimization
Coastal flood protection with budget $B = \$500M$:
$\begin{aligned}
\text{minimize} \quad & \sum_{i=1}^{7} w_i \cdot \text{ExpectedHarm}_i(d) \\[0.5em]
\text{subject to} \quad & \text{MinProtection}_j(d) \geq \text{SafetyThreshold}, \quad &&\forall j \in \text{Communities} \\
& \text{TotalCost}(d) \leq 500\text{M} \\
& \mathbb{P}[\text{Catastrophic Flood}] < 0.01
\end{aligned}$
Optimal Solution:
- Seawalls in high-density areas: $200M
- Managed retreat program: $150M
- Green infrastructure (wetlands, parks): $150M
This hybrid approach minimizes harm across all unions while respecting budget and safety constraints.
Mathematical Synthesis
Theorem 8: MathGov Completeness
The MathGov framework provides a complete mathematical specification through the tuple:
Eq. 7.1
$$\mathcal{M} = \langle \mathcal{U}, \mathbf{w}, R, \pi, S, \mathcal{RF}, \mathcal{O} \rangle$$
Framework Components
$\mathcal{U}$
Union Structure
Seven nested stakeholder categories from individual to cosmic scales.
$\mathbf{w}$
Weight Vector
Democratic allocation of importance across unions.
$R$
Ripple Dynamics
Spatiotemporal consequence propagation model.
$\pi$
Bayesian Beliefs
Posterior belief distribution over uncertain parameters.
$S$
Sentience Gradient
Moral consideration function based on consciousness indicators.
$\mathcal{RF}$
Rights Floor
Set of inviolable constraints protecting fundamental dignities.
$\mathcal{O}$
Optimization Engine
Computational layer (algorithms, solvers, heuristics) that searches for the optimal decision $d^*$ respecting all MathGov constraints.
Together, the tuple $\mathcal{M}$ provides a complete, extensible mathematical
“constitution” for ethical decision-making, ensuring transparency, rigor,
and adaptability over time.