Computational Ethics: A Field Theory of Moral Operators

Introduction: The Shift to Moral Physics

Traditional ethical discourse has long been mired in the metaphysical—debating the subjective nature of “good” and “ evil” as if they were ethereal substances. In the era of high-frequency multi-agent systems and autonomous intelligence, this approach is no longer sufficient. We must transition from moral philosophy to Moral Systems Engineering: the application of rigorous control theory and network dynamics to the problem of value alignment.

Ethics can be rigorously framed as a field theory. In this paradigm, “evil” is not a spooky metaphysical force but a measurable distortion in a multi-agent reward manifold. By treating moral valence as a structural property of system architectures, we move toward a “moral physics” that analyzes how harm and benefit propagate through complex social and computational topologies, governed by definable operators rather than ambiguous intent.

The Anatomy of the Pathological: Functional Definitions

We define “evil” functionally as a systematic pattern of structural harm. It is not merely an event, but a stable process where a system extracts value from its constituents or environment while suppressing the feedback signals that would normally inhibit such damage.

This stability is maintained through Justification Loops—recursive information structures that normalize pathology. Whether in a human mind or an algorithmic objective function, a Justification Loop reinterprets harm as necessity, efficiency, or justice. It acts as a cognitive or computational moat, preventing the system from recognizing the reality of the damage it inflicts. Thus, the study of evil becomes the study of these loops: how they form, how they decouple agents from ground truth, and how they can be dismantled.

The Operator Algebra: Graph-Theoretic Formulation

To move beyond metaphor, we reject the continuous integral formulation in favor of a discrete, graph-theoretic model. This allows us to capture the specific topology of harm—who hurts whom, and who notices. We model the moral field using a multi-agent reward manifold defined by four key components.

1. The Sparse Relevance Matrix ($W$)

Not all agents matter equally to all others at all times. We define a relevance matrix $W_{ij}(s)$ representing the weight agent $j$ assigns to agent $i$’s well-being in state $s$. This matrix is typically sparse (most agents are indifferent to distant others) and dynamic.

2. Perceived Reward Tensors ($\hat{r}$)

Objective reality is filtered through perception. Let $r_i(s)$ be the true reward of agent $i$. We define the perceived reward tensor $\hat{r}_i^{(j)}(s)$ as agent $j$’s estimate of agent $i$’s reward.

3. The Local Evil Density ($\epsilon$)

We define the instantaneous “evil” of a state transition $s \to s’$ not as abstract badness, but as the relevance-weighted sum of perceived negative reward deltas. It is the aggregate “bad news” propagating through the network:

\[\epsilon(s \to s') = \sum_{j=1}^N \sum_{i=1}^N W_{ij}(s) \cdot \text{ReLU}\left( - \Delta \hat{r}_i^{(j)} \right)\]

Where $\Delta \hat{r}i^{(j)} = \hat{r}_i^{(j)}(s’) - \hat{r}_i^{(j)}(s)$. The ReLU function $(x)+$ ensures we only count negative deviations (harm), consistent with the asymmetry of suffering.

4. The Trajectory Functional ($E[\pi]$)

The total moral cost of a policy or history $\pi$ is the discounted sum of local evil density over time. This functional $E[\pi]$ represents the system’s accumulated structural harm:

\[E[\pi] = \sum_{t=0}^{T-1} \gamma^t \epsilon(s_t \to s_{t+1})\]

Minimizing $E[\pi]$ is the primary objective of a morally aligned system, distinct from maximizing total global reward ( which can be achieved by utility monsters).

Topological Dynamics: Graphs and Substrates

The moral field does not exist on a regular grid; it operates on a dual-layer topology.

1. Sparse Relevance Graphs

Most agents are morally “blind” to one another due to structural limitations. The relevance matrix $W$ is sparse, meaning harm is produced and experienced in local, dense subgraphs. However, the Moral Graph Laplacian ($L = D - W$) allows us to model how local distortions ripple outward through the network.

2. The Continuous Media Substrate

Overlaying the sparse graph is a continuous media substrate $M(t, \phi)$. Mass media provides low-frequency coupling that fills gaps in the sparse graph. We decompose the perceived reward $\hat{r}_i^{(j)}$ into direct signals and media-filtered signals:

\[\hat{r}_i^{(j)} = \beta_{ij} r_i + (1-\beta_{ij}) \int K_{ij} M d\phi\]

Here, $\beta_{ij}$ represents direct visibility. When $\beta_{ij} \to 0$, perception is dominated by the media kernel $K_{ij}$, allowing the substrate to overwrite local reality.

3. Moral Resonance

The most dangerous systemic failures occur at the point of Moral Resonance. We formalize this as the projection of substrate modes onto the eigenvectors $v_k$ of the Moral Graph Laplacian. Resonance occurs when:

\[\langle M(\phi), v_k \rangle \gg 0\]

This alignment between a global narrative $M(\phi)$ and a structural eigenvector $v_k$ (a vulnerable social cluster) means small media inputs produce outsized, localized harm cascades, effectively “tuning” the population to a specific frequency of justification or dehumanization.

The Asymmetry Principle: Hysteresis and Repair

A critical error in early computational ethics was the assumption of symmetry between good and evil—that a unit of harm is merely the negative of a unit of benefit. In reality, they are governed by distinct operators with fundamentally different scaling laws and temporal dynamics.

1. Asymmetric Kernels

We model the impact of a stimulus $x$ using two distinct kernels:

2. The Damage State ($D$)

Harm is not instantaneous; it leaves a residue. We define a persistent Damage State $D_i(t)$ for each agent, which evolves according to a hysteresis-dominated equation:

\[\frac{dD_i}{dt} = \ell_-(\text{Harm}_i) - \lambda D_i - R_i(t)\]

Here, $\lambda$ is typically small, representing the slow natural healing of trauma or structural degradation. The term $\ell_-(\text{Harm}_i)$ drives the state up rapidly, creating a “fast-in, slow-out” dynamic characteristic of hysteresis.

3. The Repair Operator ($R$)

Crucially, the cessation of harm ($\text{Harm}_i \to 0$) does not reset $D_i$ to zero. It merely halts the accumulation. To reduce the damage state significantly, a distinct Repair Operator $R$ must be applied.

Engineering the Moral Field: Applications

This field theory provides a foundation for Moral Systems Engineering, transforming ethical constraints into concrete design requirements. We identify four high-value application domains:

  1. Harm Detection (Spectral Analysis of $W$): By computing the eigenvalues of the Moral Graph Laplacian ($L = D - W$), we can identify structural vulnerabilities. Small eigenvalues correspond to “moral blind spots”—clusters of agents weakly connected to the broader empathy network, where harm can accumulate undetected by the collective.

  2. Policy Stress-Testing (Simulating $\Delta H$ and $\Delta D$): Policies are simulated to measure their impact on the Damage State ($D$). Unlike standard cost-benefit analysis, this accounts for hysteresis. A policy is rejected if it triggers a “fast-in” harm spike that exceeds the system’s repair capacity ($R$), regardless of long-term utility.

  3. Moral Resonance Detection (Predicting Cascades): We predict dangerous cascades by monitoring the alignment between media substrate signals $M(\phi)$ and the network’s eigenvectors. When a narrative aligns with a structural fault line (eigenvector $v_k$), the system enters resonance. Intervention involves “detuning” the network by increasing cross-cluster relevance weights.

  4. Algorithmic Governance (Optimizing for Net Valence): Governance is framed as an optimization problem: maximize net valence subject to the constraint that local evil density $\epsilon$ never exceeds a critical threshold $\epsilon_{crit}$. This prevents “utility monster” scenarios where minority suffering is justified by aggregate majority pleasure.

The goal of computational ethics is not to enforce a specific morality, but to design architectures that are structurally resistant to the formation of pathological justification loops. We must stop treating ethics as a set of rules and start treating it as the engineering of a stable, low-distortion reward topology.