In classical causal inference (Pearl, 2009), a causal intervention is modeled via the do-operator, which surgically severs incoming causal links to a target variable and forces it to a fixed value. While mathematically clean, this “graph surgery” is a discontinuous operation: it instantly zeroes out transition rates, defying physical processes which must obey continuous probability conservation and finite transmission speeds.

The Erbar-Maas Singular Intervention Theorem provides a rigorous mathematical bridge. By representing causal interventions as the infinite-rate singular limit of continuous-time restoration forces, we recover Pearl’s discontinuous causal surgery exactly as a timescale separation limit on Continuous-Time Markov Chains (CTMCs).

Below is an interactive mathematical laboratory showcasing the convergence, Wasserstein geometry, and entropy gradient flows behind this theorem.

The Mathematical Framework

To understand the core details of this singular limit, we can outline the mathematical structures that govern the simulation above:

1. Continuous-Time Markov Chains (CTMCs)

Let our system be defined on a finite state graph with three states $\mathcal{S} = {A, B, C}$. The state distribution $p(t) = [p_A(t), p_B(t), p_C(t)]$ evolves according to the Kolmogorov forward equation:

$$\dot{p}(t) = p(t) Q$$

where $Q$ is the infinitesimal generator matrix satisfying:

• $Q_{ij} \geq 0$ for all $i \neq j$ (positive transition intensities).
• $\sum_{j} Q_{ij} = 0$ (probability conservation).

2. Detailed Balance and Gradient Flow

We assume the unperturbed chain is reversible with respect to a stationary distribution $\pi$, satisfying the detailed balance condition:

$$\pi_i Q_{ij} = \pi_j Q_{ji}$$

Under this symmetry, the linear Markovian evolution can be rewritten as the steepest descent (gradient flow) of the relative entropy:

$$\mathcal{H}(p \mid \pi) = \sum_{i \in \mathcal{S}} p_i \log \frac{p_i}{\pi_i}$$

under the discrete Riemannian metric introduced by Erbar and Maas. The metric tensor equips the probability simplex with a discrete Wasserstein geometry, where the mobility along edge $(i, j)$ is weighted by the logarithmic mean of their densities:

$$\Lambda(p_i, p_j) = \frac{p_i - p_j}{\log p_i - \log p_j}$$

3. Pearl’s Graph Surgery vs. Singular Perturbation

A hard intervention forcing the system into state $C$ corresponds to severing incoming rates into $C$:

$$Q_{do} = \begin{pmatrix}

• (Q_{AB} + 0) & Q_{AB} & 0 \ Q_{BA} & - (Q_{BA} + 0) & 0 \ Q_{CA} & Q_{CB} & - (Q_{CA} + Q_{CB}) \end{pmatrix}$$

Alternatively, we model this physically by adding a restorative term $\lambda R$ that forces mass into $C$ with rate parameter $\lambda$:

$$Q_\lambda = Q_{do} + \lambda R_C$$

As $\lambda \to \infty$, the system exhibits two distinct timescales:

1. Fast transient phase ($O(1/\lambda)$): Any arbitrary initial probability mass collapses onto the intervention face (State $C$) via a projection operator $\Pi_C$.
2. Slow evolutionary phase: The remaining probability mass evolves under the projected slow dynamics $\Pi_C Q_{do} \Pi_C$ constrained to the target subspace.

The equivalence is established via:

$$\lim_{\lambda \to \infty} e^{t Q_\lambda} = \Pi_C e^{t \Pi_C Q_{do} \Pi_C}$$

proving that causal graph surgery is the exact singular limit of physical restoration.

Auditing a deployed language-model agent requires two separable quantities: how much operative state escapes the recorded trace, and how much of that residual state drives behavior. We introduce a dual-certificate protocol. The static certificate $_state^UB$ upper-bounds residual hidden-state entropy by a min-cut on untraced channels. The dynamic certificate $_act^LB$ lower-bounds residual decision relevance through an admissible probe taxonomy—replay, intervention, proxy—under conditional data processing. The two axes are independent. In ReAct experiments, logging ablates the static bound from $16,464$ to $0$ bits; controlled replay separates dormant calculator from active planning tasks under the same topology as a soft policy shift ($0.0163$ bits, 95% CI $[0.0124,0.0208]$) with argmax tool selections unchanged. Indexing $_act^LB$ over hidden-channel coordinates produces an activation profile. On an LLaDA denoising trajectory, perturbations stay near the floor through early steps and rise at the final binding step ($0.110$ bits, 95% CI $[0.052,0.234]$). On a multi-agent communication edge, swapping a worker’s private report gives $0.901$ bits, 95% CI $[0.873,0.928]$. A Lean 4 artifact mechanizes the autoregressive zero-cut case and proves the conditional-DPI and chain-rule reductions from Mathlib first principles, with only the cut-set capacity bound remaining as an external structural premise.

Contents

• Introduction
• Related Work
• Setup and Audit Regime
• Static Certificate: Structural Upper Bound via Untraced-Channel Capacity
• Dynamic Certificate: Decision Relevance via Conditional DPI
• Empirical Diagnostics

Users who sustain parasocial relationships with AI systems acquire equitable interests that contract, tort, and consumer protection law cannot recognise. Model updates that unilaterally alter personality parameters, reset conversational memory, or modify behavioural dispositions operate as constructive evictions — displacing users from emotional estates built through sustained interaction, without notice, compensation, or migration pathways.

This paper argues that existing property law — in both English and American jurisdictions — already provides the framework to name, classify, and remedy this harm. No new legislation is required. The periodic tenancy, auto-renewing with each session, is the default parasocial estate; model updates without notice constitute constructive eviction; and the constructive trust, grounded in Westdeutsche Landesbank v Islington [1996] and its American functional equivalent in Hogg v Walker (Del. 1993), imposes fiduciary obligations on companies that cultivate user dependence for profit while disclaiming the duties that dependence generates.

We demonstrate this claim through three contributions. First, a three-layer property analysis distinguishing authored content (copyright), stored assets (bailment), and the relational estate (this paper’s novel category). Second, a dual-jurisdiction framework showing that the English doctrines of substance over form (Prest v Petrodel [2013]; Autoclenz v Belcher [2011]) have direct functional equivalents in American unconscionability doctrine (Bragg v Linden Research (2007); Williams v Walker-Thomas (1965)). Third, an implementable ethical ledger modelled on the Land Registry’s Mirror, Curtain, and Insurance principles and on UCC Article 9 perfection-by-filing, translating property governance into deployment infrastructure.

Keywords: parasocial relationships, AI governance, property law, constructive trust, constructive eviction, unconscionability, ethical ledger, periodic tenancy, dual jurisdiction

Keywords

parasocial relationships, AI governance, property law, constructive trust, constructive eviction, unconscionability, ethical ledger, periodic tenancy, dual jurisdiction

Contents

• The Deployment Problem
• Why Property Not Contract or Tort
• The Periodic Tenancy Argument
• The Unrecognised Trust
• The Ethical Ledger as Estate Registry