Continuous-Time Causal Inference: Aligning Pearl's Do-Calculus with Itô SDEs

Introduction

In classical causal inference, Judea Pearl's structural causal models (SCMs) assume static, discrete functional relations of the form $X_i = f_i(\text{PA}_i) + U_i$. However, high-frequency alternative datasets and DePIN physical telemetry behave as continuous-time stochastic processes. To execute interventional routing decisions, we must generalize $\text{do}$-calculus to stochastic differential equations (SDEs) integrated via Itô calculus.

This article presents the mathematical alignment of Itô-Pearl Causal Manifolds, showing how structural interventions mutate the infinitesimal generator of the system and introduce second-order diffusion correction terms.

Stochastic Differential Equations as Continuous SCMs

Let the 158-dimensional state space trajectory $h_t \in \mathbb{R}^{158}$ be governed by a system of coupled Itô SDEs:

\[ dh_{t, j} = f_j(h_t) dt + g_j(h_t) dW_{t, j}, \quad j = 1, \dots, 158 \]

where $f_j(h_t)$ represents the drift (causal mechanism) determined by the parent nodes $\text{PA}_j$, and $dW_{t, j}$ are independent Brownian motion increments representing unobserved background factors.

Defining the $\text{do}$-Operator on SDEs

An intervention $\text{do}(h_i = u)$ corresponds to a structural mutation of the SDE system. This action shears off all incoming causal links to variable $i$, forcing its state to $u$ and setting its differential to zero:

\[ dh_{t, i} = 0 \quad \text{with} \quad h_{0, i} = u \]

For all downstream variables $j \neq i$, their equations remain intact but are evaluated on the intervened state space:

\[ dh_{t, j} = f_j(h_t \setminus h_i, u) dt + g_j(h_t \setminus h_i, u) dW_{t, j} \]

The Itô-Pearl Alignment Theorem

For any downstream target variable $Y_t = \Phi(h_t)$, the evolution of the interventional expectation $E[\Phi(h_t) \mid \text{do}(h_i = u)]$ depends on the infinitesimal generator of the mutated process. By Itô's Lemma, the differential $d\Phi(h_t)$ under the mutated dynamics is:

\[ d\Phi(h_t) = \left( \mathcal{A}_{\text{do}(i)} \Phi(h_t) \right) dt + \sum_{j \neq i} g_j(h_t) \frac{\partial \Phi}{\partial h_j} dW_{t, j} \]

where $\mathcal{A}_{\text{do}(i)}$ is the mutated infinitesimal generator:

\[ \mathcal{A}_{\text{do}(i)} = \sum_{j \neq i} f_j(h_t) \frac{\partial}{\partial h_{j}} + \frac{1}{2} \sum_{j \neq i} g_j(h_t)^2 \frac{\partial^2}{\partial h_{j}^2} \]

The second-order term $\frac{1}{2} g_j^2 \frac{\partial^2 \Phi}{\partial h_j^2}$ acts as the stochastic causal correction. Neglecting this term introduces a structural bias in interventional predictions proportional to the local quadratic variation of the parent nodes.

Implementation in the 55-Layer DAG

In the V5.1 KairosGrandUnifiedDAG execution engine, this alignment is enforced in Layer 54 (Choquet Minimax Router):

  • Infinitesimal Generator Masking: The input state tensor is masked by the symmetry breaking attention shocks: $h_{\text{aligned}} = h \odot (1.0 - S_t)$. This represents the $\mathcal{A}_{\text{do}(i)}$ operator, shearing off spurious feedback columns.
  • Minimax Risk Projection: The possibility contour and expected actions are evaluated using $h_{\text{aligned}}$, ensuring that routing weights are immunized against feedback leakage from rival chains during high-volatility regimes.