We define a minimal viable agent bounded by a full Fristonian Markov Blanket explicitly grounded in the canonical cortical microcircuit. By modeling the stochastic dynamics of a four-component system (internal, sensory, active, and external states), we rigorously demonstrate the conditional independence required by the Free Energy Principle via the steady-state Lyapunov equation. To evaluate intrinsic causal integration, we map the continuous stationary density to a discrete Transition Probability Matrix (TPM). We apply Tononi's Integrated Information Theory (IIT 4.0), replacing the Earth Mover's Distance with the Intrinsic Difference metric, mathematically guaranteeing $\Phi > 0$ for recurrent corticothalamic microcircuits.
Following Friston \cite{Friston2013}, we partition the universe into four interacting states: internal ($c_t$), sensory ($s_t$), active ($a_t$), and external ($\lambda_t$). We ground this topologically in the canonical microcircuit for predictive coding \cite{Bastos2012}: $s_t$ represents L4 thalamocortical inputs, $c_t$ represents the recurrent L2/3 and L5 populations, $a_t$ represents L5 deep outputs and L6 corticothalamic feedback, and $\lambda_t$ represents the environmental hidden states.
The continuous dynamics are governed by a coupled system of Stochastic Differential Equations (SDEs) driven by standard Wiener processes:
Crucially, there is no direct coupling between $c_t$ and $\lambda_t$, and sensory states $s_t$ do not depend on internal states $c_t$. This structural asymmetry breaks the v-structure, preventing $s_t$ from acting as a collider, ensuring that conditioning on the blanket does not inadvertently open an information path between $c_t$ and $\lambda_t$. Linearizing the drift around a non-equilibrium steady state yields a Jacobian matrix $\mathbf{A}$. The stationary covariance $\boldsymbol{\Sigma}$ is determined by the Helmholtz decomposition $\mathbf{A}=(\mathbf{Q}-\mathbf{D})\boldsymbol{\Sigma}^{-1}$, where $\mathbf{Q}$ is the anti-symmetric solenoidal flow and $\mathbf{D}$ is the diffusion tensor. Provided the solenoidal flow preserves the boundary topology, the precision matrix is block-sparse ($\boldsymbol{\Sigma}^{-1}_{c\lambda}=0$), ensuring $p(c, \lambda\mid s, a)= p(c \mid s, a)p(\lambda\mid s, a)$ and rigorously proving the Markov blanket.
To evaluate Tononi's $\Phi$, we assess the intrinsic cause-effect power of the internal states $c_t$. We derive a discrete Transition Probability Matrix $\text{TPM}(c' \mid c)$ from the exact Fokker-Planck stationary distribution $p(\mathbf{x})$ over a minimal timescale $\Delta t$, applying maximum entropy priors to the boundary conditions \cite{Albantakis2023}.
Using the IIT 4.0 framework \cite{Albantakis2023, Oizumi2014}, we measure the irreducible intrinsic information across the Minimum Information Partition (MIP) using the Intrinsic Difference (ID) between the intact Cause-Effect Structure (CES) and the partitioned CES:
Because the internal cortical microcircuit $(c_t)$ possesses strong recurrent loops (e.g., L2/3 $\to$ L5 and L5 $\to$ L2/3), the localized block of the Lyapunov covariance $\boldsymbol{\Sigma}_{cc}$ is strictly irreducible under any bisection. Consequently, the intrinsic difference is strictly positive, mathematically guaranteeing $\Phi > 0$ for biological cortical columns.
