Variational framework for morphogenesis from two coupled vector fields: OMP Model

The OMP model is a variational framework for morphogenesis defined on a bounded regular domain $\Omega \subset \mathbb{R}^3$. It couples two vector fields $\vec A$ and $\vec B$ through a local energy 
\[
E_{\text{loc}} = (1-r)(\|\vec C\|_\varepsilon - 1)^2 + r\|\vec C - \vec R_{\text{target}}\|^2,
\]
where $\vec C = \alpha \vec A (*)_k \beta \vec B$ is a stochastic mixture of geometric modes via a differentiable Gumbel-Softmax relaxation. The order parameter $\rho = \sigma(\theta - E_{\text{loc}})$ quantifies emergent matter density. The barycentric constraint $\alpha + \beta = \rho$ is enforced via $b = \sigma(u)$, $\alpha = \rho(1-b)$, $\beta = \rho b$, ensuring $C^\infty$ smoothness.

A Lyapunov functional $\mathcal{L}(t) = \frac12 \int_\Omega (\rho-1)^2 d\Omega$ guarantees stability for the simplified system. Dynamics combine gradient descent, feedback $\lambda \rho \vec R_{\text{eff}}$, and diffusion $\mu \Delta$. Local existence is proved in $H^2$ with Neumann conditions via analytic semigroup theory (Pazy, Henry). The model unifies self-organization ($r=0$) and guided design ($r=1$).

A new perspective is introduced: intention-driven design through a map $\Phi : \mathcal{I} \to \mathcal{R}_{\text{target}}$, leading to the scheme $\text{Intention} \xrightarrow{\Phi} \vec R_{\text{target}} \xrightarrow{\text{OMP}} \text{stable emergence}$. This positions OMP as a dynamic design language where geometry emerges from informational equilibrium rather than direct specification. Numerical simulations on a $64^3$ grid and an interactive JavaScript/WebGL prototype are provided.