Variational Geometric Network Framework

The SimplesModel is built on the premise that geometry itself can act as an agent of action and materialization. Here, a Simple is defined as a smooth, closed, oriented two-dimensional surface \( \Sigma \) embedded in \( \mathbb{R}^3 \), whose evolution is governed by its intrinsic state \( (\Sigma, H, K) \).

The Energy Landscape of an Agent

The "agency" of a Simple is mathematically expressed through a surface energy functional \( E[\Sigma] \). This functional determines the agent's preference for certain configurations over others:

\[ E[\Sigma] = \int_{\Sigma} \gamma \, dA + \int_{\Sigma} k \, (H - H_0)^2 \, dA \]

In this expression, \( \gamma \) represents surface tension and \( k \) represents bending rigidity. The agent permits transitions only to states where the first variation of energy is non-positive (\( \delta E \le 0 \)), replacing linear time with a variational ordering of states.

Force as Geometric Bias

The variational structure induces a Geometric Force Density \( \mathcal{F} \), characterized as the functional derivative of the energy. This represents the "intrinsic bias" of the agent at any given surface point:

\[ \mathcal{F} = \gamma H - 2k\Delta_{\Sigma}(H - H_0) - 2k(H - H_0)(H^2 - K) \]

This force is composed of a surface-tension term (\( \gamma H \)) and a bending term involving the Laplace–Beltrami operator \( \Delta_{\Sigma} \). These terms dictate how the surface responds to local perturbations and manages higher-order geometric smoothing.

The Constraint of Area Conservation

A defining characteristic of this framework is Per-Simple Area Conservation. Every agent must maintain a constant total area \( A_0 = \int_{\Sigma} dA \). This constraint transforms local geometric changes into global redistributions.

Mathematically, this is handled via a Lagrange multiplier \( \lambda \), resulting in a constrained force \( \mathcal{F}_{\mathrm{constrained}} = \mathcal{F} + \lambda \). Because \( \lambda \) is determined globally, any local deformation forces an instantaneous, compensatory adjustment elsewhere on the surface.

Network Interactions and Path Dependence

When multiple Simples interact, the total network energy \( E_{\mathrm{tot}} \) includes interaction terms \( E_{\mathrm{int}} \) arising from shared films. As proved in the framework, any transition that modifies the curvature field alters the set of future accessible states \( \mathcal{S}(\Sigma_{n+1}) \). This ensures that the system is history-dependent: each state transition selects one branch of evolution and simultaneously generates a new set of future possibilities.