Using the nonlinear information field I(x,t)I(x,t)I(x,t), we define an effective energy density functional as:
E(x,t)=12(I2+c2I2+I4)\mathcal{E}(x,t) = \frac{1}{2} \left( |\dot{I}|^2 + c^2 |\nabla I|^2 + \lambda |I|^4 \right)E(x,t)=21(I2+c2I2+I4)
We then associate an effective curvature scalar Reff(x,t)R_{\text{eff}}(x,t)Reff(x,t) heuristically proportional to the spatial Laplacian of the energy density:
Reff(x,t)2E(x,t)R_{\text{eff}}(x,t) \propto -\nabla^2 \mathcal{E}(x,t)Reff(x,t)2E(x,t)
This approach treats energy localization as a curvature-generating mechanism, mapping dense regions to positive curvature (geometric wells) and depleted zones to negative curvature (geometric voids).
2. Numerical Mapping: Energy to Effective Geometry
Simulations reveal that spatio-temporal energy concentration due to blink excitation leads to:
Peaked energy zones with Gaussian-like radial profiles
Strong gradients in E(x,t)\mathcal{E}(x,t)E(x,t), leading to sharp peaks in Reff(x,t)R_{\text{eff}}(x,t)Reff(x,t)
Emergent structures resembling gravitational potential wells, but arising purely from information wave interactions
These geometric analogs are not imposed, but dynamically self-organized due to nonlinear resonant feedback loops.
3. Temporal Stability and Dynamical Geometry
We find that curvature analogs:
Oscillate around equilibrium values after blink excitation
Show ringing modes similar to gravitational wave reverberations
Can stabilize into quasi-static geometry if the damping \gamma is sufficiently high
Exhibit topology-preserving memory: once a curvature pattern emerges, it resists decay unless actively damped
4. Role of Nonlinearity and Scaling
