Substrate Dynamics

Quicklinks#

• docs README
• docs api integration examples
• docs api README
• docs api schema overview
• docs api using the schemas
• docs experiments faraday paradox experiment
• docs experiments README
• docs experiments replication checklist
• docs experiments resonance alignment tests
• docs experiments rotating conductor tests
• docs methods dimensional layers
• docs methods field equations
• docs methods operator definitions
• docs methods README
• docs methods triadic fields
• docs onboarding model map
• docs onboarding reading guide
• docs onboarding triadic quickstart
• docs onboarding verification tests
• docs overview comparison to gr models
• docs overview glossary
• docs overview introduction
• docs overview README
• docs overview resonance primitives
• docs overview theoretical background
• docs simulations boundary conditions
• docs simulations numerical methods
• docs simulations README
• docs simulations solver_architecture
• docs simulations validation metrics
• docs simulations core README
• previous folder

1. Evolution Framework#

Field evolution follows a general update rule:

1. Apply operators
2. Integrate in time
3. Enforce boundaries
4. Record diagnostics

This loop continues until termination criteria are met.

2. Operator Application#

Operators act on fields in a defined sequence or composition:

• diffusion smooths scalar and vector fields
• alignment drives directional coherence
• coupling links fields across layers
• resonance activation triggers envelope dynamics
• decay stabilizes the system

Operator composition may be linear, nonlinear, or gated by resonance conditions.

3. Time Integration#

The substrate supports multiple integration schemes:

• explicit Euler (rapid prototyping)
• Runge–Kutta (higher stability)
• semi-implicit methods (stiff operators)

Timestep selection respects stability constraints such as CFL conditions.

4. Boundary Conditions#

Boundary handlers enforce:

• Dirichlet conditions (fixed values)
• Neumann conditions (fixed gradients)
• periodic boundaries (looped domains)
• custom experimental boundaries (e.g., rotating-frame transforms)

Boundaries are applied after operator updates to maintain physical consistency.

5. Stability and Control#

Stability is maintained through:

• damping
• normalization
• resonance envelope clipping
• timestep adaptivity

These mechanisms prevent runaway growth and ensure coherent evolution.

6. Emergent Behavior#

Dynamic interactions among fields and operators can produce:

• coherent resonance pockets
• rotating or oscillatory patterns
• alignment waves
• paradox-class responses

These emergent structures are central to the substrate’s expressive power.