T3: Three-Dimensional Cohesion Dynamics

Educational Visualization

This is a conceptual visualization, not a simulation of physical systems. It demonstrates substrate mechanics (relaxation, closure, mismatch) in a 3D setting to illustrate surface geometry and curvature flow from discrete constraints. This toy model does not claim physical fidelity and runs entirely in-browser for pedagogical purposes.

This page presents an interactive demonstration of the T3 Cohesion Dynamics toy model: a three-dimensional finite grid with binary spins, 6-neighbor adjacency, and fully emergent 3D geometric behavior.

Interactive Demo

Grid Size (N):

Randomize▶ Auto-RelaxStep CD

Scenario:FreeDivergentPinned Bubble

Free boundaries - All spins random, relaxation removes domain walls → cohesion

α (x-weight):1.0

β (y-weight):1.0

γ (z-weight):1.0

Step n:0

Mismatch M(X):0.0

Surface Area |W(X)|:0

Weights (α,β,γ):(1.0, 1.0, 1.0)

Status:At Closure ✓

About the T3 Three-Dimensional Toy Model

This interactive demo implements the T3 3D Cohesion Dynamics toy model on an N×N×N cubic grid with 6-neighbor adjacency. Each site carries a spin ±1.

Key Concepts:

• Domain Walls as Surfaces: In 3D, domain walls form 2D surfaces (rendered using marching cubes). These can be closed bubbles, spanning sheets, or complex topological structures.
• Mismatch M(X): Weighted sum of domain-wall faces. Each edge contributes α (x-direction), β (y-direction), or γ (z-direction).
• Relaxation: Majority-rule updates that never increase M(X). This produces approximate mean curvature flow on domain-wall surfaces.
• Anisotropy: Different weights (α, β, γ) create directional preferences in surface tension, demonstrating how the W-vector concept might work.
• Divergent Surfaces: In the "Divergent" scenario, a spanning surface cannot be removed → infinite height for regions containing it.

Controls:

• Rotate view: Left mouse drag
• Zoom: Mouse wheel
• Pan: Right mouse drag
• Step CD: Single relaxation step
• Auto-Relax: Continuous relaxation

Toy Model Overview

This toy model implements T3 — A 3D Toy Model of Cohesion Dynamics as specified in the research documentation. The model demonstrates how mismatch dynamics on a discrete 3D substrate naturally produce surface geometry, curvature flow, and topological phenomena.

The Discrete 3D Substrate

We work on a finite N×N×N cubic grid:

$$V = \{1,\dots,N\} \times \{1,\dots,N\} \times \{1,\dots,N\}$$

with 6-neighbor adjacency (face-sharing cubes):

$$(i,j,k) \sim (i\pm 1,j,k), \quad (i,j,k) \sim (i,j\pm 1,k), \quad (i,j,k) \sim (i,j,k\pm 1)$$

Each site carries a binary spin:

$$X_{i,j,k} \in \{-1,+1\}$$

This is the minimal substrate that can exhibit:

• 2D domain-wall surfaces (not just curves)
• Topological features (bubbles, cavities, knots)
• Curvature-driven dynamics
• Volume-based divergence (irreducible spanning surfaces)

Mismatch in 3D with Anisotropic Weights

The local mismatch at site $(i,j,k)$ is a weighted sum of disagreements with neighbors:

$$m_{i,j,k}(X) = \alpha \cdot \mathbf{1}_{\{X_{i,j,k}\neq X_{i+1,j,k}\}} + \beta \cdot \mathbf{1}_{\{X_{i,j,k}\neq X_{i,j+1,k}\}} + \gamma \cdot \mathbf{1}_{\{X_{i,j,k}\neq X_{i,j,k+1}\}}$$

where:

• $\alpha$ = weight for x-direction edges (default 1.0)
• $\beta$ = weight for y-direction edges (default 1.0)
• $\gamma$ = weight for z-direction edges (default 1.0)

The total mismatch is:

$$M(X) = \sum_{i,j,k} m_{i,j,k}(X)$$

(Note: Each edge is counted twice, so $M(X) = 2 \times$ total weighted domain-wall face count.)

Anisotropic interpretation: The weights $(\alpha, \beta, \gamma)$ can be viewed as a primitive 3-component tolerance vector analogous to the $W$-vector in the full CD theory. Different weights create directional preferences in surface tension.

Domain Walls as 2D Surfaces

A domain-wall face exists whenever two adjacent spins differ.

In 3D, these faces form 2D surfaces:

• Closed bubbles (topologically spheres)
• Spanning sheets (connect opposite boundaries)
• Nested shells (concentric layers)
• Branching structures (surfaces meeting at edges)

The demo uses marching cubes (specifically, the SurfaceNets algorithm) to extract and render these surfaces as triangular meshes.

This is the first toy model where domain walls are true surfaces with area, curvature, and topology.

Height in Terms of Surface Area

For a finite region $R \subseteq V$, height is:

$$H(R;X) = \inf\{M(R;X) - M(R;Y) \mid X \rightsquigarrow_R Y,\ Y \text{ a closure in } R\}$$

Geometric interpretation:

Height equals the minimal weighted surface area reduction needed to reach a locally stable configuration inside $R$.

Examples:

• A small bump on a flat surface has height proportional to the bump’s area
• A closed bubble of surface area $A$ has height proportional to $A$
• A topologically trapped bubble may have infinite height if it cannot shrink to zero within $R$
• A spanning surface forced by boundary conditions has infinite height

This gives height a direct surface-geometric meaning — the natural 3D generalization of perimeter from T2.

Relaxation Dynamics and Mean Curvature Flow

We use a 6-neighbor majority-rule relaxation:

1. At site $v=(i,j,k)$, compute neighbor counts:

   • $n_+ = |\{u \sim v \mid X_u = +1\}|$
   • $n_- = |\{u \sim v \mid X_u = -1\}|$

2. Propose:

   $$X_v' = \begin{cases} +1 & n_+ > n_-, \\ -1 & n_- > n_+, \\ X_v & n_+ = n_-. \end{cases}$$

3. Accept the flip iff $M(X') \le M(X)$

This rule is:

• Local (depends only on immediate neighbors)
• Mismatch-nonincreasing (preserves or reduces $M$)
• CD-conformant (permitted by substrate axioms)

Synchronous CD Step

The demo uses a synchronous batch update:

1. Compute all proposed flips from $X^{(n)}$
2. Accept only those that do not increase $M$
3. Apply them simultaneously to obtain $X^{(n+1)}$

Effect on Domain-Wall Surfaces

A flip at $v$ can only affect mismatch along faces incident to $v$.

• At convex protrusions, the majority rule flips $v$ to smooth the bump
• At concave indentations, it fills the dent
• Small bubbles shrink and disappear
• Large surfaces smooth toward minimal area

Thus, relaxation produces:

Approximate discrete mean curvature flow: surfaces move inward with speed proportional to local curvature.

This is the 3D analogue of the curvature flow observed in T2, and it directly validates the continuum PDE predictions in Paper B.

Divergence in 3D: Spanning Surfaces and Infinite Height

The 2D toy model (T2) produced divergent curves forced by boundary conditions.

In 3D, the minimal divergent regions become 2-dimensional spanning surfaces.

Divergent Scenario: Fixed Opposite Faces

Fix boundaries:

• Face $x=0$: all spins $+1$
• Face $x=N-1$: all spins $-1$
• Other faces: free

Every configuration must contain at least one $+/-$ interface spanning the grid from $x=0$ to $x=N-1$. No local flips can eliminate all such mismatches.

Thus any volume $R$ containing a forced spanning surface satisfies:

$$H(R;X) = \infty$$

This is a literal realization of the “divergent surface” / “horizon precursor” from Paper F.

Minimal Divergent Surface

A set of faces $\Gamma$ is a minimal divergent surface if:

1. Every configuration consistent with the boundary conditions has domain-wall faces on at least one face of $\Gamma$
2. For any strict subset $\Gamma' \subsetneq \Gamma$, there exists a configuration with no domain walls on $\Gamma'$

Example: A planar sheet of faces perpendicular to $x$, spanning from the $+1$ boundary to the $-1$ boundary.

This reproduces the abstract “codimension-1 horizon” notion from the foundational papers through an explicit 3D construction.

Incorporating Tolerance: The ($\alpha$, $\beta$, $\gamma$) Weight Vector

The anisotropic weights $(\alpha, \beta, \gamma)$ provide a concrete realization of the multi-component tolerance concept.

Consequences of varying weights:

• Isotropic ($\alpha = \beta = \gamma$): Surfaces have uniform tension in all directions → spherical bubbles minimize area
• Anisotropic ($\alpha \neq \beta$ or $\beta \neq \gamma$): Surfaces prefer orientations with lower weights → stretched or flattened bubbles
• Strong anisotropy (e.g., $\alpha = 2$, $\beta = \gamma = 0.5$): Surfaces resist forming perpendicular to the $x$-axis → “pinning” in certain directions

This is the first primitive mechanism demonstrating how:

• Different tolerance components control different directional responses
• A 3-vector of weights naturally emerges in a 3D substrate
• Surface geometry responds to anisotropy → connection to emergent metric structure

This directly supports the $W$-vector formalism in the full theory.

Canonical Scenarios

The demo provides three canonical scenarios:

Scenario: Free

• All boundaries are free (no constraints)
• Random initial spins throughout the volume
• Behavior: Domain-wall surfaces shrink and vanish → complete cohesion
• Height: Finite, equals initial weighted surface area
• Outcome: Relaxation converges to uniform configuration (all $+1$ or all $-1$)

Scenario: Divergent

• Face $x=0$ fixed at $+1$
• Face $x=N-1$ fixed at $-1$
• All other boundaries free
• Behavior: A spanning surface perpendicular to $x$ persists → minimal divergent surface
• Height: Infinite for any region containing the forced interface
• Outcome: Relaxation smooths the surface but cannot eliminate it

This scenario demonstrates:

• Irreducible mismatch imposed by topology
• Codimension-1 divergence (a 2D surface in 3D space)
• Horizon-like structure (regions separated by an unremovable boundary)

Scenario: Pinned Bubble

• A spherical shell of radius $\approx N/5$ centered in the grid is pinned to $+1$
• Interior region initialized randomly or to $-1$
• Exterior region initialized to $-1$
• Behavior: Domain walls wrap around the pinned obstacle and may stabilize
• Height: Finite but constrained by the pinned shell — cannot fully relax
• Outcome: Metastable configurations with trapped surfaces

This scenario demonstrates:

• Topological constraints from pinned obstacles
• Cavity formation (bubbles trapped inside shells)
• Metastability (local closures that are not globally minimal)

What T3 Reveals About Cohesion Dynamics

1. Domain-Wall Surfaces Generate True Curvature Flow

In 3D, domain walls are actual surfaces with:

• Area (proportional to height)
• Curvature (driving relaxation dynamics)
• Topology (spheres, tori, spanning sheets)

Relaxation produces discrete mean curvature flow — this validates the PDE structure predicted in Paper B.

2. Divergent Surfaces Arise from Topology, Not Axioms

No additional axioms needed:

• Boundary conditions + topological constraints automatically produce spanning surfaces that cannot be removed
• These surfaces exhibit infinite height
• They separate regions → horizon-like structure

This strengthens the conceptual foundation for the “divergence surface as proto-horizon” idea in Paper F.

3. Anisotropy Creates Directional Surface Tension

The 3-component weight vector $(\alpha, \beta, \gamma)$:

• Produces directional preferences in surface orientation
• Creates anisotropic surface tension
• Leads to non-spherical minimal surfaces

This is exactly the structure needed to connect to:

• The 3-component $W$-vector in the full theory
• Emergent metric structure
• Lorentzian vs Euclidean signature

4. Topology Becomes Observable and Computable

3D allows:

• Closed bubbles (topologically $S^2$)
• Nested shells (concentric spheres)
• Knotted loops of domain-wall edges
• Non-contractible surfaces (in periodic boundary conditions)

Height directly measures topological information encoded in the substrate state.

5. 3D is the First Substrate Where Geometry Fully Emerges

The substrate does not assume geometry, yet:

• Domain walls have surface area
• Height corresponds to area reduction
• Relaxation approximates mean curvature flow
• Anisotropy produces metric-like structure

Thus T3 demonstrates:

Surface geometry and curvature-driven dynamics can emerge from mismatch rules alone, without assuming a manifold or metric.

This is the crucial stepping stone for the continuum theory in Papers A and B.

Experiments to Try

Use the interactive demo to explore:

1. Free Scenario:

   • Watch random configurations relax to uniformity
   • Observe bubbles shrinking and disappearing
   • Notice surface smoothing over multiple steps

2. Divergent Scenario:

   • See the persistent spanning surface that cannot be eliminated
   • Observe relaxation smoothing the surface without removing it
   • Try different grid sizes to see how the surface scales

3. Pinned Bubble Scenario:

   • Watch how domain walls wrap around the pinned shell
   • Observe trapped cavities and metastable states
   • Notice how relaxation is constrained by the obstacle

4. Anisotropy Effects:

   • Set $\alpha = 2.0$, $\beta = \gamma = 0.5$ → surfaces avoid perpendicular to $x$
   • Set $\gamma = 2.0$, $\alpha = \beta = 0.5$ → surfaces avoid perpendicular to $z$
   • Observe how bubbles become ellipsoidal instead of spherical

5. Curvature Flow:

   • Watch convex bumps smooth out
   • See small bubbles shrink faster than large ones (curvature $\propto 1/r$)
   • Observe surface-area reduction over steps

6. Step-by-Step Relaxation:

   • Use “Step CD” to see individual synchronous updates
   • Track mismatch reduction per step
   • Notice when “At Closure ✓” appears

7. Grid Size Effects:

   • Start with $N=10$ for fast updates
   • Try $N=20$ (default) for realistic structure
   • Push to $N=30$ or $N=40$ for complex topology (slower but impressive)

Next Steps: From Toy Models to Formal Theory

With T1, T2, and T3 complete, we now have:

• T1 (1D): Domain-wall points, basic mismatch mechanics
• T2 (2D): Domain-wall curves, emergent length and curvature
• T3 (3D): Domain-wall surfaces, emergent area and mean curvature flow

These toy models provide:

✅ Concrete mechanisms for all abstract CD primitives
✅ Geometric emergence from discrete mismatch dynamics
✅ Divergent surfaces from topology (not axioms)
✅ Anisotropic tolerance via weight vectors
✅ Curvature-driven relaxation → continuum PDE structure

The next phase is:

1. Extract candidate theorems from observed phenomena in T1–T3
2. Formalize the continuum limit — how discrete → continuous PDEs
3. Prove existence and uniqueness of relaxation paths
4. Characterize divergence rigorously using topological invariants
5. Connect to Lorentzian geometry via multi-channel anisotropy

T3 is not just an illustration — it literally demonstrates the phenomena that will become the basis for rigorous theorems in Papers A and B.

References

This implementation is based on:

“T3 — 3D Cohesion Dynamics Demo Specification” (Research Documentation)

• Section 1: Overview and Design Philosophy
• Section 2: Data Model (3D Grid, Anisotropic Weights)
• Section 3: Mismatch Functional
• Section 4: Relaxation Rule (CD-conformant)
• Section 5: Domain Walls (Surface Extraction)
• Section 6: Rendering (Marching Cubes, react-three-fiber)
• Section 7: Scenarios (Free, Divergent, Pinned Bubble)
• Section 8: Performance Constraints
• Section 9: UI Layout and Controls
• Section 10: Scientific Significance

This model extends the concepts from T1 (1D) and T2 (2D) to the full 3D case, providing the final experimental foundation before formal continuum results.