T2: Two-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 2D setting to illustrate how geometric behavior emerges 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 T2 Cohesion Dynamics toy model: a two-dimensional finite grid with binary spins, 4-neighbor adjacency, and emergent geometric behavior.

Interactive Demo

Grid Size (N):

Randomize▶ Auto-RelaxStep CD

Canonical Scenario:Scenario F (Free)Scenario D (Divergent)Scenario P (Pinned)

Free boundaries - Domain walls shrink and vanish → cohesion

Anisotropy ε:0.00

α=1.00 (vertical)β=1.00 (horizontal)

Step n:0

Mismatch M(X):0.0

Domain Walls |W(X)|:0

Scenario:F

Anisotropy (α,β):(1.00, 1.00)

Status:At Closure ✓

About the T2 Two-Dimensional Toy Model

This interactive demo implements the T2 Two-Dimensional Cohesion Dynamics toy model on an N×N grid with 4-neighbor adjacency. Each site carries a spin ±1.

Key Concepts:

• Domain Walls: Edges between sites with different spins (shown as thick borders). In 2D, domain walls form curves - closed loops, spanning lines, or branching structures.
• Local Mismatch m_i,j: Sum of disagreements with 4-neighbors, weighted by edge costs.
• Total Mismatch M(X): Sum of all local mismatches, divided by 2 (since edges are counted twice).
• Synchronous CD Step: All sites propose flips via majority rule simultaneously. Only flips that don't increase M(X) are accepted.
• Height: In 2D, height corresponds to domain-wall perimeter reduction. Minimal perimeter needed to reach local stability.

Canonical Scenarios:

• Scenario F (Free): No constraints. Domain walls shrink and vanish → full cohesion.
• Scenario D (Divergent): Left edge fixed at +1, right edge at -1. Every configuration must contain a spanning vertical interface - a minimal divergent curve. This is a 1D surface that cannot be removed by any relaxation sequence.
• Scenario P (Pinned): A +1 island is pinned (cannot flip) in a sea of -1. Domain walls wrap around the obstacle and cannot fully relax.

Tolerance & Anisotropy:

• Weighted Mismatch: Vertical edges have weight α, horizontal edges have weight β.
• Anisotropy ε: Controls directional preference. When α > β, vertical walls are more costly → walls prefer horizontal orientation. When β > α, horizontal walls are more costly → walls prefer vertical orientation.
• This is the first mechanism showing how tolerance introduces directional structure, connecting to the W-vector concept in the full theory.

Emergent Geometry:

• Curvature Flow: Majority rule causes convex kinks to smooth out and concave dents to fill in. This produces approximate mean curvature flow - interfaces move with speed proportional to curvature.
• Domain Wall Geometry: Walls have length, can form loops, and exhibit curvature. Geometry emerges from mismatch rules alone, without assuming a manifold.

Interactions:

• Click a cell to select it and see detailed analysis
• Double-click a cell to flip it immediately (except pinned cells)
• Step CD: Perform one synchronous relaxation step
• Auto-Relax: Continuously relax to closure
• Blue cells = +1, Orange cells = -1

Toy Model Overview

This toy model implements T2 — A 2D Toy Model of Cohesion Dynamics as specified in the research documentation. The model demonstrates how mismatch dynamics on a discrete 2D substrate naturally produce geometric phenomena.

The Discrete 2D Substrate

We work on a finite N×N grid:

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

with 4-neighbor adjacency:

$$(i,j) \sim (i\pm 1,j), \qquad (i,j) \sim (i,j\pm 1)$$

Each site carries a binary spin:

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

This is the simplest substrate that can exhibit nontrivial 2D domain-wall geometry.

Mismatch in 2D

The local mismatch at site $(i,j)$ counts disagreements with neighbors:

$$m_{i,j}(X) = \sum_{(k,\ell)\sim(i,j)} \mathbf{1}_{\{X_{i,j}\neq X_{k,\ell}\}}$$

The total mismatch is:

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

(Note: Each edge is counted twice, so $M(X) = 2|W(X)|$ where $W(X)$ is the set of domain-wall edges.)

A domain wall is an edge $\{u,v\}$ with different spins:

$$W(X) = \{\{u,v\} \mid X_u \neq X_v\}$$

Unlike in 1D, domain walls in 2D form curves:

• Closed loops
• Spanning lines
• Branching structures
• Curvature-bearing shapes

This is the first appearance of geometric behavior emerging from mismatch alone.

Height in Terms of Domain-Wall Geometry

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 domain-wall perimeter reduction needed to reach a locally stable configuration inside $R$.

Examples:

• A single kink on a straight domain wall has height ≈ 1
• A closed loop of perimeter $L$ has height proportional to $L$
• A topologically trapped loop may have infinite height if it cannot be removed by any local sequence within $R$

This gives height a direct geometric meaning - something the abstract theory only has implicitly.

Relaxation Dynamics and Emergent Curvature Flow

We use a majority-rule relaxation:

1. At site $v=(i,j)$, 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 neighbors)
• Mismatch-nonincreasing (preserves or reduces $M$)
• Permitted by the substrate axioms

Synchronous CD Step

The demo uses a synchronous 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 Walls

A flip at $v$ can only affect mismatch along edges $(u,v)$.

• At convex kinks, the majority rule flips $v$ to smooth the bump
• At concave dents, it fills the indentation
• Small islands shrink and disappear

Thus, relaxation produces:

Approximate curvature flow: interfaces move inward with speed proportional to curvature.

This mirrors the PDE behavior expected in the continuum theory and gives a concrete mechanism for smoothness emerging from mismatch.

Divergence in 2D: Domain-Wall Surfaces That Cannot Be Removed

The 1D toy model produced a single divergent point (an interface forced by boundary conditions).

In 2D, the minimal divergent regions become 1-dimensional surfaces.

Fixed-Boundary Example (Scenario D)

Fix boundaries:

• Left edge: all $+1$
• Right edge: all $-1$
• Top/bottom: free

Every configuration must contain at least one $+/-$ interface spanning the grid vertically. No local flips can eliminate all such mismatches.

Thus any strip $R$ containing a spanning interface satisfies:

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

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

Minimal Divergent Curve

A set of edges $\Gamma$ is a minimal divergent curve if:

1. Every configuration consistent with the boundary conditions has mismatch on at least one edge of $\Gamma$
2. For any strict subset $\Gamma' \subsetneq \Gamma$, there exists a configuration with no mismatches on $\Gamma'$

Example: A straight vertical line of edges between a $+1$ left boundary and $-1$ right boundary.

This reproduces the abstract notion from Paper A through an explicit construction.

Incorporating Tolerance: Weighted Mismatch and Anisotropy

To connect to the multi-component notion of tolerance, introduce weighted mismatch:

$$m_{i,j}(X) = \sum_{u\sim(i,j)} \left(1 + \varepsilon \cdot \eta_{(i,j);u}\right) \mathbf{1}_{\{X_{i,j} \neq X_u\}}$$

where:

• $\varepsilon$ is a tolerance parameter
• $\eta$ is an anisotropy field

Simplified Anisotropy Model

In the demo, we use:

$$m_{i,j}(X) = \alpha \cdot \text{(vertical disagreements)} + \beta \cdot \text{(horizontal disagreements)}$$

where:

• $\alpha = 1 + \varepsilon$ (weight for vertical edges)
• $\beta = 1 - \varepsilon$ (weight for horizontal edges)

Consequences:

• $\varepsilon = 0$: Isotropic domain-wall tension
• $\varepsilon > 0$: Vertical walls more costly → walls prefer horizontal orientation
• $\varepsilon < 0$: Horizontal walls more costly → walls prefer vertical orientation
• Large anisotropy causes pinning — walls resist bending in certain directions

This is the first primitive mechanism from which a 3-component tolerance vector $W$ might generalize: different directions, offsets, or curvature responses correspond to different tolerance channels.

Canonical Scenarios

The demo provides three canonical boundary scenarios:

Scenario F: Free Boundaries

• All boundaries are free (no constraints)
• Random initial spins
• Behavior: Domain walls shrink and vanish → complete cohesion
• Height: Finite, equals initial mismatch

Scenario D: Divergent Spanning Interface

• Left edge fixed at $+1$
• Right edge fixed at $-1$
• Top/bottom free
• Behavior: Vertical interface persists → minimal divergent curve
• Height: Infinite for any region containing the forced interface

Scenario P: Pinned Obstacle

• A central $+1$ “island” is pinned (cannot flip)
• Surrounding region initialized to $-1$
• Behavior: Domain walls wrap around the obstacle and pin
• Height: Finite but constrained by the obstacle

What T2 Teaches Us About Cohesion Dynamics

1. Height Becomes Geometrically Meaningful

In 2D, height exactly matches:

• Domain-wall perimeter reduction (finite height), or
• The impossibility of removing a compulsory contour (infinite height)

This justifies height’s role in the foundational theory.

2. Divergent Surfaces Arise Naturally

No axioms needed — topological constraints and boundary conditions produce them automatically.

This strengthens the conceptual grounding of the “horizon” idea.

3. Relaxation Produces Curvature-Like Dynamics

This gives a substrate-level mechanism for the emergence of hyperbolic/elliptic structure in the continuum theory.

4. Tolerance Introduces Directional Structure

Weighted mismatch already produces:

• Anisotropic tension
• Pinning
• Different curvature responses in different directions

This is exactly the structure needed to connect to the 3-component $W$-vector envisioned in the larger theory.

5. 2D Already Shows the Seeds of Geometry

The substrate does not assume geometry, yet:

• Domain walls have length
• Height corresponds to perimeter
• Relaxation approximates curvature flow

Thus T2 demonstrates:

Geometry-like behavior can emerge from mismatch rules alone, without assuming a manifold.

This is a crucial stepping stone for the continuum theory.

Experiments to Try

Use the interactive demo to explore:

1. Scenario F (Free): Watch random configurations relax to uniformity
2. Scenario D (Divergent): Observe the persistent spanning interface that cannot be eliminated
3. Scenario P (Pinned): See how domain walls wrap and stabilize around obstacles
4. Anisotropy Effects: Adjust ε to make walls prefer certain orientations
5. Curvature Flow: Watch convex kinks smooth out and small islands shrink
6. Domain Wall Geometry: Observe loops, curves, and branching structures
7. Step-by-Step Relaxation: Use “Step CD” to see individual synchronous updates
8. Closure Detection: Notice when “At Closure ✓” appears - no more decreasing moves available

Next Steps: T3 (3D Toy Model)

T3 will extend T2 into 3D:

• Domain walls become surfaces
• Height becomes surface area
• Divergent regions become volumes with unresolvable boundary surfaces
• Tolerance produces anisotropic surface tension
• Relaxation approximates mean curvature flow in 3D
• The $W$-vector can be interpreted directly as a 3-component mismatch response

This will provide the final experimental foundation needed before formal continuum results.

References

This implementation is based on:

“T2 — A 2D Toy Model of Cohesion Dynamics” (Research Specification)

• Section 1: The Discrete 2D Substrate
• Section 2: Mismatch in 2D
• Section 3: Height in Terms of Domain-Wall Geometry
• Section 4: Relaxation Dynamics and Emergent Curvature Flow
• Section 5: Divergence in 2D
• Section 6: Incorporating Tolerance
• Section 7: Minimal Divergent Surfaces in 2D
• Section 8: What T2 Teaches Us About the CD Programme

This model extends the concepts from T1 (1D) and provides concrete mechanisms for geometric emergence from discrete mismatch dynamics.