T1: One-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 simplified 1D setting. The tolerance parameter W shown here is an admissibility window that constrains which configurations may participate in joint closure—it does not drive structural formation or encode physical constants.

This toy model runs entirely in-browser and illustrates pedagogical concepts from the T1 paper series.

This page presents an interactive demonstration of the first Cohesion Dynamics toy model: a one-dimensional finite chain with binary symbols and nearest-neighbour constraints.

Interactive Demo

Chain Size (N):

RandomSingle WallTwo WallsAlternating

§8 Boundary Constraints:Free (No constraints)Fixed Opposite (X₁=+1, Xₙ=-1)

§9 Tolerance ε:

Threshold W:

At threshold (ε = W)

← UndoRedo →▶ Auto-RelaxStep Relax

Mismatch M(X):0

Weighted M_ε(X):0.00

Height H(V;X):0

Domain Walls:0

Status:At Closure ✓

About This Toy Model

This interactive demo implements the T1 One-Dimensional Cohesion Dynamics toy model, now extended with sections 8-10. It visualizes a finite chain of 8 sites, each carrying a binary symbol (+1 or -1).

Key Concepts:

• Domain Walls (⚡): Appear between sites with different values. Each wall contributes 1 unit to the mismatch M(X).
• Mismatch M(X): Total count of domain walls. Measures how far the configuration is from perfect agreement.
• Relaxation Moves: Site flips that don't increase mismatch (ΔM ≤ 0). Sites with available relaxation moves are highlighted.
• Closure: A configuration where no strictly mismatch-decreasing moves are available. The system is locally stable.
• Height H(V;X): For the full chain, equals M(X). Represents the minimum number of domain walls that must disappear during relaxation to closure.

§8 Divergent Configuration (Irreducible Mismatch):

• Fixed-Opposite Boundaries: When enabled, X₁=+1 and Xₙ=-1 are locked (🔒).
• Irreducible Mismatch: At least 1 domain wall must exist between opposite boundaries. This demonstrates a minimal divergent region - an irreducible inconsistency.
• Try it: Enable fixed-opposite mode and watch relaxation. The system reaches closure but cannot eliminate all domain walls. One remains as an obstruction.

§9 Tolerance Parameter ε (W-like Mechanism):

• Weighted Mismatch: M_ε(X) = ε × D(X), where ε controls domain wall cost.
• Small ε: Domain walls are "cheap" → slow relaxation, extended smoothing times.
• Large ε: Domain walls are "expensive" → rapid relaxation toward uniformity.
• Threshold W: When ε < W, system remains cohesive. When ε > W, domain walls become effectively rigid, behavior resembles divergence surfaces.
• Try it: Vary ε and W to see how tolerance affects the system's evolution character.

§10 Continuum Limit (Shocks & Diffusion):

• Domain Walls → Shocks: As discretization spacing h→0, domain walls persist as finite discontinuities in the continuum field φ(x).
• Relaxation → Diffusion: Local flips reduce discrete curvature (X_i-1 - 2X_i + X_i+1 ≈ h²φ_xx), analogous to the diffusion equation ∂_tφ = κ∂_xxφ.
• Insight: This 1D toy model illustrates how discrete relaxation dynamics can emerge as continuous PDEs with second spatial derivatives in the continuum limit.

Interactions:

• Click a site to select it and see flip analysis
• Double-click a site to flip it immediately (except locked boundaries)
• Step Relax: Perform one relaxation move (greedy: picks first ΔM < 0, else first ΔM = 0)
• Auto-Relax: Automatically relaxes to closure

Try These Experiments:

• Start with "Alternating" pattern - maximum mismatch! Watch it relax to closure.
• Create "Two Walls" and observe how domain walls can be annihilated.
• Notice: uniform configurations (all +1 or all -1) are closures with M = 0.
• Observe that Height H = M for the full chain in this 1D model.
• §8: Enable fixed-opposite boundaries and see the irreducible domain wall that cannot be eliminated.
• §9: Try ε = 0.5 (small) vs ε = 3.0 (large) and observe relaxation speed differences.
• §9: Set ε > W and watch the tolerance indicator change to "Divergent-like".

Toy Model Overview

This toy model implements the T1 One-Dimensional Substrate as specified in “Cohesion Dynamics Toy Models I: A One-Dimensional Substrate”. The model demonstrates:

The Substrate

• Vertices: A finite chain $V = \\{1,2,\\dots,N\\}$
• Edges: Nearest-neighbor connections $E = \\{\\{i,i+1\\} : 1 \\le i \\le N-1\\}$
• Alphabet: Binary symbols $\\Sigma = \\{-1,+1\\}$
• Configuration: A map $X : V \\to \\Sigma$

Key Primitives

1. Edge Constraints

For each edge $e = \\{i,i+1\\}$:

C_e(X) = \\begin{cases} 0 & \\text{if } X_i = X_{i+1} \\text{ (cohesive)},\\\\[4pt] 1 & \\text{if } X_i \\ne X_{i+1} \\text{ (frustrated/domain wall)}. \\end{cases}

2. Global Mismatch

The mismatch functional counts domain walls:

$$M(X) = \\sum_{e \\in E} C_e(X) = \\sum_{i=1}^{N-1} \\mathbf{1}_{\\{X_i \\ne X_{i+1}\\}}$$

Properties:

• $M(X) \\in \\{0,1,2,\\dots,N-1\\}$
• $M(X) = 0$ if and only if $X$ is uniform (all sites agree)

3. Relaxation Moves

A single-site flip at $j$ changes $X_j \\to -X_j$. The local mismatch change is:

$$\\Delta M_j(X) = M(Y) - M(X)$$

where $Y$ is the configuration after flipping site $j$.

A flip is a relaxation move if and only if:

$$\\Delta M_j(X) \\le 0$$

That is, relaxation moves are exactly those that do not increase global mismatch.

4. Closure

A configuration $Y$ is a closure if:

• No site admits a strictly mismatch-decreasing flip ($\\Delta M_j < 0$ for all $j$)

In this 1D model:

• Closure exists for every configuration
• Uniform configurations (all +1 or all -1) are closures with $M = 0$

5. Height

For the full chain $V$, the height is:

$$H(V;X) = M(X)$$

The height equals the number of domain walls in the initial configuration. It represents the minimum mismatch reduction achievable through relaxation to closure.

Mathematical Properties

Acyclicity

Since $M(X)$ is integer-valued and bounded below by 0, and each relaxation move satisfies $\\Delta M \\le 0$:

• Any relaxation sequence must terminate in finitely many steps
• No relaxation cycles with net mismatch decrease can exist
• The relaxation graph is acyclic

Closure Uniqueness

• If only $\\Delta M < 0$ moves are allowed: closure is unique (up to global sign)
• If $\\Delta M = 0$ moves are allowed: multiple closures may exist with the same $M$ value
  • Domain walls can “slide” without changing total mismatch
  • This nonuniqueness motivates the concept of provenance in the full theory

Height Properties

In this 1D toy model, height has a particularly simple form:

• For the full chain: $H(V;X) = M(X)$ (number of domain walls)
• For any interval $R$: $H(R;X) = M(R;X)$ (internal domain walls)
• Height is not an independent degree of freedom - it’s fully determined by mismatch

Explicit Example: 4-Site Chain

Consider $N=4$ with configuration $X = (+1, -1, -1, +1)$:

Mismatch Calculation

• Edge $\\{1,2\\}$: $+1$ vs $-1$ → domain wall → $C_{\\{1,2\\}}(X)=1$
• Edge $\\{2,3\\}$: $-1$ vs $-1$ → cohesive → $C_{\\{2,3\\}}(X)=0$
• Edge $\\{3,4\\}$: $-1$ vs $+1$ → domain wall → $C_{\\{3,4\\}}(X)=1$

Total mismatch: $M(X) = 2$

Relaxation Analysis

Checking each site:

Site $j$	After Flip	New $M$	$\\Delta M_j$	Allowed?
1	$(-1,-1,-1,+1)$	1	-1	✓ Yes
2	$(+1,+1,-1,+1)$	2	0	✓ Yes
3	$(+1,-1,+1,+1)$	2	0	✓ Yes
4	$(+1,-1,-1,-1)$	1	-1	✓ Yes

All sites admit relaxation moves. Sites 1 and 4 decrease mismatch.

Relaxation to Closure

One possible sequence:

1. Flip site 1: $X \\rightsquigarrow (-1,-1,-1,+1)$ with $M=1$
2. Flip site 4: $(-1,-1,-1,+1) \\rightsquigarrow (-1,-1,-1,-1)$ with $M=0$

The final configuration $(-1,-1,-1,-1)$ is uniform and thus a closure.

Height: $H(V;X) = 2$ (the two domain walls that were eliminated).

8. Divergent Configurations (Irreducible Mismatch)

Boundary-Constrained Chain

When we impose fixed boundary conditions with $X_1 = +1$ and $X_N = -1$ (locked endpoints), an interesting phenomenon emerges:

Every configuration must contain at least one domain wall.

No matter how the interior sites are configured, the chain begins with $+1$ and ends with $-1$, guaranteeing at least one transition point where $X_k = +1$ and $X_{k+1} = -1$.

Irreducible Mismatch

This creates an irreducible mismatch:

$$\inf\\{ D(Y) : X \rightsquigarrow Y \\} = 1$$

Relaxation can shift the domain wall left or right, but cannot eliminate it.

The height of the entire chain becomes:

$$H([1,N];X) = 1$$

Minimal Divergent Region

The single edge $\\{k,k+1\\}$ containing the unavoidable domain wall is a minimal divergent region in the obstruction sense:

• Its height is $H(\\{k,k+1\\};X) = 1$
• Any strict subregion has height $0$
• No relaxation can eliminate its mismatch

This demonstrates how irreducible inconsistency manifests in the substrate.

Try it in the demo: Enable “Fixed Opposite” boundary mode and observe that relaxation cannot reach $M = 0$.

9. Tolerance Parameter $\varepsilon$ (W-like Mechanism)

To model tolerance - a key concept in CD - we introduce a parameter $\varepsilon > 0$ that controls mismatch weighting.

Weighted Mismatch

Define:

$$m_i^{(\varepsilon)}(X) = \varepsilon \cdot \mathbf{1}_{\\{X_i \ne X_{i+1}\\}}$$

The total weighted mismatch is:

$$M_\varepsilon(X) = \sum_{i=1}^{N-1} m_i^{(\varepsilon)}(X) = \varepsilon \cdot D(X)$$

Relaxation moves must remain $M_\varepsilon$-nonincreasing.

Behavior as $\varepsilon$ Varies

Small $\varepsilon$ (domain walls cheap):

• Mismatch decreases have little effect on total cost
• Relaxation evolves slowly
• Many configurations remain rough for extended times
• Interpretation: Low tolerance → long coherent updates → extended smoothing

Large $\varepsilon$ (domain walls expensive):

• Disagreements are heavily penalized
• Relaxation rapidly removes mismatches
• Closure reached quickly
• Interpretation: High tolerance → strong pressure for uniformity

Tolerance Threshold $W$

When the system has a tolerance ceiling $W$:

• If $\varepsilon < W$: mismatch is “within tolerance” → cohesion maintained
• If $\varepsilon > W$: mismatch becomes unacceptable → behavior resembles divergence

In this 1D model:

• $\varepsilon < W$: domain walls move but don’t cause partition
• $\varepsilon > W$: domain walls become effectively rigid, like divergence surfaces

Tolerance controls whether evolution remains cohesive or fragments.

Try it in the demo: Adjust $\varepsilon$ and $W$ to see how tolerance affects system behavior.

10. Continuum Limit (Shocks & Diffusion)

We provide a minimal continuum interpretation of the discrete dynamics.

Domain Walls → Shock Discontinuities

Let discretization spacing be $h = 1/N$. Define the piecewise-constant interpolant:

$$\phi_h(x) = X_i \quad \text{for } x \in [(i-1)h, ih)$$

A domain wall is a jump in $\phi_h$. As $h \to 0$:

• Its width shrinks to $0$
• Its effect persists as a finite discontinuity

Domain walls converge to shocks in the continuum field $\phi(x,t)$.

Relaxation → Diffusion Equation

For small $h$, relaxation depends on $X_{i-1}, X_i, X_{i+1}$. Taylor expansion gives:

$$X_{i-1} - 2X_i + X_{i+1} \approx h^2 \phi_{xx}(x)$$

Relaxation reduces this discrete curvature:

• If $\phi_{xx} > 0$, $X_i$ tends to decrease
• If $\phi_{xx} < 0$, $X_i$ tends to increase

In the scaling limit:

$$\partial_t \phi = \kappa \partial_{xx}\phi$$

for some $\kappa > 0$ depending on flip rules.

Interpretation: Relaxation → continuous diffusion.

Relevance for Paper B

Although diffusion is parabolic, this toy model illustrates:

• Mismatch → gradient energy
• Relaxation → curvature reduction
• Discrete dynamics → second spatial derivatives in continuum
• Closure cycles → emergent continuous time

This provides intuition for why more complex CD systems can produce:

• Second-order PDEs
• Continuum fields
• Geometric propagation
• And ultimately, Lorentzian causal structure in higher-dimensional, multi-channel models

Significance for CD Theory

This toy model serves as a concrete laboratory for testing CD primitives:

1. Substrate Structure: Demonstrates how graph topology + alphabet + constraints define a CD substrate
2. Mismatch Functional: Shows a simple, computable measure of incompatibility
3. Relaxation Dynamics: Illustrates local evolution that preserves or improves cohesion
4. Closure: Provides explicit examples of stable configurations
5. Height: Demonstrates a well-defined measure of “distance from compatibility”
6. Divergence (§8): Shows irreducible mismatch and minimal divergent regions
7. Tolerance (§9): Illustrates how weighted mismatch controls system evolution character
8. Continuum Limit (§10): Connects discrete relaxation to continuous PDEs

While 1D is deliberately simple, it establishes patterns that extend to higher dimensions where:

• Frustration can prevent full relaxation
• Height captures nontrivial topological information
• Multiple closure branches and provenance become essential

Experiments to Try

Use the interactive demo above to explore:

1. Maximum Mismatch: Try the “Alternating” pattern - creates maximum domain walls
2. Relaxation Paths: Use “Step Relax” to see individual moves or “Auto-Relax” to reach closure
3. Closure Detection: Notice when the system reports “At Closure ✓”
4. Domain Wall Dynamics: Watch how domain walls move and annihilate
5. ΔM Analysis: Click sites to see whether their flip would increase, decrease, or preserve mismatch
6. Different Chain Sizes: Try $N=3$ (minimal) up to $N=20$ (more complex)
7. §8 Irreducible Mismatch: Enable “Fixed Opposite” boundaries and observe the domain wall that cannot be eliminated
8. §9 Small Tolerance: Set $\varepsilon = 0.5$ and watch slow, gradual relaxation
9. §9 Large Tolerance: Set $\varepsilon = 3.0$ and observe rapid mismatch reduction
10. §9 Threshold Effects: Try $\varepsilon < W$ (cohesive) vs $\varepsilon > W$ (divergent-like)
11. §10 Continuum Hints: Look for the continuum interpretation panel when domain walls are present

References

This implementation is based on:

“Cohesion Dynamics Toy Models I: A One-Dimensional Substrate” (v0.1, Series T1)

• Section 2: One-Dimensional CD Substrate
• Section 3: Local Constraints and Mismatch
• Section 4: Local Relaxation Moves
• Section 5: Acyclicity and Closure
• Section 6: Height in the 1D Toy Model
• Section 7: Explicit Examples
• Section 8: A Simple Divergent Configuration (Irreducible Mismatch)
• Section 9: Introducing a Tolerance Parameter $\varepsilon$ (A 1D “W-like” Mechanism)
• Section 10: Baby Continuum-Limit Analysis (Domain Walls → Shocks; Relaxation → Diffusion)

Later papers in this series (T2, T3) will extend to 2D and 3D lattices where richer phenomena emerge.