T4: Continuum Admission and Divergence

This page presents interactive demonstrations of the T4 Cohesion Dynamics toy model: the first model that explicitly distinguishes between substrate evolution (what could happen locally) and continuum admission (what is allowed to exist globally).

Overview

Earlier toy models (T1–T3) explored mismatch, relaxation, and divergence within the substrate. T4 introduces W-tolerance as an admission criterion, showing that:

Some locally consistent states are inadmissible to the continuum and therefore do not exist in the rendered world.

This allows divergence to appear not merely as trapped inconsistency, but as loss of representation.

Conceptual Summary

• The substrate may propose local updates freely.
• Each proposed update is checked against a tolerance vector W.
• Only admissible updates are committed to the continuum.
• Inadmissible regions:
  • stop updating,
  • are no longer rendered,
  • leave behind only boundary constraints.

This introduces error correction, horizon formation, and existence filtering without invoking forces, geometry, or 3D structure.

Formal Specification

1. Substrate

Let the substrate be a graph G = (V, E):

• V: cells (1D chain or 2D grid)
• E: adjacency relations

Each cell $v \in V$ carries a local state:

$$X_v \in \{-1, +1\}$$

2. Mismatch

Define local mismatch at $v$:

$$m(v) = \sum_{u \sim v} \mathbf{1}[X_u \neq X_v]$$

Define regional mismatch over $R \subseteq V$:

$$M(R) = \sum_{v \in R} m(v)$$

Mismatch measures local inconsistency pressure but does not determine admissibility.

3. Substrate Relaxation (Proposal Phase)

At each discrete step:

• Each cell proposes an update $X_v \to X'_v$
• Proposals must be mismatch-nonincreasing:

$$m'(v) \le m(v)$$

This defines possible substrate evolution.

No commitment occurs yet.

4. Tolerance Vector W

Each cell or region has a tolerance budget:

$$W = (W_{\text{shape}}, W_{\text{clock}}, W_{\text{spin}})$$

In T4, only a scalar proxy is used:

$$\|W\| = W_{\text{total}}$$

$W$ defines the maximum admissible inconsistency for continuum participation.

5. Height and Admissibility

Define height of a region $R$:

$$H(R) = M(R) - M_{\min}(R)$$

where $M_{\min}(R)$ is the minimal mismatch achievable by any closure in $R$.

Admissibility condition:

$$H(R) \le \|W\|$$

6. Continuum Commitment Rule

A proposed update affecting region $R$ is:

• Committed if $H(R) \le \|W\|$
• Rejected otherwise

Rejected regions:

• do not update,
• are excluded from rendering,
• may only influence neighbors via boundary constraints.

7. Rendering Rule (Continuum Semantics)

Only committed cells are rendered.

Inadmissible cells are:

• hidden,
• visually removed,
• treated as non-participating.

Boundaries between admissible and inadmissible regions remain visible.

8. Divergence (Continuum Definition)

A region $R$ is divergent if:

$$\forall t > t_0,\quad H(R, t) > \|W\|$$

Divergent regions:

• cannot re-enter the continuum,
• represent loss of existence, not just trapped inconsistency.

T4-1D: One-Dimensional Demo

The 1D demo shows how a divergent interior disappears from a 1D continuum.

Setup

• 1D chain of N cells
• Fixed boundary spins (+1 on left, −1 on right)
• Small W budget

Expected Behavior

• A domain wall forms
• Interior region exceeds W
• Interior cells disappear
• Only boundary/domain wall remains rendered

Visual Effect

• Cells vanish
• A “gap” appears
• Boundary becomes permanent

Key Insight

Divergence removes regions from the world rather than trapping them.

T4-1D: Continuum Admission in One Dimension

Substrate evolution with W-tolerance admission criterion

Mismatch M:0

Height H:0

Tolerance W:2.0

Admissible:✓ Yes

At Closure:✓ Yes

Steps:0

Chain Size N:12

Tolerance W:2.0

Boundary Mode:FreeFixed Opposite (Divergent)

Reset RandomStep CDAuto Relax

What You're Seeing

Fixed Opposite Boundaries: Left boundary locked at +1, right at -1. This forces at least one domain wall to exist (M_min = 1).

Height = M - M_min = 0 - 1 = 0

T4-2D: Two-Dimensional Demo

The 2D demo demonstrates horizon formation via admissibility filtering.

Setup

• 2D grid
• Opposing fixed boundaries
• Local relaxation enabled
• Global W budget enforced

Expected Behavior

• Domain wall curves form
• Interior pocket exceeds W
• Interior cells are excluded from rendering
• Boundary loop persists

Visual Effect

• “Holes” form in space
• Only enclosing surfaces remain
• Clear separation between substrate dynamics and continuum existence

T4-2D: Continuum Admission in Two Dimensions

Horizon formation via W-tolerance admissibility filtering

Mismatch M:0

Height H:0

Tolerance W:4.0

Admissible:✓ Yes

At Closure:✓ Yes

Steps:0

Grid Size N:16×16

Tolerance W:4.0

Boundary Mode:Free BoundariesDivergent (Fixed Opposite)

Reset RandomStep CDAuto Relax

What You're Seeing

Free Boundaries: No constraints. Domain walls can shrink and vanish completely.

Height = M - M_min = 0 - 0 = 0

All regions are admissible. The continuum matches the substrate.

Key insight: In T4, the continuum is not identical to the substrate. W-tolerance acts as an admission filter. Regions with excessive height are excluded from existence, creating gaps, holes, and horizon-like boundaries without invoking forces or energy minimization.

What T4 Demonstrates

✔ Key Concepts

• Continuum ≠ substrate: The substrate can propose states that don’t exist in the continuum
• W as admission constraint: Tolerance defines what can exist, not just what’s energetically favorable
• Divergence as loss of existence: Not just trapped inconsistency, but actual removal from reality
• Error correction without forces: Admissibility filtering provides a selection mechanism
• Horizons as admissibility boundaries: Regions are separated by existence thresholds

✖ What T4 Does NOT Claim

• Realistic gravity
• Binding or constructors
• 3D necessity
• Full spacetime emergence

Those belong to later work (Paper C and beyond).

Why One Model with Two Demos

• The theory is identical in 1D and 2D
• Only the adjacency graph changes
• Keeping one T4 paper avoids fragmentation
• Two demos reinforce dimensional independence of continuum admissibility

Programmatically:

• One theoretical framework
• Two renderers / initial conditions
• Demonstrates universality of the admission mechanism

Experiments to Try

In the 1D Demo

1. Default scenario: Watch the forced domain wall create inadmissible interior
2. Adjust W: Try larger W values to see more cells remain admissible
3. Observe gaps: Notice how cells literally vanish from the continuum
4. Boundary persistence: See how only the boundary region remains visible

In the 2D Demo

1. Free boundaries: Start with no constraints and watch normal relaxation
2. Divergent boundaries: Enable fixed opposite boundaries to force spanning interfaces
3. Adjust W: Vary tolerance to control how much interior can exist
4. Observe holes: Watch regions disappear from the rendered world
5. Boundary loops: See how only the enclosing surfaces persist

Significance for CD Theory

T4 is the first toy model with continuum semantics. It demonstrates:

1. Substrate vs Continuum Separation: Not all substrate states are real
2. Admission Mechanism: W-tolerance provides a filter on existence
3. Divergence Redefinition: From “trapped inconsistency” to “loss of representation”
4. Error Correction Primitive: Without invoking forces or energy minimization
5. Horizon Formation: As an emergent consequence of admissibility boundaries

This bridges the gap between:

• Earlier toy models (T1–T3): Pure substrate dynamics
• Future work: Full continuum theory with constructors and binding

References

This implementation is based on the T4 specification:

“T4 — Continuum Admission and Divergence” (Research Brief)

• Section 1: Conceptual Summary
• Section 2: Formal Specification
• Section 3: T4-1D Programmer Brief
• Section 4: T4-2D Programmer Brief
• Section 5: Why One T4 Model with Two Demos
• Section 6: What T4 Proves (and What It Doesn’t)

This model extends the concepts from T1 (1D), T2 (2D), and T3 (3D) by introducing the crucial distinction between substrate dynamics and continuum existence.