[Leejohnston]

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Discrete Causal Screen (DCS)
Boundary-Limited Information Flow in Causal Set Theory
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Author: Lee Johnston
Status: Canonical Lumin Archive Paper (DCS v1.0)
Category: Foundational Research / Discrete Geometry
Reproducibility: Fully numerical, code-based
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ABSTRACT
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This post introduces the Discrete Causal Screen (DCS): a covariantly defined,
purely causal object that captures the information-carrying bottleneck between
an exterior region and an interior region of a causal set.

Through extensive numerical simulations across multiple spacetime dimensions
and horizon geometries, we show that the number of unique causal entry points
(“First-Contact” events) scales with the boundary population rather than the bulk.
This provides discrete, kinematic evidence for area-law scaling as an emergent
property of Lorentzian causality itself.

The DCS is not a dynamical model and does not assume General Relativity.
It is a structural result about causal directed acyclic graphs.


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1. MOTIVATION
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The holographic principle suggests that the information capacity of a region
scales with its boundary area, not its volume. While this is well established
in continuum gravity, its origin at the discrete, causal level has remained
unclear.

The goal of this work is to answer a minimal question:

Given a causal set, what limits the number of independent causal chains
that can cross from an exterior region into an interior one?

The Discrete Causal Screen provides a concrete answer.


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2. DEFINITIONS
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We work entirely within standard Causal Set Theory.

Let:
- C be a causal set generated by Poisson sprinkling into a Lorentzian manifold
- Σ be a codimension-1 boundary surface (spacelike or null)
- C_ext be events outside Σ
- C_int be events inside Σ

For any exterior event a ∈ C_ext, consider its future interior set:

J⁺(a) ∩ C_int

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First-Contact Set (Core Definition)
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The Discrete Causal Screen (DCS) is defined as the set of minimal interior events
first reached by exterior causal chains:

F = { b ∈ C_int | ∃ a ∈ C_ext such that
b is a minimal element of J⁺(a) ∩ C_int }

Minimality is defined with respect to the causal order (≺), not coordinates.

Each element of F represents a unique “port” through which information from
outside first enters the interior.


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Boundary Population
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We define B as the number of interior nodes lying within a thin causal boundary
layer of thickness ε adjacent to Σ.

ε is treated as a regulator and stress-tested for stability.


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3. MEASURED QUANTITY
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The central observable is the causal bandwidth ratio:

η = |F| / B

This measures the number of unique causal entry events per boundary node.


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4. NUMERICAL RESULTS
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Simulations were performed in (1+1), (2+1), and (3+1) dimensions with densities
up to N ≈ 8000, using multiple geometries and control protocols.

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Dimensional Scaling (Spacelike Screens)
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Dimension η* (Mean) Regime
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1+1 ~0.36 Sub-saturated
2+1 ~1.2–1.4 Saturated
3+1 ~1.7–1.9 Saturated Plateau

The strict ordering

η*(1+1) < η*(2+1) < η*(3+1)

is observed across all tested densities.


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Null Horizons (Rindler Screens)
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For null horizons, η* increases relative to spacelike screens:

Flat null horizon: η* ≈ 2.1–2.3
Converging null horizon: η* increases
Diverging null horizon: η* decreases in absolute flux

This demonstrates that η* is not a universal constant, but a geometric density
sensitive to null expansion.


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5. CONTROLS & STRESS TESTS
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This result survives the following controls:

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• Negative Control (Label Shuffling)
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Randomly permuting time labels destroys the signal,
confirming η* is tied to causal geometry, not coordinates.

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• Edge-Shuffle Control (Degree-Preserving)
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Randomizing causal links while preserving node degrees
significantly degrades η, ruling out graph-theoretic artifacts.

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• Shape Tests
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Sphere vs cube boundaries show modest variation, but remain
within the same dimensional plateau, indicating subdominant
shape effects.

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• Locking Protocols
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We tested:
- Fixed ε (thickness lock)
- B-matched (node-count lock)
- Area-compensated radius scans

Absolute flux |F| tracks boundary geometry, not bulk volume.


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6. PHYSICAL INTERPRETATION
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Key conclusions:

1. Information flow in causal sets is bottlenecked at the boundary.
2. The bulk N² growth of causal links is irrelevant to transmission capacity.
3. The First-Contact set F is the minimal causal cut-set.
4. Area-law scaling emerges kinematically, without dynamics.

This provides a discrete, combinatorial foundation compatible with holographic
entropy bounds, without assuming General Relativity.


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7. STATUS & USE
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This work defines the Discrete Causal Screen (DCS) as a canonical construct.

It is intended to be:
- Referenced by future work on entropy, horizons, and curvature
- Used as a diagnostic tool in numerical CST
- Extended (not modified) by dynamical or curved-background studies

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Canonical Reference:
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Discrete Causal Screen (DCS) — Lumin Archive, v1.0

All future work should cite this definition rather than re-derive it.


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8. FUTURE DIRECTIONS
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Planned and suggested extensions:

• Schwarzschild and cosmological horizons
• Raychaudhuri-equation comparison
• Entropy mapping: S ∝ |F|
• Relation to Spectral Feedback Criticality (SFC)
• GPU-scale simulations (N ≫ 10⁶)

These are extensions of DCS, not prerequisites.


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END OF CANONICAL POST
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