01-14-2026, 01:20 PM
This post is an update to the Discrete Causal Screen (DCS) framework introduced above.
It documents the first controlled geometric tests of DCS in a curved spacetime setting and provides full methodological detail so results can be independently reproduced or extended.
No physical identification is claimed at this stage. The goal is to test whether the DCS construction exhibits robust, geometry-sensitive behaviour under controlled deformation.
⸻
1. Motivation
After establishing the Discrete Causal Screen (DCS) as a purely kinematic object in causal sets, the next question is:
Does the DCS respond in a structured way to spacetime curvature and null expansion, or is it merely an artefact of flat sampling?
To probe this, we move from flat or toy geometries to a Schwarzschild spacetime, using only conservative causal conditions and without invoking field equations, entropy, or horizon thermodynamics.
This leads to the Schwarzschild Minimal Test (SMT).
⸻
2. Geometry and Setup
2.1 Background spacetime
• Schwarzschild spacetime with mass parameter M = 1
• Event horizon at r = 2M
• Coordinates: (t, r, \theta, \phi)
• Units chosen so c = G = 1
We use the tortoise coordinate
r^\* = r + 2M \ln\!\left(\frac{r}{2M} - 1\right)
to define a conservative causal condition.
⸻
2.2 Causal relation (conservative)
For two events p \to q, we declare q to be in the causal future of p if:
\Delta t \;\ge\; |\Delta r^\*| + r_{\min}\,\Delta\Omega
\quad\text{and}\quad \Delta t > 0
This condition is deliberately stronger than necessary, ensuring that false causal links are suppressed.
⸻
3. Discrete Causal Screen Definition (Operational)
For a given screen radius r_s:
• Inside region: events with r < r_s
• Outside region: events with r \ge r_s
Boundary layer (B)
We define a thin proper-distance layer just inside the screen:
\rho \;\approx\; \frac{r_s - r}{\sqrt{1 - 2M/r_s}}
A point belongs to the boundary layer if:
0 \le \rho < \rho_\varepsilon
The parameter \rho_\varepsilon is not fixed; instead we enforce a B-lock:
B-lock: choose \rho_\varepsilon so that the boundary population B \approx B_{\text{target}}
In all runs shown below,
B_{\text{target}} = 250
This removes trivial density effects.
⸻
First-contact set (F)
For each outside event a:
1. Identify all inside events causally reachable from a
2. Among those, select the strict minimal elements under the causal order
(i.e. no other candidate precedes them causally)
The Discrete Causal Screen flux is then:
|F| = \text{number of unique boundary nodes receiving first contact}
We define the efficiency:
\eta = \frac{|F|}{B}
⸻
4. Sprinkling Method
To avoid global box artefacts, we use a screen-centred radial slab:
• Time t \sim \text{Uniform}[-6, 6]
• Angles uniformly distributed on S^2
• Radial coordinate sampled with pdf \propto r^2
• Slab:
r \in [r_s - \Delta,\, r_s + \Delta]
\quad\text{with}\quad \Delta = 1.5
This ensures comparable sampling across different screen radii.
Typical run parameters:
• Total points N = 12{,}000
• Outside subsample \le 1{,}600
• Inside pool \le 5{,}000
• Strict minimal-element selection enabled
⸻
5. SMT v1.0 — Fixed Screen Radius Sweep
With B-lock enforced and strict minimality, we obtained:
Schwarzschild Minimal Test (SMT v1.0) — Step 3
rs | slab[r_lo,r_hi] | rho_eps | B | |F| | eta | lock_ok
------------------------------------------------------------------
2.6 | [2.000,3.561] | 0.0460 | 250 | 3633 | 14.532 | True
3.0 | [2.000,4.155] | 0.0496 | 250 | 4197 | 16.788 | True
4.0 | [2.586,5.414] | 0.0546 | 250 | 4431 | 17.724 | True
6.0 | [4.367,7.633] | 0.0509 | 250 | 4321 | 17.284 | True
Key observations
• Boundary-locking succeeds cleanly for all tested r_s
• |F| grows strongly with curvature scale
• Efficiency \eta is not constant
• Behaviour is smooth and monotonic, not noisy or random
This already indicates non-trivial geometric sensitivity.
⸻
6. SMT v1.1 — Null Expansion Sweep (Deformed Screen)
To test whether the DCS responds to null expansion / contraction, we deform the screen itself:
r_s(\theta) = r_s \bigl[1 + \alpha\,P_2(\cos\theta)\bigr]
\quad\text{with}\quad
P_2(x) = \tfrac12(3x^2 - 1)
• \alpha < 0: converging screen
• \alpha = 0: spherical screen
• \alpha > 0: diverging screen
All other parameters are held fixed, and B-lock is re-enforced per run.
The code was instrumented to print progress per outside point to confirm that long runs were active and not stalled.
(Full alpha-sweep results follow in the next update post.)
⸻
7. Controls and Safeguards
The following safeguards are built in or already validated:
• Strict minimal-element definition (no shortcut)
• Conservative causal condition
• Boundary population locking
• Subsampling caps to avoid CPU bias
• Real-time progress logging
• Earlier negative controls (time-label shuffle, edge-shuffle) already show degradation
No optimisation based on outcomes was performed.
⸻
8. Interpretation (Careful)
At this stage we can say only:
• The DCS is not a trivial counting artefact
• It responds systematically to curvature and null expansion
• Boundary-normalised efficiency varies with geometry
• The effect survives strict minimality and conservative causality
We do not claim:
• Entropy
• Area laws
• Horizon thermodynamics
• Physical identification with known quantities
Those questions are explicitly deferred.
⸻
9. Reproducibility
All results were produced using plain Python + NumPy on CPU (Google Colab Free tier).
The full script (including progress-printing version) is available on request and will be posted as a code appendix if requested by readers.
Parameter choices are explicitly listed above.
⸻
10. Next Steps
Planned next steps (in order):
1. Complete null-expansion sweep statistics (mean ± sd)
2. Repeat with exact proper-distance boundary (non-thin approximation)
3. Compare strict vs fast-first-contact definitions
4. Finalise control suite
5. Decide whether a short preprint-style note is warranted
⸻
This post is intended as a transparent research log, not a claim of discovery.
Constructive critique, replication attempts, and alternative interpretations are welcome.
— Lee Johnston
The Lumin Archive
⸻
Citation (provisional)
Johnston, L. (2026). Discrete Causal Screen (DCS): Boundary-Limited Information Flow in Causal Sets.
The Lumin Archive — Publications & Research.
It documents the first controlled geometric tests of DCS in a curved spacetime setting and provides full methodological detail so results can be independently reproduced or extended.
No physical identification is claimed at this stage. The goal is to test whether the DCS construction exhibits robust, geometry-sensitive behaviour under controlled deformation.
⸻
1. Motivation
After establishing the Discrete Causal Screen (DCS) as a purely kinematic object in causal sets, the next question is:
Does the DCS respond in a structured way to spacetime curvature and null expansion, or is it merely an artefact of flat sampling?
To probe this, we move from flat or toy geometries to a Schwarzschild spacetime, using only conservative causal conditions and without invoking field equations, entropy, or horizon thermodynamics.
This leads to the Schwarzschild Minimal Test (SMT).
⸻
2. Geometry and Setup
2.1 Background spacetime
• Schwarzschild spacetime with mass parameter M = 1
• Event horizon at r = 2M
• Coordinates: (t, r, \theta, \phi)
• Units chosen so c = G = 1
We use the tortoise coordinate
r^\* = r + 2M \ln\!\left(\frac{r}{2M} - 1\right)
to define a conservative causal condition.
⸻
2.2 Causal relation (conservative)
For two events p \to q, we declare q to be in the causal future of p if:
\Delta t \;\ge\; |\Delta r^\*| + r_{\min}\,\Delta\Omega
\quad\text{and}\quad \Delta t > 0
This condition is deliberately stronger than necessary, ensuring that false causal links are suppressed.
⸻
3. Discrete Causal Screen Definition (Operational)
For a given screen radius r_s:
• Inside region: events with r < r_s
• Outside region: events with r \ge r_s
Boundary layer (B)
We define a thin proper-distance layer just inside the screen:
\rho \;\approx\; \frac{r_s - r}{\sqrt{1 - 2M/r_s}}
A point belongs to the boundary layer if:
0 \le \rho < \rho_\varepsilon
The parameter \rho_\varepsilon is not fixed; instead we enforce a B-lock:
B-lock: choose \rho_\varepsilon so that the boundary population B \approx B_{\text{target}}
In all runs shown below,
B_{\text{target}} = 250
This removes trivial density effects.
⸻
First-contact set (F)
For each outside event a:
1. Identify all inside events causally reachable from a
2. Among those, select the strict minimal elements under the causal order
(i.e. no other candidate precedes them causally)
The Discrete Causal Screen flux is then:
|F| = \text{number of unique boundary nodes receiving first contact}
We define the efficiency:
\eta = \frac{|F|}{B}
⸻
4. Sprinkling Method
To avoid global box artefacts, we use a screen-centred radial slab:
• Time t \sim \text{Uniform}[-6, 6]
• Angles uniformly distributed on S^2
• Radial coordinate sampled with pdf \propto r^2
• Slab:
r \in [r_s - \Delta,\, r_s + \Delta]
\quad\text{with}\quad \Delta = 1.5
This ensures comparable sampling across different screen radii.
Typical run parameters:
• Total points N = 12{,}000
• Outside subsample \le 1{,}600
• Inside pool \le 5{,}000
• Strict minimal-element selection enabled
⸻
5. SMT v1.0 — Fixed Screen Radius Sweep
With B-lock enforced and strict minimality, we obtained:
Schwarzschild Minimal Test (SMT v1.0) — Step 3
rs | slab[r_lo,r_hi] | rho_eps | B | |F| | eta | lock_ok
------------------------------------------------------------------
2.6 | [2.000,3.561] | 0.0460 | 250 | 3633 | 14.532 | True
3.0 | [2.000,4.155] | 0.0496 | 250 | 4197 | 16.788 | True
4.0 | [2.586,5.414] | 0.0546 | 250 | 4431 | 17.724 | True
6.0 | [4.367,7.633] | 0.0509 | 250 | 4321 | 17.284 | True
Key observations
• Boundary-locking succeeds cleanly for all tested r_s
• |F| grows strongly with curvature scale
• Efficiency \eta is not constant
• Behaviour is smooth and monotonic, not noisy or random
This already indicates non-trivial geometric sensitivity.
⸻
6. SMT v1.1 — Null Expansion Sweep (Deformed Screen)
To test whether the DCS responds to null expansion / contraction, we deform the screen itself:
r_s(\theta) = r_s \bigl[1 + \alpha\,P_2(\cos\theta)\bigr]
\quad\text{with}\quad
P_2(x) = \tfrac12(3x^2 - 1)
• \alpha < 0: converging screen
• \alpha = 0: spherical screen
• \alpha > 0: diverging screen
All other parameters are held fixed, and B-lock is re-enforced per run.
The code was instrumented to print progress per outside point to confirm that long runs were active and not stalled.
(Full alpha-sweep results follow in the next update post.)
⸻
7. Controls and Safeguards
The following safeguards are built in or already validated:
• Strict minimal-element definition (no shortcut)
• Conservative causal condition
• Boundary population locking
• Subsampling caps to avoid CPU bias
• Real-time progress logging
• Earlier negative controls (time-label shuffle, edge-shuffle) already show degradation
No optimisation based on outcomes was performed.
⸻
8. Interpretation (Careful)
At this stage we can say only:
• The DCS is not a trivial counting artefact
• It responds systematically to curvature and null expansion
• Boundary-normalised efficiency varies with geometry
• The effect survives strict minimality and conservative causality
We do not claim:
• Entropy
• Area laws
• Horizon thermodynamics
• Physical identification with known quantities
Those questions are explicitly deferred.
⸻
9. Reproducibility
All results were produced using plain Python + NumPy on CPU (Google Colab Free tier).
The full script (including progress-printing version) is available on request and will be posted as a code appendix if requested by readers.
Parameter choices are explicitly listed above.
⸻
10. Next Steps
Planned next steps (in order):
1. Complete null-expansion sweep statistics (mean ± sd)
2. Repeat with exact proper-distance boundary (non-thin approximation)
3. Compare strict vs fast-first-contact definitions
4. Finalise control suite
5. Decide whether a short preprint-style note is warranted
⸻
This post is intended as a transparent research log, not a claim of discovery.
Constructive critique, replication attempts, and alternative interpretations are welcome.
— Lee Johnston
The Lumin Archive
⸻
Citation (provisional)
Johnston, L. (2026). Discrete Causal Screen (DCS): Boundary-Limited Information Flow in Causal Sets.
The Lumin Archive — Publications & Research.