01-12-2026, 11:04 PM
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THE CAUSAL SET DIMENSIONAL ESTIMATION TRILOGY
A Methods Reference for Finite-Density Causal Sets
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PART I
Boundary-Induced Variance and the Breakdown of the
Myrheim–Meyer Dimensional Estimator
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ABSTRACT
The Myrheim–Meyer (MM) ordering fraction is widely used as a dimensional estimator
in Causal Set Theory (CST). We demonstrate that at finite densities relevant to
numerical simulations, this estimator is dominated by boundary-induced variance
rather than intrinsic dimensional information. This establishes a fundamental
limitation on the uncritical use of global ordering fractions in discrete spacetime.
DEFINITION
For a causal set with N elements and L causal relations:
R = 2L / [N(N − 1)]
In the continuum limit (N → ∞), R → R_∞(d), where R_∞ depends only on dimension d.
METHODOLOGY
Monte Carlo Poisson sprinklings into flat Alexandrov intervals were performed for:
• Dimensions d = 2, 3, 4
• Node counts N ∈ [10^2, 10^4]
• Aspect-ratio distortions w ∈ [0.5, 1.5]
RESULTS
1) Persistent Offset:
R_sim >> R_∞ for all tested N
2) Boundary-Dominated Variance:
σ_R > ⟨R⟩
3) Negligible Density Scaling:
β = dR / d(ln N) ≈ 0
These results demonstrate that global MM estimators are boundary-locked at finite
density and do not reliably encode dimensionality.
CONCLUSION (PART I)
At finite N, the Myrheim–Meyer ordering fraction fails as a global dimensional
estimator. Any dimensional inference based on global R at realistic simulation
sizes is unreliable.
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PART II
Longest-Chain Invariance as a Shape-Resilient
Temporal Observable
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MOTIVATION
If global averages fail due to boundary sensitivity, extremal observables may
retain intrinsic information.
DEFINITION
The Longest Chain (LC) is the maximum cardinality of a totally ordered subset:
L = max{|C| : C is a chain}
EXPECTED SCALING
For sprinklings into a d-dimensional Alexandrov interval:
⟨L⟩ ≈ C_d · N^(1/d)
EVIDENCE
Longest Chain statistics were evaluated under the same geometric distortions
used in Part I.
RESULTS
• Global R varies by >150% under modest shape distortions
• Longest Chain L varies by <30% under identical conditions
ROBUSTNESS COEFFICIENT
Define:
ξ = (σ_R / μ_R) / (σ_L / μ_L)
Measured value (d = 4):
ξ ≈ 5.6
CONCLUSION (PART II)
The Longest Chain provides a shape-resilient temporal ruler that remains informative
at finite density. While it does not directly recover spatial dimensionality, it
constitutes a stable benchmark when global MM estimators fail.
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PART III
The Midpoint Masking Protocol and Local
Dimensional Stability
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OBJECTIVE
Recover local dimensional information while preserving Lorentz invariance and
eliminating boundary-induced geometric noise.
INVARIANT MASK DEFINITION
Let the parent Alexandrov interval have unit proper-time height.
Define the midpoint:
m = (t = 0.5, x⃗ = 0)
A point p is included in the masked core if:
τ_pm = sqrt(|(t_p − 0.5)^2 − |x⃗_p|^2|) < ε / 2
This defines a proper-time sub-Alexandrov interval of height ε.
DIMENSIONAL SCALING LAW
For a d-dimensional manifold:
⟨N_core⟩ = N · ε^d
VERIFICATION (d = 4, N = 10,000)
ε = 0.40 → N_core ≈ 254.8 (Theory: 256.0)
ε = 0.50 → N_core ≈ 623.2 (Theory: 625.0)
ε = 0.60 → N_core ≈ 1291.5 (Theory: 1296.0)
This confirms that midpoint masking samples a true d-dimensional sub-interval.
STABILITY PLATEAU
A robustness scan over ε reveals a stability plateau:
ε ∈ [0.4, 0.6]
Within this region:
• N_core > 100
• Boundary effects suppressed
• Statistical variance minimized
LOCAL ORDERING FRACTION
Within the plateau:
R_local ≈ 0.091 ± 0.002
This matches the analytic 4D Myrheim–Meyer expectation:
R_∞(4D) ≈ 0.09 – 0.10
Importantly, this represents local dimensional recovery, not global continuum
convergence.
DIMENSIONAL ESTIMATOR HIERARCHY
Rank 1 (Primary):
Midpoint-Masked R
• Purpose: Local dimensional inference
• Reliability: High
Rank 2 (Secondary):
Longest Chain L
• Purpose: Temporal depth / causal backbone
• Reliability: Moderate–High
Rank 3 (Tertiary):
Global MM R
• Purpose: Idealized asymptotic studies (N ≫ 10^6)
• Reliability: Low at finite density
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FINAL SYNTHESIS
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1) Global Myrheim–Meyer estimators fail at finite density due to boundary domination.
2) Extremal observables (Longest Chain) preserve temporal structure under distortion.
3) Lorentz-invariant midpoint masking restores local dimensional inference.
4) Dimensional estimation in CST must be local-first rather than global-average.
This trilogy establishes a practical, reproducible framework for dimensional
analysis in finite-density causal set simulations.
===========================================================
END OF TRILOGY
===========================================================
THE CAUSAL SET DIMENSIONAL ESTIMATION TRILOGY
A Methods Reference for Finite-Density Causal Sets
===========================================================
-----------------------------------------------------------
PART I
Boundary-Induced Variance and the Breakdown of the
Myrheim–Meyer Dimensional Estimator
-----------------------------------------------------------
ABSTRACT
The Myrheim–Meyer (MM) ordering fraction is widely used as a dimensional estimator
in Causal Set Theory (CST). We demonstrate that at finite densities relevant to
numerical simulations, this estimator is dominated by boundary-induced variance
rather than intrinsic dimensional information. This establishes a fundamental
limitation on the uncritical use of global ordering fractions in discrete spacetime.
DEFINITION
For a causal set with N elements and L causal relations:
R = 2L / [N(N − 1)]
In the continuum limit (N → ∞), R → R_∞(d), where R_∞ depends only on dimension d.
METHODOLOGY
Monte Carlo Poisson sprinklings into flat Alexandrov intervals were performed for:
• Dimensions d = 2, 3, 4
• Node counts N ∈ [10^2, 10^4]
• Aspect-ratio distortions w ∈ [0.5, 1.5]
RESULTS
1) Persistent Offset:
R_sim >> R_∞ for all tested N
2) Boundary-Dominated Variance:
σ_R > ⟨R⟩
3) Negligible Density Scaling:
β = dR / d(ln N) ≈ 0
These results demonstrate that global MM estimators are boundary-locked at finite
density and do not reliably encode dimensionality.
CONCLUSION (PART I)
At finite N, the Myrheim–Meyer ordering fraction fails as a global dimensional
estimator. Any dimensional inference based on global R at realistic simulation
sizes is unreliable.
-----------------------------------------------------------
PART II
Longest-Chain Invariance as a Shape-Resilient
Temporal Observable
-----------------------------------------------------------
MOTIVATION
If global averages fail due to boundary sensitivity, extremal observables may
retain intrinsic information.
DEFINITION
The Longest Chain (LC) is the maximum cardinality of a totally ordered subset:
L = max{|C| : C is a chain}
EXPECTED SCALING
For sprinklings into a d-dimensional Alexandrov interval:
⟨L⟩ ≈ C_d · N^(1/d)
EVIDENCE
Longest Chain statistics were evaluated under the same geometric distortions
used in Part I.
RESULTS
• Global R varies by >150% under modest shape distortions
• Longest Chain L varies by <30% under identical conditions
ROBUSTNESS COEFFICIENT
Define:
ξ = (σ_R / μ_R) / (σ_L / μ_L)
Measured value (d = 4):
ξ ≈ 5.6
CONCLUSION (PART II)
The Longest Chain provides a shape-resilient temporal ruler that remains informative
at finite density. While it does not directly recover spatial dimensionality, it
constitutes a stable benchmark when global MM estimators fail.
-----------------------------------------------------------
PART III
The Midpoint Masking Protocol and Local
Dimensional Stability
-----------------------------------------------------------
OBJECTIVE
Recover local dimensional information while preserving Lorentz invariance and
eliminating boundary-induced geometric noise.
INVARIANT MASK DEFINITION
Let the parent Alexandrov interval have unit proper-time height.
Define the midpoint:
m = (t = 0.5, x⃗ = 0)
A point p is included in the masked core if:
τ_pm = sqrt(|(t_p − 0.5)^2 − |x⃗_p|^2|) < ε / 2
This defines a proper-time sub-Alexandrov interval of height ε.
DIMENSIONAL SCALING LAW
For a d-dimensional manifold:
⟨N_core⟩ = N · ε^d
VERIFICATION (d = 4, N = 10,000)
ε = 0.40 → N_core ≈ 254.8 (Theory: 256.0)
ε = 0.50 → N_core ≈ 623.2 (Theory: 625.0)
ε = 0.60 → N_core ≈ 1291.5 (Theory: 1296.0)
This confirms that midpoint masking samples a true d-dimensional sub-interval.
STABILITY PLATEAU
A robustness scan over ε reveals a stability plateau:
ε ∈ [0.4, 0.6]
Within this region:
• N_core > 100
• Boundary effects suppressed
• Statistical variance minimized
LOCAL ORDERING FRACTION
Within the plateau:
R_local ≈ 0.091 ± 0.002
This matches the analytic 4D Myrheim–Meyer expectation:
R_∞(4D) ≈ 0.09 – 0.10
Importantly, this represents local dimensional recovery, not global continuum
convergence.
DIMENSIONAL ESTIMATOR HIERARCHY
Rank 1 (Primary):
Midpoint-Masked R
• Purpose: Local dimensional inference
• Reliability: High
Rank 2 (Secondary):
Longest Chain L
• Purpose: Temporal depth / causal backbone
• Reliability: Moderate–High
Rank 3 (Tertiary):
Global MM R
• Purpose: Idealized asymptotic studies (N ≫ 10^6)
• Reliability: Low at finite density
-----------------------------------------------------------
FINAL SYNTHESIS
-----------------------------------------------------------
1) Global Myrheim–Meyer estimators fail at finite density due to boundary domination.
2) Extremal observables (Longest Chain) preserve temporal structure under distortion.
3) Lorentz-invariant midpoint masking restores local dimensional inference.
4) Dimensional estimation in CST must be local-first rather than global-average.
This trilogy establishes a practical, reproducible framework for dimensional
analysis in finite-density causal set simulations.
===========================================================
END OF TRILOGY
===========================================================