PREGEOMETRIC RELATIONAL DYNAMICS AND UNIVERSAL CAUSAL STRUCTURE A PERMUTATION-INVARIANT PHASE-SELECTION WITNESS: Quasilocality, Cone Locking, and the Boundary of Temporal Orientation
Abstract
This research presents a constructive mathematical model of how quasilocal geometry and effective causal structure can arise from permutation-invariant relational dynamics without predefined spatial coordinates, distances, orientations, or embeddings. Starting from finite undirected graphs, the model introduces an invariant energy Eₙ(A) = Σᵢ(deg_A(i) − 2)² + (k(A) − 1), together with stochastic edge-flip evolution. Its exact ground states are connected cycles, and their equilibrium selection is established through quantitative probability bounds. Under the intrinsic scaling aₙ = ℓ/N, these graph configurations converge to the continuous circle S¹ in the Gromov–Hausdorff–Prokhorov sense. The result provides an explicit example of geometry emerging from relational organisation, within a deliberately specified one-dimensional model.
A second central result concerns the emergence of a common propagation cone across multiple field sectors. An O(r)-equivariant relaxation mechanism drives initially unequal positive stiffness coefficients toward a common scalar limit according to B(s) = b̄Iᵣ + e^(−γs)(B₀ − b̄Iᵣ). Consequently, distinct propagation speeds converge exponentially toward v = √b̄. Following graph equilibration and stiffness relaxation, the selected geometries remain metastable with controlled probability, while their normalised wave responses approach the continuum equation ∂²ᵤq − b̄∂²ₓq = 0. Finite-graph response tails do not vanish exactly, but their properly normalised contributions disappear uniformly outside buffered causal cones in the continuum limit. Exact enumeration of 1,024 graphs for N = 5 and 32,768 graphs for N = 6 provides independent finite-size consistency checks of the analytical construction.
The most significant conceptual distinction concerns effective causal orientation versus the fundamental arrow of time. The limiting system supports an oriented causal partial order, (t,x) ≼ (s,y), whenever s ≥ t and d(x,y) ≤ v(s − t). However, this orientation depends on a prescribed forward evolution, nonequilibrium preparation, and retarded initial conditions. The work explicitly demonstrates why an undirected relational structure and a time-reversal-invariant equilibrium state do not independently determine a preferred temporal direction.
The central contribution is therefore a mathematically controlled existence result, connecting relational phase selection, intrinsic metric emergence, multicomponent cone locking, and operational causality while identifying the remaining requirements for a closed microscopic theory of spacetime emergence. Rather than claiming a derivation of physical 3+1-dimensional spacetime or gravitational dynamics, the study establishes a quantitative benchmark for future investigations into pregeometric dynamics, emergent locality, causal reconstruction, and temporal orientation.
Related
- Managers' Freelancer Mindset Orientation and relational evaluations as drivers of pay decisions for external workers
- Orientation engineering is a universal strategy for ferroelectric trans-switching
- Universal Shift Congruences for Weighted Permutation Sequences
- Given the lattice: Relational Coherence and the Keeping of the Living, a Universal Protection Theory
- Pregeometric 5D Elastodynamics and Trans-Brane Einstein-Rosen Connectors: Canonical Synthesis of the FCD-RBv6 Model
BibTeX
@misc{preece2026pregeometric,
title = {{PREGEOMETRIC RELATIONAL DYNAMICS AND UNIVERSAL CAUSAL STRUCTURE A PERMUTATION-INVARIANT PHASE-SELECTION WITNESS: Quasilocality, Cone Locking, and the Boundary of Temporal Orientation}},
author = {Andrei Preece and Boris Batenin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23432446},
url = {https://zenodo.org/records/23432446},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - PREGEOMETRIC RELATIONAL DYNAMICS AND UNIVERSAL CAUSAL STRUCTURE A PERMUTATION-INVARIANT PHASE-SELECTION WITNESS: Quasilocality, Cone Locking, and the Boundary of Temporal Orientation
AU - Andrei Preece
AU - Boris Batenin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23432446
UR - https://zenodo.org/records/23432446
AB - This research presents a constructive mathematical model of how quasilocal geometry and effective causal structure can arise from permutation-invariant relational dynamics without predefined spatial coordinates, distances, orientations, or embeddings. Starting from finite undirected graphs, the model introduces an invariant energy Eₙ(A) = Σᵢ(deg_A(i) − 2)² + (k(A) − 1), together with stochastic edge-flip evolution. Its exact ground states are connected cycles, and their equilibrium selection is established through quantitative probability bounds. Under the intrinsic scaling aₙ = ℓ/N, these graph configurations converge to the continuous circle S¹ in the Gromov–Hausdorff–Prokhorov sense. The result provides an explicit example of geometry emerging from relational organisation, within a deliberately specified one-dimensional model. A second central result concerns the emergence of a common propagation cone across multiple field sectors. An O(r)-equivariant relaxation mechanism drives initially unequal positive stiffness coefficients toward a common scalar limit according to B(s) = b̄Iᵣ + e^(−γs)(B₀ − b̄Iᵣ). Consequently, distinct propagation speeds converge exponentially toward v = √b̄. Following graph equilibration and stiffness relaxation, the selected geometries remain metastable with controlled probability, while their normalised wave responses approach the continuum equation ∂²ᵤq − b̄∂²ₓq = 0. Finite-graph response tails do not vanish exactly, but their properly normalised contributions disappear uniformly outside buffered causal cones in the continuum limit. Exact enumeration of 1,024 graphs for N = 5 and 32,768 graphs for N = 6 provides independent finite-size consistency checks of the analytical construction. The most significant conceptual distinction concerns effective causal orientation versus the fundamental arrow of time. The limiting system supports an oriented causal partial order, (t,x) ≼ (s,y), whenever s ≥ t and d(x,y) ≤ v(s − t). However, this orientation depends on a prescribed forward evolution, nonequilibrium preparation, and retarded initial conditions. The work explicitly demonstrates why an undirected relational structure and a time-reversal-invariant equilibrium state do not independently determine a preferred temporal direction. The central contribution is therefore a mathematically controlled existence result, connecting relational phase selection, intrinsic metric emergence, multicomponent cone locking, and operational causality while identifying the remaining requirements for a closed microscopic theory of spacetime emergence. Rather than claiming a derivation of physical 3+1-dimensional spacetime or gravitational dynamics, the study establishes a quantitative benchmark for future investigations into pregeometric dynamics, emergent locality, causal reconstruction, and temporal orientation.
ER -