Corpus record: PFT:EMERGENCE_OF_LORENTZ_INVARIANT_STRUCTURE_FROM_DISCRETE_TRANSPORT_SYMMETRY_CONTINUUM_LIMIT
Availability: patternfieldtheory zenodo academia
Publication check: pending database verification. Public approval: no.
Emergence of Lorentz Invariant Structure from Discrete Transport Symmetry - Continuum Limit of Propagation in the Hexagonal Allen Orbital Lattice
2026-05-08
A relativistic invariant interval is derived as the continuum limit of isotropic discrete propagation on a hexagonal adjacency lattice. Finite propagation rate, directional symmetry, and uniform update structure produce a hyperbolic wave equation governing coarse scale transport. The invariant interval emerges as a geometric constraint imposed by bounded propagation. Lorentz transformations arise as the symmetry group preserving the transport metric.
Discrete Transport Geometry
Definition 1 (Allen Orbital Lattice). The Allen Orbital Lattice \(\mathrm{AOL}\) is a hexagonal adjacency graph with uniform step length \(a\) and update time \(\tau\).
Propagation occurs through admissible local transitions.
Definition 2 (Maximum Transport Rate). \[c = \frac{a}{\tau}.\]
No propagation exceeds this rate.
—
Isotropic Update Symmetry
Each lattice node has six equivalent propagation directions.
Proposition 1. Transport is directionally symmetric under rotations preserving hexagonal adjacency.
Thus large scale propagation is isotropic.
—
Discrete Evolution Equation
Let \(\psi(\mathbf{x},t)\) represent transport amplitude.
Update rule:
\[\psi(\mathbf{x}, t+\tau) = \frac{1}{6} \sum_{i=1}^{6} \psi(\mathbf{x} + \mathbf{e}_i a, t).\]
—
Continuum Expansion
Taylor expand to second order.
Spatial expansion:
\[\psi(\mathbf{x} + \mathbf{e}_i a, t) = \psi + a (\mathbf{e}_i \cdot \nabla)\psi + \frac{a^2}{2} (\mathbf{e}_i \cdot \nabla)^2 \psi.\]
Sum over symmetric directions eliminates first order term.
Second order term yields Laplacian:
\[\psi(\mathbf{x}, t+\tau) = \psi + \frac{a^2}{6}\nabla^2 \psi.\]
—
Temporal Expansion
Expand left side:
\[\psi(\mathbf{x}, t+\tau) = \psi + \tau \partial_t \psi + \frac{\tau^2}{2} \partial_t^2 \psi.\]
—
Matching Orders
Equate expansions:
\[\tau \partial_t \psi + \frac{\tau^2}{2} \partial_t^2 \psi = \frac{a^2}{6}\nabla^2 \psi.\]
Divide by \(\tau^2\):
\[\partial_t^2 \psi + \frac{2}{\tau}\partial_t \psi = \frac{a^2}{3\tau^2}\nabla^2 \psi.\]
For propagation dominated regime, damping negligible:
\[\partial_t^2 \psi = \frac{a^2}{3\tau^2}\nabla^2 \psi.\]
—
Wave Speed
Define effective propagation speed:
\[c^2 = \frac{a^2}{3\tau^2}.\]
Thus:
\[\partial_t^2 \psi = c^2 \nabla^2 \psi.\]
This is the hyperbolic wave equation.
—
Invariant Interval
Wave equation preserved under transformations maintaining:
\[c^2 t^2 - x^2 - y^2 - z^2.\]
Definition 3 (Transport Interval). \[s^2 = c^2 t^2 - r^2.\]
This interval remains invariant under allowed coordinate changes preserving propagation structure.
—
Lorentz Transformation
Consider transformation between coordinate frames moving at velocity \(v\) along \(x\).
Require invariance:
\[c^2 t'^2 - x'^2 = c^2 t^2 - x^2.\]
Solution yields:
\[x' = \gamma(x - vt)\]
\[t' = \gamma\left(t - \frac{vx}{c^2}\right)\]
\[\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.\]
—
Interpretation
Finite propagation rate enforces hyperbolic geometry of spacetime description.
Lorentz transformations preserve transport interval.
Thus relativistic kinematics arise from bounded discrete propagation.
—
Structural Consequence
Propagation bound defines causal structure.
Isotropic adjacency defines spatial symmetry.
Continuum limit defines wave equation.
Wave equation defines invariant interval.
Invariant interval defines Lorentz symmetry group.
—
Unified Role in Transport Hierarchy
Discrete step scale sets maximal propagation.
Propagation symmetry generates continuum transport law.
Continuum transport law generates invariant geometry.
Invariant geometry governs large scale dynamics.
—
Glossary
Propagation Bound — maximal structural update rate
Isotropic Symmetry — directional equivalence
Continuum Limit — coarse scale transport description
Invariant Interval — preserved propagation metric
Lorentz Transformation — coordinate change preserving interval
—
References
Einstein, A. (1905). Special theory of relativity.
Minkowski, H. (1908). Space and time.
Jackson, J. D. (1999). Classical electrodynamics.
Weinberg, S. (1995). Quantum theory of fields.
—
Document Timestamp and Provenance
This document derives Lorentz invariant structure from isotropic bounded propagation in the Allen Orbital Lattice. It establishes the emergence of relativistic kinematics as the continuum limit of discrete transport symmetry.