A limit theorem linking continuous plate models to the topology of the plate boundary network
Abstract
Plate tectonics is described in two registers: a discrete one, in which rigid plates tile the sphere and meet at trivalent junctions, and a continuous one, in which Bercovici and Wessel (1994) replace hard plate outlines by smooth shape functions of finite boundary half-width δ*. We prove that the two registers are connected by a limit theorem. Generalizing the shape functions to a signed-geodesic-distance construction, we show that they converge almost everywhere, exponentially in 1/δ*, to the plate indicator functions; that the sharp plate mosaic is a finite regular trivalent CW decomposition of the sphere under an explicit structural hypothesis; and that a component-counted weighted Euler characteristic χwδ*(τ), built from the super-level sets of the shape functions, equals V−E+F=2 for all δ* below an explicit threshold δ0(τ), with its face, edge and vertex terms converging separately to the numbers of plates, boundary arcs and triple junctions. The theorem is proved unconditionally on the signed-distance offset cover under positive-reach, separation and bounded-sector hypotheses, and transferred to the normalized partition-of-unity field for thresholds τ<1/3 by a comparison lemma whose junction-ball step, a planar single-crossing property, is certified numerically. The residual |χw−2| defines a topological diffuseness index for diffuse plate boundaries. For the present-day PB2002 network the hypotheses are measured to be non-vacuous, with δ0(0.15)≈2.8 km. The theorem is the mathematical foundation of the topological audit of plate reconstructions presented in a companion paper (Kim, 2026, submitted to Geoscience Frontiers).
Related
- A Physics-Constrained Implicit Profile Network for Continuous Reconstruction of Tropical Cyclone Near-Surface Wind Profiles
- A unifying principle linking water uptake and nutrient delivery in trees with implications for soilless vascular delivery
- Functorial invariants for chaos topology from data
- When Does Stress Build Strength? Linking Stress Physiology to Resilience in the Age of Rapid Environmental Change
- Engineering cyanobacterial resilience for cadmium bioremediation: linking Cd speciation, cellular detoxification, and synthetic biology.
BibTeX
@misc{kim2026limit,
title = {{A limit theorem linking continuous plate models to the topology of the plate boundary network}},
author = {Seung-Sep Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39187},
eprint = {2609.39187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39187},
}
RIS
TY - UNPB
TI - A limit theorem linking continuous plate models to the topology of the plate boundary network
AU - Seung-Sep Kim
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39187
UR - https://arxiv.org/abs/2609.39187
AB - Plate tectonics is described in two registers: a discrete one, in which rigid plates tile the sphere and meet at trivalent junctions, and a continuous one, in which Bercovici and Wessel (1994) replace hard plate outlines by smooth shape functions of finite boundary half-width δ^*. We prove that the two registers are connected by a limit theorem. Generalizing the shape functions to a signed-geodesic-distance construction, we show that they converge almost everywhere, exponentially in 1/δ^*, to the plate indicator functions; that the sharp plate mosaic is a finite regular trivalent CW decomposition of the sphere under an explicit structural hypothesis; and that a component-counted weighted Euler characteristic χ_w^δ^*(τ), built from the super-level sets of the shape functions, equals V−E+F=2 for all δ^* below an explicit threshold δ_0(τ), with its face, edge and vertex terms converging separately to the numbers of plates, boundary arcs and triple junctions. The theorem is proved unconditionally on the signed-distance offset cover under positive-reach, separation and bounded-sector hypotheses, and transferred to the normalized partition-of-unity field for thresholds τ<1/3 by a comparison lemma whose junction-ball step, a planar single-crossing property, is certified numerically. The residual |χ_w−2| defines a topological diffuseness index for diffuse plate boundaries. For the present-day PB2002 network the hypotheses are measured to be non-vacuous, with δ_0(0.15)≈2.8 km. The theorem is the mathematical foundation of the topological audit of plate reconstructions presented in a companion paper (Kim, 2026, submitted to Geoscience Frontiers).
ER -