Finite-color reconstruction of groups and coherent configurations
Abstract
Every group has a canonical coloring of its unordered pairs by inverse pairs of differences. We prove that this coloring can be coarsened to four colors without changing its color-preserving permutation group, for groups of arbitrary cardinality. Four is the least uniform bound. A related construction applies to finite homogeneous coherent configurations: if every basic relation has valency at most d, where d≥ 2, a coarsening with at most 2d+1 off-diagonal colors in the symmetric case, or 2d+3 off-diagonal arc colors in general, has the original configuration as its coherent closure. Two-dimensional Weisfeiler-Leman refinement recovers that partition within 2⌈log2 n⌉ rounds on n points, with the specified atomic initialization. Affine-plane schemes show that linear dependence on valency is necessary. We apply the group realization to classify groups admitting naturalizing invariant metrics and show that, for a suitable such metric, naming the identity determines the compatible multiplication uniquely. Finally, using additional Cayley-realization and finite-classification inputs and explicit certificates, we determine the least number of metric values for every group. The direct universal construction and the finite coherent-reconstruction theorem do not depend on those computations.
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BibTeX
@misc{sutherland2026finitecolor,
title = {{Finite-color reconstruction of groups and coherent configurations}},
author = {Alex J. Sutherland},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33938},
eprint = {2609.33938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33938},
}
RIS
TY - UNPB
TI - Finite-color reconstruction of groups and coherent configurations
AU - Alex J. Sutherland
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.33938
UR - https://arxiv.org/abs/2609.33938
AB - Every group has a canonical coloring of its unordered pairs by inverse pairs of differences. We prove that this coloring can be coarsened to four colors without changing its color-preserving permutation group, for groups of arbitrary cardinality. Four is the least uniform bound. A related construction applies to finite homogeneous coherent configurations: if every basic relation has valency at most d, where d≥ 2, a coarsening with at most 2d+1 off-diagonal colors in the symmetric case, or 2d+3 off-diagonal arc colors in general, has the original configuration as its coherent closure. Two-dimensional Weisfeiler-Leman refinement recovers that partition within 2⌈log_2 n⌉ rounds on n points, with the specified atomic initialization. Affine-plane schemes show that linear dependence on valency is necessary. We apply the group realization to classify groups admitting naturalizing invariant metrics and show that, for a suitable such metric, naming the identity determines the compatible multiplication uniquely. Finally, using additional Cayley-realization and finite-classification inputs and explicit certificates, we determine the least number of metric values for every group. The direct universal construction and the finite coherent-reconstruction theorem do not depend on those computations.
ER -