One Extra Dimension Suffices for the Complete-Graph Squared-Stress
Abstract
Let all pairwise distances of an arbitrary configuration of points in ℝℓ be known exactly. We prove that every second-order stationary point of the complete-graph squared-stress objective over configurations in ℝk has zero residual whenever k > ℓ. The result holds for every number of points, without a genericity assumption, and allows repeated points and degenerate ground-truth configurations. It resolves the one-extra-dimension conjecture for complete, uniformly weighted data. The proof combines positivity of the residual stress, polar normalization of the candidate configuration, a row-wise rank-nullity argument, and propagation of zero curvature through the full Hessian. The theorem, including the passage from derivatives of the original quartic objective to its matrix formulation, has been formalized in Lean 4. The accompanying verification records include a kernel check with zero trust level and a transitive axiom audit.
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BibTeX
@misc{yan2026extra,
title = {{One Extra Dimension Suffices for the Complete-Graph Squared-Stress}},
author = {Lilin Yan and Hongwei Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06531},
eprint = {2610.06531},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06531},
}
RIS
TY - UNPB
TI - One Extra Dimension Suffices for the Complete-Graph Squared-Stress
AU - Lilin Yan
AU - Hongwei Zhao
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06531
UR - https://arxiv.org/abs/2610.06531
AB - Let all pairwise distances of an arbitrary configuration of points in ℝ^ℓ be known exactly. We prove that every second-order stationary point of the complete-graph squared-stress objective over configurations in ℝ^k has zero residual whenever k > ℓ. The result holds for every number of points, without a genericity assumption, and allows repeated points and degenerate ground-truth configurations. It resolves the one-extra-dimension conjecture for complete, uniformly weighted data. The proof combines positivity of the residual stress, polar normalization of the candidate configuration, a row-wise rank-nullity argument, and propagation of zero curvature through the full Hessian. The theorem, including the passage from derivatives of the original quartic objective to its matrix formulation, has been formalized in Lean 4. The accompanying verification records include a kernel check with zero trust level and a transitive axiom audit.
ER -