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A Counterexample to the Exact Eight-Point Angular-Energy Conjecture

  • Jinze Zhao
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Abstract

We give an exact counterexample to the eight-point angular-energy conjecture stated by Mouchet in 2005 and published in 2006. The conjecture specifies both a radical value for the global maximum and a square antiprism with unit-side horizontal squares separated by h0=√(1+2√2)/2. We prove that the same family at the rational separation 489/500 has a strictly larger angular sum. The proof reduces the sum of all $168$ angle cosines to a one-variable formula, factors its derivative, and certifies its sign using rational inequalities. We also characterize the unique maximizing separation within this family. All comparisons needed for the counterexample are exact. The result disproves the exact stated conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{zhao2026counterexample,
  title = {{A Counterexample to the Exact Eight-Point Angular-Energy Conjecture}},
  author = {Jinze Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03207},
  eprint = {2610.03207},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03207},
}
RIS
TY  - UNPB
TI  - A Counterexample to the Exact Eight-Point Angular-Energy Conjecture
AU  - Jinze Zhao
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03207
UR  - https://arxiv.org/abs/2610.03207
AB  - We give an exact counterexample to the eight-point angular-energy conjecture stated by Mouchet in 2005 and published in 2006. The conjecture specifies both a radical value for the global maximum and a square antiprism with unit-side horizontal squares separated by h_0=√(1+2√2)/2. We prove that the same family at the rational separation 489/500 has a strictly larger angular sum. The proof reduces the sum of all $168$ angle cosines to a one-variable formula, factors its derivative, and certifies its sign using rational inequalities. We also characterize the unique maximizing separation within this family. All comparisons needed for the counterexample are exact. The result disproves the exact stated conjecture.
ER  -