Makeev’s universal-cover conjecture
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Makeev’s universal-cover conjecture is tested in dimensions four and five through Kuperberg’s equivalent function formulation. The counterexample target is an explicit smooth odd function on S^(d−1) that agrees with no linear functional on any rotated normalized A_d root system, for sufficiently small nonzero ε.
Why it matters and what counts as progress
Why it matters: It probes the first two unresolved dimensions of the proposed geometric covering property.
What counts as progress: An explicit body violating the universal-cover property in an unresolved dimension, or a theorem identifying the dimensions where it survives.
How the problem developed
- 1999 · Partial result [1]
Greg Kuperberg Established the relevant inscribed-polytope conclusion in dimension three; the homological obstruction did not settle higher dimensions.
- August 2026 · Counterexamples [2]
ChatGPT Reported counterexamples to the smooth odd-function formulation in dimensions four and five; the manuscript’s conclusion is limited to those dimensions.
AI contribution
Counterexamples to Makeev's regular-simplex conjecture in dimensions four and five
In each of dimensions four and five, for every sufficiently small nonzero ε, an explicit smooth odd function on S^(d−1) cannot agree with a linear functional on any rotated normalized A_d root system. Through Kuperberg’s equivalence, these give counterexamples to Makeev’s universal-cover conjecture in those dimensions.
Sources
Website timeline
Publication dates and editorial changes.
ChatGPT-authored construction challenges Makeev’s universal-cover conjecture