A Chain-Level Borsuk--Ulam Obstruction Proof of Norine's Antipodal-Coloring Conjecture

Hehui Wu, Ningyuan Yang

For every n≥2, a two-colouring of the edges of the n-cube in which antipodal edges have opposite colours contains a monochromatic path from some vertex to its antipode.

Sources for this record

What was already known

Norine posed the conjecture in 2008. Earlier results covered the stronger geodesic version through dimension five, dimension six by computation and later by hand, and SAT cases through dimension eight.

AI involvement AI-centered

Wu and Yang credit GPT 5.6 Sol Ultra with the central chain-level obstruction idea. They developed the reduction, chain construction and final manuscript; Codex assisted drafting.

The authors attribute the central proof idea to the model, while they develop and revise the complete argument.

Importance Notable

The proof extends the finite-dimensional cases to every cube dimension.

Limitations

The authors revised and approved the full proof, but do not explicitly report a completed independent check of its chain-map argument.

Technical verification Reported

A complete proof is presented, but the authors report revision and approval rather than an explicit completed human or machine check of this argument. Reported is the missing-check default.

Human understanding Full

The human authors explain how the colouring yields a rook-labeling chain map and why the chain-level Borsuk–Ulam obstruction rules it out.

Correction and publication history