Math Conjectures

Gottschalk Surjunctivity Conjecture

OPENMajorConjectureProposed 1973 · Full conjecture

Canonical statement

For every group GG and every finite alphabet AA, every continuous GG-equivariant injective map τ:AG→AG\tau:A^G\to A^G, with AGA^G carrying the product topology and the shift action, is surjective.
View source LaTeX
For every group \(G\) and every finite alphabet \(A\), every continuous \(G\)-equivariant injective map \(\tau:A^G\to A^G\), with \(A^G\) carrying the product topology and the shift action, is surjective.
DYN-015:[Gottschalk1973Surjunctive][Pestov2008SoficHyperlinear][OpenAI2026TenAdvances]
Dynamical systemsAlgebra and representation theory

Notes

Gottschalk called a group surjunctive when every injective cellular automaton over it is surjective [Gottschalk1973Surjunctive]. Sofic groups are surjunctive, making this a major downstream test of finite approximation [Pestov2008SoficHyperlinear]. The group announced as non-sofic on August 1, 2026 has no established surjunctivity classification [OpenAI2026TenAdvances], so that announcement does not resolve this broader dynamical conjecture.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.