Gottschalk's surjunctivity conjecture

For a group $G$ and a finite alphabet $A$ , a cellular automaton on $A^G$ is a map $\tau(x)(g)=\phi((x(gu))_{u\in M})$ given by a finite memory set $M\subset G$ and a local rule $\phi:A^M\to A$ . $G$ is surjunctive if every injective cellular automaton on $A^G$ , for every finite $A$ , is surjective. Gromov and Weiss proved that sofic groups are surjunctive, and surjunctivity implies Kaplansky's direct finiteness over finite fields. Gottschalk (1973) conjectured that every group is surjunctive. Is every group surjunctive?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Symbolic dynamics on groups; cellular automata
Posed by: Walter Gottschalk, Some general dynamical notions, Recent Advances in Topological Dynamics, LNM 318 (1973)
Year posed: 1973
Years open: 53y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 48 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

The group $G$ and elements $a,b\in K[G]$ of the characteristic-two direct-finiteness counterexample ( $K$ finite, $G$ finitely presented with odd torsion) define a cellular automaton on $K^G$ that is injective but not surjective, so $G$ is not surjunctive. The odd-characteristic companion (September 26) gives the same conclusion over a field of order $p^4$ for a finitely generated group with torsion. The automata are linear; no torsion-free non-surjunctive group is claimed in these papers (the torsion-free companion does not discuss automata).

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The cellular automaton is obtained from the direct-finiteness counterexample by the standard algebra-to-automaton transfer, proved in full in Section 6 of the principal manuscript.

Verification

No independent mathematician has checked this yet. Checked here: the September 23 manuscript's Section 6 was read against Gottschalk's conjecture as stated there; the coefficients of $b$ give a linear cellular automaton on $K^G$ ( $K$ finite) with left inverse from $a$ , injective and not surjective. The proof was not refereed. The Lean statements attached to this manuscript do not include the automaton; the formal statement that does is the odd-characteristic companion's ComparatorChallenges/OddKaplansky.lean (listed in formalization.yaml, declaration OAI.OddKaplansky.main_theorem), read here: for the least prime factor $p$ of $(\binom{1200}{600}!)^2+1$ , a field of order $p^4$ , a finitely generated group with a nontrivial torsion element, and $a,b$ with $ab=1$ , $ba\ne1$ , such that the convolution map $x\mapsto(g\mapsto\sum_u b_u x(gu))$ on all configurations $G\to K$ is injective and not surjective. That states a counterexample to the conjecture (alphabet $K$ , memory set the support of $b$ ). Not rebuilt here.

Gottschalk's surjunctivity conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion