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Gottschalk's surjunctivity conjecture

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).

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Gottschalk's surjunctivity conjecture · Gottschalk surjunctivity

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Gottschalk's surjunctivity conjecture

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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.