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Berkholz's question: is Weisfeiler-Leman equivalence with the dimension as part of the input EXPTIME-complete?

Theorem 1.1 (principal manuscript): deciding joint-update $k$ -WL equivalence with $k\ge2$ in binary and explicit adjacency matrices is EXPTIME-complete under polynomial-time many-one reductions, also for connected simple uncoloured graphs of equal order and maximum degree three. Companions claim: deciding whether a given dimension identifies a graph (WL dimension at most $k$ ) is EXPTIME-complete, also with $k$ in unary; for every large fixed $k$ , deciding $k$ -WL equivalence needs $n^{\Omega(k)}$ deterministic sequential time unconditionally (multitape Turing machines and log-word RAMs, diameter-two graphs); and parity lifts give $n^{\Omega(k)}$ exclusions under ETH. No randomized or nonuniform lower bounds are claimed, and the separate-update convention is treated only in the fixed-dimension companions.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 25, 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: Berkholz's question: is Weisfeiler-Leman equivalence with the dimension as part of the input EXPTIME-complete? · Variable-dimension Weisfeiler-Leman equivalence

Confidence: Not scored

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VibeMathed record: Berkholz's question: is Weisfeiler-Leman equivalence with the dimension as part of the input EXPTIME-complete?

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the history section of the principal manuscript were read against Berkholz's question as Seppelt records it. Lean: lean/docs/133.md links ComparatorChallenges/VariableWL (theorem OAI.VariableWL.main, solution module OAI.Combinatorics.VariableWL.Completeness, file exists at the pinned commit), not in formalization.yaml. The statement was read: with Turing machines, EXPTIME and polynomial-time many-one reductions defined from scratch, both the language of (G, H, k) with k >= 2 in binary and joint-update k-WL equivalence, and its connected subcubic equal-order restriction, are EXPTIME-complete. That is the headline. Not rebuilt here. The result uses the joint-update convention.