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The stable commutator length of a relator is not a one-relator group invariant

A negative answer to Heuer and Löh's question: the isomorphism type of a one-relator group $\langle S \mid r\rangle$ does not determine $\mathrm{scl}_S(r)$ . The witnesses are $r=\mathtt{aabABabABBAbaabABBAb}$ and $r'=\mathtt{aabABabABabABBAbaBAb}$ , both length 20 in $F_2'$ , with $\langle a,b \mid r\rangle\cong\langle a,b\mid r'\rangle$ but $\mathrm{scl}(r)=1$ against $\mathrm{scl}(r')=1/2$ . The mechanism is what made the search finite: $\mathrm{scl}$ is an $\operatorname{Aut}(F_2)$ -invariant, so a pair in *different* orbits whose one-relator groups happen to be isomorphic has its two scl values unconstrained by each other. The search was for that configuration among words of length at most 20. Scope: it settles the question as posed and nothing wider. It does not say which invariants do determine scl, and this is a single pair rather than a construction giving arbitrary gaps.

Exact FrontierDelta

disproved→disproved

Scope and record

Occurred: Oct 29, 2026

Delta type: REGISTRY REVISION

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-co-developed. 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: The stable commutator length of a relator is not a one-relator group invariant · SCL is not invariant of relator in 1-relator groups

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

Claude Opus 5
model · ai model contributor · Anthropic

Harmonic Aristotle (Lean formalisation)
model · ai model contributor · Harmonic

Artem Semidetnov
human · human collaborator

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Named evidence

VibeMathed record: The stable commutator length of a relator is not a one-relator group invariant

specialist registry · passed

Unreviewed, lowered from Lean-verified on inspection. A labelling correction, not a doubt about the mathematics. The theorem has two halves - the groups are isomorphic, and the relators have different scl - and only the first is formalised. The Lean file says so itself: "(The paper computes scl r = 1 ≠ 1/2 = scl r' with scallop; see scl/.)" So the half that makes this a counterexample is a computation, not a machine-checked proof, and Lean-verified additionally requires the formal statement to be independently anchored, which an author's own repository is not. What the Lean does establish, and it is not nothing: one file, 282 lines, with no sorry, no admit, no native_decide and no declared axiom. It builds the two homomorphisms explicitly and proves they compose to the identity in both directions, so the isomorphism is constructed rather than asserted. Not built here - no Lean toolchain on this machine, and it pins v4.28.0. The scl values were not reproduced here either. They are reproducible in principle: scl/setup_scallop.sh fetches and patches Alden Walker's scallop, and compute_scl.py rechecks every value the paper quotes against it, exiting non-zero on disagreement. It needs a C++11 compiler with GLPK and GMP, none present on this machine. Checked independently on 30 August 2026: both relators have exponent sum zero in both generators, so both do lie in $F'$ as the question requires; both are cyclically reduced, length 20, distinct, and match the abstract exactly.

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The stable commutator length of a relator is not a one-relator group invariant

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean formalisation, search pipeline and scl recomputation

code · pending

code link supplied by the VibeMathed registry.

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Heuer and Löh, where the question is posed

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean formalisation, search pipeline and scl recomputation

code · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

Corrects The stable commutator length of a relator is not a one-relator group invariant →

Supersedes The stable commutator length of a relator is not a one-relator group invariant →

Lean formalisation, search pipeline and scl recomputation evidence for this event

This event attributed to Claude Opus 5

This event attributed to Artem Semidetnov

The stable commutator length of a relator is not a one-relator group invariant evidence for this event

VibeMathed record: The stable commutator length of a relator is not a one-relator group invariant evidence for this event

Heuer and Löh, where the question is posed evidence for this event

This event attributed to Harmonic Aristotle (Lean formalisation)

The stable commutator length of a relator is not a one-relator group invariant parent of this event

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