A spurious contents-page match and an entropy obstruction to strict local rigidity

Definitions and mathematical statement

This permanent record contains no recoverable conjecture. The supplied verification identifies a parser error: a contents-page "Andre--Oort Conjecture" entry followed by page number 21 was mistaken for "Conjecture 21", and the extraction then captured local-rigidity introductory prose ending before an actual conjecture. Neither the following Conjecture 1 nor the later genuine Conjecture 21 is substituted here; both reportedly have separate canonical records. Thus no new open-problem statement is invented for this ID.

The captured context defines local smooth rigidity for ${\mathbb{Z}}^k$ actions by $C^\infty$ conjugacy for sufficiently $C^1$-close smooth actions, and for ${\mathbb{R}}^k$ actions only up to an automorphism of ${\mathbb{R}}^k$. It calls an action Anosov if an element is normally hyperbolic relative to the orbit foliation, and calls the described actions algebraic when induced by automorphisms or translations on infrahomogeneous Lie-group spaces. Its irreducibility discussion informally excludes elementary rank-one reductions, without a full definition. Historical claims distinguish topological structural stability of Anosov diffeomorphisms from smooth higher-rank rigidity, and note a large-$C^\ell$ perturbation requirement in the cited partially hyperbolic toral result.

The aid develops a separate contextual lemma explaining the ${\mathbb{R}}^k$ parameter gauge. Let $k\geq1$ and let $\alpha:{\mathbb{R}}^k\to\operatorname{Diff}^\infty(M)$ be a smooth action on a compact finite-dimensional manifold. Suppose an element $\alpha(v_0)$ has finite positive topological entropy $H$. For rational $c=p/q>0$, $c\ne1$, define $\alpha_c(v)=\alpha(cv)$. Then \[ h_{\mathrm{top}}(\alpha_c(v_0))=cH\ne H, \] since $\alpha(cv_0)^q=\alpha(v_0)^p$ and entropy scales under positive integral iterates. Therefore no homeomorphism can conjugate the two actions preserving every parameter $v$. On the other hand $\alpha_c=\alpha\circ(c\operatorname{id}_{{\mathbb{R}}^k})$, so they are equivalent after the allowed $GL(k,{\mathbb{R}})$ reparametrization.

As $c\to1$, these reparametrized actions converge to $\alpha$ in the smooth compact-open action topology, meaning uniform convergence of each finite derivative order on compact parameter sets times $M$. Hence every neighborhood in that topology contains such a rational reparametrization. A stronger topology uniform over all of ${\mathbb{R}}^k$ is not assumed. The lemma shows why strict parameter-preserving local rigidity can fail even though the usual parameter-gauge version is appropriate. It does not refute any correctly gauged rigidity conjecture, apply the scalar reparametrization to ${\mathbb{Z}}^k$, or supply the missing R. Spatzier conjecture mentioned by the final lead-in.

Short English statement

This is a spurious contents-page/context record, not a mathematical conjecture. The draft preserves that verdict and does not replace it with a neighboring problem. The supplied contextual entropy lemma explains why local rigidity of real higher-rank actions allows parameter automorphisms: arbitrarily small scalar reparametrizations change positive entropy while remaining equivalent under that gauge.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-other-notes.json, notes[67]; canonical tag "conjecture", despite the supplied invalid-record verdict.
• AIM workshop Emerging applications of measure rigidity, https://aimath.org/WWN/measrigid/measrigid.pdf, document dated 1 November 2004, scribe A. Gorodnik. The contents-page numbers and page headers are extraction context, not a numbered conjecture statement.
• Captured references [93] for an early higher-rank rigidity result, [97] for natural irreducible algebraic Anosov actions, and [39] for partially hyperbolic toral automorphisms with a large-$C^\ell$ perturbation requirement. Complete bibliography entries are not supplied. R. Spatzier is named only as communicator of the next, uncaptured conjecture.
• Supplied research attempt 1, registry updated 14 August 2026: invalid_statement, not_a_problem, low novelty confidence, machine-generated and not peer reviewed. Statement verification says unrecoverable and forbids substitution of the genuine adjacent conjectures. Registry status nevertheless remains open with OPEN-TRIAGE classification.
• The entropy lemma is supplied contextual mathematical development, not a recovered original problem or an external theorem citation.