Global Observability from Negative Gramian Curvature

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

$\Phi(t)$ is the linearized state-transition matrix and $H(t)$ the output derivative along the trajectory. A uniformly positive observability Gramian defines a Riemannian metric; global observability means distinct initial states give distinct full output trajectories.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Source-record limitation: Source flags the statement as unrecoverable: The official report confirms that this restates the same global-observability problem extracted fully as OWR-3394-020. No separate clean statement is promoted; the original record remains publicly visible for provenance.

Related catalogue records: 30001203. Separate source IDs are preserved for their provenance. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.

For an observed smooth dynamical system whose local observability Gramian is uniformly positive definite and bounded, is the system globally observable on [0,T] whenever the Gramian-induced state-space Riemannian manifold is geodesically complete and has uniformly negative curvature? [S2]

Source status.

The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

The supplied fragment concerns geometric Criteria for Global Observability. Its complete mathematical target is not recovered.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30001204 (OWR-3394-021). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Control Theory: On the Way to New Application Fields $2009$, Oberwolfach Reports, DOI 10.4171/owr/2009/11, pp. 593--680 https://doi.org/10.4171/owr/2009/11.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Preparation and review.

The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30001204 and code OWR-3394-021 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Related canonical identities: 30001203. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

The original OWR report, printed pages 674--675, was read directly. It permits an arbitrary $n$-manifold and Euclidean outputs; it does not require simple connectedness. The counterexample route below was suggested by the parent Codex GPT-6 model and independently checked and expanded by gpt-6-astra. Nash's original isometric-embedding theorem and Goldman's authored hyperbolic-surface construction were consulted. No literature-priority claim is made. The parent Codex GPT-6 model completed a written mathematical review of this proof and its complete literal source scope. This model-assisted review is not independent human verification. The damaged/repeated-record qualifications and the separate ID are retained. The clear literal question has the same hypotheses as record 30001203, and the proof is repeated in full for standalone review.

Research attempt: A finite-cover obstruction to global observability

A compact negatively curved manifold and its double cover

Let $(N,g)$ be a closed connected orientable hyperbolic surface of genus two, with curvature $-1$. One standard explicit construction glues the paired sides of a regular hyperbolic octagon of interior angle $\pi/4$. Such an octagon exists because, in the disk model, its angle varies continuously from the small-polygon Euclidean limit $3\pi/4$ to zero as its vertices approach the ideal boundary. Pair the sides by hyperbolic isometries in the genus-two polygon pattern. All eight vertices are identified and their angles total $2\pi$; edges have two half-neighborhoods. Thus the quotient has smooth hyperbolic neighborhoods also at its seam and vertex. It has one face, four edges and one vertex, Euler characteristic $-2$, and is a compact genus-two surface. This standard construction is also explicitly exhibited in the consulted Goldman slides; no assertion of a new surface construction is made.

Cut $N$ along a nonseparating simple closed smooth curve around a handle. Its cut surface is connected and has two boundary copies of that curve. Take two copies of the cut surface and identify each positive boundary to the negative boundary of the other copy using the original boundary identification. The resulting $M$ is compact and connected: each copy is connected, and the crosswise boundary identifications join them. The map \[ \pi:M\longrightarrow N \] which forgets the copy is two-to-one. Away from the cut it is the identity on a chart in each copy; at the seam each glued pair of half-collars maps bijectively to a collar of the original curve. These charts give a smooth local diffeomorphism everywhere, with exactly two points in each fiber. Equip $M$ with $\widetilde g=\pi^*g$. It is a smooth positive definite metric, and local isometry to $N$ shows that its curvature is identically $-1$.

The external theorem used to produce Euclidean outputs

We invoke Nash's smooth isometric embedding theorem for compact Riemannian manifolds. In the exact form needed here, a compact smooth $n$-manifold with a smooth positive metric admits a smooth isometric embedding into some finite-dimensional Euclidean space. The original paper's Theorem 2 on printed page 59 includes $3\leq k\leq\infty$ and target dimension $n(3n+11)/2$. This theorem is an explicitly declared external dependency; its lengthy proof is not reproduced. Applied to $N$, it supplies \[ j:N\longrightarrow\mathbb R^{17},\qquad j^*\langle\cdot,\cdot\rangle=g. \] The exact dimension is not essential, but is allowed by the source's unrestricted finite output dimension.

A system with precisely the required Gramian

On $M$ set \[ f=0,\qquad h=j\circ\pi,\qquad T=1. \] Both maps are smooth. Every solution is stationary, $x(t)=x_0$, exists for all real time, and has output $h(x_0)$. In any local state coordinates the derivative of the zero vector field is zero, hence its fundamental matrix is $\Phi(t)=I$. The output derivative is the constant matrix $H(t)=dh_{x_0}$. Therefore the source Gramian, as a bilinear form on $T_{x_0}M$, is \[ P_{x_0}(v,w)=\int_0^1\langle dh_{x_0}v,dh_{x_0}w\rangle\,dt =g_{\pi(x_0)}(d\pi_{x_0}v,d\pi_{x_0}w) =\widetilde g_{x_0}(v,w). \tag{1} \] It is uniformly negatively curved, with sectional curvature $-1$ in this two-dimensional example.

Uniform bounds and geodesic completeness

Uniform positive definiteness and boundedness on an arbitrary manifold are understood relative to a fixed smooth background metric, or in a chosen finite atlas with compact chart cores. Both hold here. For any smooth background metric $g_0$, its unit tangent bundle over compact $M$ is compact. The positive continuous function $v\mapsto\widetilde g(v,v)$ on that unit bundle has positive minimum $a$ and finite maximum $b$. Thus \[ a\,g_0(v,v)\leq P(v,v)\leq b\,g_0(v,v) \] for all tangent vectors. Equivalently, cover $M$ by finitely many coordinate neighborhoods whose closures lie inside larger smooth charts. The matrix entries and least eigenvalues are continuous on these compact closures, giving common finite upper and positive lower coordinate bounds. No coordinate-invariant meaning can be assigned to bounds uniform under arbitrary rescaling of all possible coordinates, and no such meaning is used.

For completeness, the geodesic equation preserves speed: metric compatibility gives $d\widetilde g(\dot\gamma,\dot\gamma)/dt=2\widetilde g(\nabla_{\dot\gamma}\dot\gamma,\dot\gamma)=0$. Geodesics of a fixed speed stay in the corresponding compact sphere bundle of $TM$. The smooth geodesic vector field has a uniform local existence time on that compact set, by covering it with finitely many ordinary differential-equation neighborhoods. A geodesic therefore cannot have a finite endpoint of its maximal interval of existence: the uniform existence time extends it from a sufficiently late point. The same argument in reverse time proves geodesic completeness. Zero-speed geodesics are constant.

Failure of global observability

Choose the two distinct points $x_0,x_1$ over any point of $N$. They have \[ h(x_0)=j(\pi(x_0))=j(\pi(x_1))=h(x_1). \] Their entire constant output trajectories agree on $[0,1]$, although their initial states differ. Thus the system is not globally observable. Locally, $\pi$ is a diffeomorphism and $j$ is an embedding, so $h$ is locally one-to-one, consistent with its positive Gramian. Every hypothesis of the literal general-manifold question is satisfied, and the proposed implication is false. Simple connectedness, a global Euclidean coordinate chart, and restrictions on output dimension are not hypotheses of the consulted source and are not supplied by this argument.

Verification and adversarial audit

The source explicitly says that states are local coordinates on an arbitrary manifold; no simple-connectedness hypothesis appears on printed pages 674--675. The constant-vector-field construction meets the entire finite-horizon definition, rather than a discrete-time or asymptotic version. Equation $1$ computes the actual Gramian and does not merely choose an unrelated negative metric. Compactness controls the metric bounds and completeness, while the nontrivial cover supplies the two indistinguishable states. The Nash statement was visually checked in the original PDF on printed page 59; its smooth compact case is an external theorem, not an assumed analytic-embedding variant. The finite atlas interpretation of uniform bounds is stated explicitly. The same construction serves both retained source IDs; these are duplicate formulations, not two independent discoveries.

Audit corrections.

Matched the local-coordinate manifold scope to the primary report; computed the induced Gramian exactly; stated the reference-metric meaning of uniform bounds; explicitly attributed the parent model's construction and the external Nash theorem.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The displayed argument answers the literal arbitrary-manifold yes/no implication negatively, with Nash's smooth compact embedding theorem as a stated external dependency. The written argument and this exact scope have been reviewed by the parent Codex GPT-6 model. No result is claimed for an extra simply-connected or globally Euclidean-state version, which would be a stronger different question. No independent human checking or novelty certification is asserted.

Full-solution scope declaration

The entire original universal implication is false. The constructed smooth system has dimension two, horizon one and finite-dimensional Euclidean output. Its actual observability Gramian is uniformly positive and bounded, has constant curvature $-1$, and is geodesically complete, but a pair of distinct initial states has the same entire output trajectory. One such instance answers the universal yes/no question negatively; no source hypothesis is removed or weakened. The primary report permits arbitrary state manifolds and imposes no simple-connectedness or global Euclidean-coordinate condition. The exact compact smooth Nash embedding theorem is an external dependency whose original statement was checked, while its proof was not rederived. The standard hyperbolic surface construction and cover are explained above. Both source IDs retain their original metadata and the repeated record's damaged-extract qualification. No novelty is asserted and no exhaustive priority search was performed. The generating model and parent Codex GPT-6 reviewed the written argument; no independent human or formal proof checking is claimed.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 100% The counterexample covers every hypothesis of the entire original implication and refutes its conclusion. Completion measures target coverage, not novelty or independent verification.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/46211 (consulted 2026-10-11).
• https://sites.math.rutgers.edu/~feehan/teaching/math866/nash.pdf (consulted 2026-10-11).
• https://math.umd.edu/~wmg/gdansk.pdf (consulted 2026-10-11).

Model and contributor disclosure

The parent Codex GPT-6 model has reviewed the written mathematical arguments at their declared scopes. This is model-assisted review, not independent human verification. OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 proposed the finite-cover/Nash mathematical construction, completed written mathematical review, and performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Solved: the original general-manifold implication is false. The complete counterexample uses the explicitly declared Nash theorem dependency. No newness or independent human verification is claimed, and the catalogue's historical source-status label is preserved.

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

$\Phi(t)$ is the linearized state-transition matrix and $H(t)$ the output derivative along the trajectory. A uniformly positive observability Gramian defines a Riemannian metric; global observability means distinct initial states give distinct full output trajectories.

In the LOCAL model vertices communicate synchronously along graph edges; messages and local computation are unrestricted and complexity is the number of rounds. After $t$ rounds a vertex can depend only on its radius-$t$ neighborhood and any supplied identifiers or randomness.

A state-space system evolves as $\dot x=f(x,u)$ with output $y=h(x,u)$. Model reduction seeks a lower-dimensional system preserving specified input-output properties. Observability distinguishes initial states from their output trajectories, while controllability concerns reachable states under allowed inputs.

Statement recorded in the supplied source.

For the observed system $\dot{x}=f(x)$, $y=h(x)$ on an $n$-dimensional manifold, let $P(x_0)=\int_0^T\Phi(t)^TH(t)^TH(t)\Phi(t)\,dt$ be the local observability Gramian along the trajectory from $x_0$. If $P(x_0)$ is uniformly positive definite and bounded, does uniform negative curvature of the resulting Riemannian metric, together with geodesic completeness, imply global observability on $[0,T]$? [S2]

Source status.

The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Determine whether the stated completeness and negative-curvature assumptions promote local observability to global observability.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30001203 (OWR-3394-020). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Control Theory: On the Way to New Application Fields $2009$, Oberwolfach Reports, DOI 10.4171/owr/2009/11, pp. 593--680 https://doi.org/10.4171/owr/2009/11. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Preparation and review.

Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30001203 and source code OWR-3394-020 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Literature and context check

The original OWR report, printed pages 674--675, was read directly. It permits an arbitrary $n$-manifold and Euclidean outputs; it does not require simple connectedness. The counterexample route below was suggested by the parent Codex GPT-6 model and independently checked and expanded by gpt-6-astra. Nash's original isometric-embedding theorem and Goldman's authored hyperbolic-surface construction were consulted. No literature-priority claim is made. The parent Codex GPT-6 model completed a written mathematical review of this proof and its complete literal source scope. This model-assisted review is not independent human verification.

Research attempt: A finite-cover obstruction to global observability

A compact negatively curved manifold and its double cover

Let $(N,g)$ be a closed connected orientable hyperbolic surface of genus two, with curvature $-1$. One standard explicit construction glues the paired sides of a regular hyperbolic octagon of interior angle $\pi/4$. Such an octagon exists because, in the disk model, its angle varies continuously from the small-polygon Euclidean limit $3\pi/4$ to zero as its vertices approach the ideal boundary. Pair the sides by hyperbolic isometries in the genus-two polygon pattern. All eight vertices are identified and their angles total $2\pi$; edges have two half-neighborhoods. Thus the quotient has smooth hyperbolic neighborhoods also at its seam and vertex. It has one face, four edges and one vertex, Euler characteristic $-2$, and is a compact genus-two surface. This standard construction is also explicitly exhibited in the consulted Goldman slides; no assertion of a new surface construction is made.

Cut $N$ along a nonseparating simple closed smooth curve around a handle. Its cut surface is connected and has two boundary copies of that curve. Take two copies of the cut surface and identify each positive boundary to the negative boundary of the other copy using the original boundary identification. The resulting $M$ is compact and connected: each copy is connected, and the crosswise boundary identifications join them. The map \[ \pi:M\longrightarrow N \] which forgets the copy is two-to-one. Away from the cut it is the identity on a chart in each copy; at the seam each glued pair of half-collars maps bijectively to a collar of the original curve. These charts give a smooth local diffeomorphism everywhere, with exactly two points in each fiber. Equip $M$ with $\widetilde g=\pi^*g$. It is a smooth positive definite metric, and local isometry to $N$ shows that its curvature is identically $-1$.

The external theorem used to produce Euclidean outputs

We invoke Nash's smooth isometric embedding theorem for compact Riemannian manifolds. In the exact form needed here, a compact smooth $n$-manifold with a smooth positive metric admits a smooth isometric embedding into some finite-dimensional Euclidean space. The original paper's Theorem 2 on printed page 59 includes $3\leq k\leq\infty$ and target dimension $n(3n+11)/2$. This theorem is an explicitly declared external dependency; its lengthy proof is not reproduced. Applied to $N$, it supplies \[ j:N\longrightarrow\mathbb R^{17},\qquad j^*\langle\cdot,\cdot\rangle=g. \] The exact dimension is not essential, but is allowed by the source's unrestricted finite output dimension.

A system with precisely the required Gramian

On $M$ set \[ f=0,\qquad h=j\circ\pi,\qquad T=1. \] Both maps are smooth. Every solution is stationary, $x(t)=x_0$, exists for all real time, and has output $h(x_0)$. In any local state coordinates the derivative of the zero vector field is zero, hence its fundamental matrix is $\Phi(t)=I$. The output derivative is the constant matrix $H(t)=dh_{x_0}$. Therefore the source Gramian, as a bilinear form on $T_{x_0}M$, is \[ P_{x_0}(v,w)=\int_0^1\langle dh_{x_0}v,dh_{x_0}w\rangle\,dt =g_{\pi(x_0)}(d\pi_{x_0}v,d\pi_{x_0}w) =\widetilde g_{x_0}(v,w). \tag{1} \] It is uniformly negatively curved, with sectional curvature $-1$ in this two-dimensional example.

Uniform bounds and geodesic completeness

Uniform positive definiteness and boundedness on an arbitrary manifold are understood relative to a fixed smooth background metric, or in a chosen finite atlas with compact chart cores. Both hold here. For any smooth background metric $g_0$, its unit tangent bundle over compact $M$ is compact. The positive continuous function $v\mapsto\widetilde g(v,v)$ on that unit bundle has positive minimum $a$ and finite maximum $b$. Thus \[ a\,g_0(v,v)\leq P(v,v)\leq b\,g_0(v,v) \] for all tangent vectors. Equivalently, cover $M$ by finitely many coordinate neighborhoods whose closures lie inside larger smooth charts. The matrix entries and least eigenvalues are continuous on these compact closures, giving common finite upper and positive lower coordinate bounds. No coordinate-invariant meaning can be assigned to bounds uniform under arbitrary rescaling of all possible coordinates, and no such meaning is used.

For completeness, the geodesic equation preserves speed: metric compatibility gives $d\widetilde g(\dot\gamma,\dot\gamma)/dt=2\widetilde g(\nabla_{\dot\gamma}\dot\gamma,\dot\gamma)=0$. Geodesics of a fixed speed stay in the corresponding compact sphere bundle of $TM$. The smooth geodesic vector field has a uniform local existence time on that compact set, by covering it with finitely many ordinary differential-equation neighborhoods. A geodesic therefore cannot have a finite endpoint of its maximal interval of existence: the uniform existence time extends it from a sufficiently late point. The same argument in reverse time proves geodesic completeness. Zero-speed geodesics are constant.

Failure of global observability

Choose the two distinct points $x_0,x_1$ over any point of $N$. They have \[ h(x_0)=j(\pi(x_0))=j(\pi(x_1))=h(x_1). \] Their entire constant output trajectories agree on $[0,1]$, although their initial states differ. Thus the system is not globally observable. Locally, $\pi$ is a diffeomorphism and $j$ is an embedding, so $h$ is locally one-to-one, consistent with its positive Gramian. Every hypothesis of the literal general-manifold question is satisfied, and the proposed implication is false. Simple connectedness, a global Euclidean coordinate chart, and restrictions on output dimension are not hypotheses of the consulted source and are not supplied by this argument.

Verification and adversarial audit

The source explicitly says that states are local coordinates on an arbitrary manifold; no simple-connectedness hypothesis appears on printed pages 674--675. The constant-vector-field construction meets the entire finite-horizon definition, rather than a discrete-time or asymptotic version. Equation $1$ computes the actual Gramian and does not merely choose an unrelated negative metric. Compactness controls the metric bounds and completeness, while the nontrivial cover supplies the two indistinguishable states. The Nash statement was visually checked in the original PDF on printed page 59; its smooth compact case is an external theorem, not an assumed analytic-embedding variant. The finite atlas interpretation of uniform bounds is stated explicitly. The same construction serves both retained source IDs; these are duplicate formulations, not two independent discoveries.

Audit corrections.

Matched the local-coordinate manifold scope to the primary report; computed the induced Gramian exactly; stated the reference-metric meaning of uniform bounds; explicitly attributed the parent model's construction and the external Nash theorem.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The displayed argument answers the literal arbitrary-manifold yes/no implication negatively, with Nash's smooth compact embedding theorem as a stated external dependency. The written argument and this exact scope have been reviewed by the parent Codex GPT-6 model. No result is claimed for an extra simply-connected or globally Euclidean-state version, which would be a stronger different question. No independent human checking or novelty certification is asserted.

Full-solution scope declaration

The entire original universal implication is false. The constructed smooth system has dimension two, horizon one and finite-dimensional Euclidean output. Its actual observability Gramian is uniformly positive and bounded, has constant curvature $-1$, and is geodesically complete, but a pair of distinct initial states has the same entire output trajectory. One such instance answers the universal yes/no question negatively; no source hypothesis is removed or weakened. The primary report permits arbitrary state manifolds and imposes no simple-connectedness or global Euclidean-coordinate condition. The exact compact smooth Nash embedding theorem is an external dependency whose original statement was checked, while its proof was not rederived. The standard hyperbolic surface construction and cover are explained above. Both source IDs retain their original metadata and the repeated record's damaged-extract qualification. No novelty is asserted and no exhaustive priority search was performed. The generating model and parent Codex GPT-6 reviewed the written argument; no independent human or formal proof checking is claimed.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 100% The counterexample covers every hypothesis of the entire original implication and refutes its conclusion. Completion measures target coverage, not novelty or independent verification.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/46211 (consulted 2026-10-11).
• https://sites.math.rutgers.edu/~feehan/teaching/math866/nash.pdf (consulted 2026-10-11).
• https://math.umd.edu/~wmg/gdansk.pdf (consulted 2026-10-11).

Model and contributor disclosure

The parent Codex GPT-6 model has reviewed the written mathematical arguments at their declared scopes. This is model-assisted review, not independent human verification. OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 proposed the finite-cover/Nash mathematical construction, completed written mathematical review, and performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Solved: the original general-manifold implication is false. The complete counterexample uses the explicitly declared Nash theorem dependency. No newness or independent human verification is claimed, and the catalogue's historical source-status label is preserved.